Minimal Surfaces from a Complex Analytic Viewpoint (Springer Monographs in Mathematics) [1st ed. 2021] 3030690555, 9783030690557

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Springer Monographs in Mathematics

Antonio Alarcón Franc Forstnerič Francisco J. López

Minimal Surfaces from a Complex Analytic Viewpoint

Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea; Mathematical Institute, University of Warwick, Coventry, UK Katrin Wendland, Research group for Mathematical Physics, Albert Ludwigs University of Freiburg, Freiburg, Germany Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NY, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NY, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

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Antonio Alarcón Franc Forstnerič Francisco J. López •

•

Minimal Surfaces from a Complex Analytic Viewpoint

123

Antonio Alarcón Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR) Universidad de Granada Granada, Spain Francisco J. López Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR) Universidad de Granada Granada, Spain

Franc Forstnerič Faculty of Mathematics and Physics University of Ljubljana Ljubljana, Slovenia Institute of Mathematics Physics and Mechanics Ljubljana, Slovenia

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-69055-7 ISBN 978-3-030-69056-4 (eBook) https://doi.org/10.1007/978-3-030-69056-4 Mathematics Subject Classification: 53-02, 32-02 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

As suggested by the title, this book treats the classical theory of minimal surfaces in Euclidean spaces by complex analytic methods. The connection between these two subjects goes back to the time of Bernhard Riemann and Karl Weierstrass, and it was pioneered in the modern era by Robert Osserman. Nevertheless, a scope of new possibilities has been discovered in the last decade through collaboration of the authors. The chief influence from the complex analytic side comes from Oka theory, alias the homotopy principle in complex analysis. This theory, which originates in the works of Kiyoshi Oka (1939) and Hans Grauert (1958), studies the existence of analytic solutions to nonlinear problems in complex analysis and geometry in the absence of topological obstructions. It turns out that Oka theory combined with convex integration theory, invented by Mikhail Gromov, furnishes powerful tools for constructing minimal surfaces. A major part of the book is based on results of the authors and our collaborators obtained since 2012. Before describing the content, let us say a few things about the subject itself. The study of locally area minimizing surfaces, or minimal surfaces for short, is one of the most classical and beautiful topics in mathematics having roots in ancient civilizations. The modern beginning of the theory is the work of Leonhard Euler who in 1744 studied area minimizing surfaces in Euclidean 3-space R3 and showed that the only surfaces of rotation with this property are planes and catenoids. The theory was put on solid ground by Joseph-Louis Lagrange who in 1760 developed variational methods and found a differential equation characterizing minimality of graphs of smooth functions on domains in the plane R2 . In 1776, Jean Baptiste Meusnier showed that a surface in R3 locally satisfies Lagrange’s minimal graphs equation if and only if its mean curvature vanishes at every point, and this is equivalent to vanishing of the metric Laplacian on the surface. Another major contribution was made by Joseph Plateau who in 1873 studied soap films spanning a given contour in R3 and showed by experiments that they are minimal surfaces. The Plateau problem, whether any closed Jordan curve in R3 spans a minimal surface, was solved affirmatively by Jesse Douglas and Tibor Rad´o in 1932. Minimal surfaces, and their higher-dimensional analogues in Riemannian manifolds, are a common playground of several branches of modern mathematics, v

vi

Preface

chief among them being differential geometry, geometric measure theory, and partial differential equations. Complex analysis enters the subject in the special case of minimal surfaces in flat Euclidean spaces by the Enneper–Weierstrass formula, which relates conformal minimal immersions into Rn to holomorphic maps with values in a quadric hypersurface in Cn , the null quadric. However, in this most classical case, complex analytic methods seem to be among the strongest ones available, as is hopefully demonstrated by the present book. Another point of contact between these two fields is Wirtinger’s theorem that every compact complex submanifold of a K¨ahler manifold is an area minimizer in its homology class. A major part of the contemporary theory focuses on classification problems, the study of complete minimal surfaces contained in bounded domains of the space (the Calabi–Yau problem), the study of surfaces of finite total Gaussian curvature, which connects the subject to algebraic geometry, and value distribution theory of the Gauss map of minimal surfaces, which is related to Nevanlinna theory. Complex analytic techniques have recently been applied to the construction of superminimal surfaces in certain Riemannian four-manifolds via the Penrose twistor theory. All these topics are to various degrees presented in the book. Let us now succinctly describe the contents. In Chapter 1 we briefly review the theory of real and complex manifolds, including the basics of holomorphic approximation theory and Oka theory. We focus on results that are used in the book, thereby helping the reader navigate through a potentially overwhelming amount of background material with minimal effort. In Chapter 2 we present the basic notions of the theory of minimal surfaces in Euclidean spaces Rn , and of the closely related holomorphic null curves in complex Euclidean spaces Cn . We recall the notion of curvature of an immersed surface, derive Lagrange’s formulas for the first and the second variation of the area, and present the classical Enneper–Weierstrass representation formula, which provides the key connection between conformal minimal immersions M → Rn from an open Riemann surface and holomorphic maps with vanishing periods from M into the punctured null quadric in Cn . This brings complex analysis into the picture in a natural and essential way, and is the basis for subsequent developments. We also recall the notion of the Gauss map, begin discussing complete minimal surfaces of finite total curvature, recall some basic instances of the maximum principle for minimal surfaces, and discuss the isoperimetric inequality. We conclude with a survey of some classical and a few recent examples of minimal surfaces. In Chapter 3 we develop the first major collection of new results concerning oriented minimal surfaces in Rn and holomorphic null curves in Cn for any n ≥ 3. All of them have been obtained since 2012 by complex analytic methods. The main new results presented in this chapter include Runge, Mergelyan, and Carleman type approximation theorems for conformal minimal surfaces, analogues of the Weierstrass and Mittag-Leffler interpolation theorems, general position theorems, construction of proper minimal surfaces in Euclidean spaces, and the homotopy theory for the space of conformal minimal immersions from a given open Riemann surface into Rn . Analogues of many of these results for nonorientable minimal surfaces in Euclidean spaces are presented in our separate publication [29].

Preface

vii

Chapter 4 begins with a brief survey of some classical results concerning minimal surfaces of finite total Gaussian curvature; these are parameterized by finitely punctured compact Riemann surfaces and their Weierstrass data extend meromorphically to the punctures. We then proceed to develop the approximation and interpolation theory for such surfaces, in analogy to what is done in Chapter 3 for general minimal surfaces, but using algebraic-geometric techniques. We also obtain an upper bound on the number of intersections of such surfaces in R3 with affine lines and present recent results on the optimal hitting problem. The study of the Gauss map of minimal surfaces is continued and substantially expanded in Chapter 5. We show that every natural candidate is the Gauss map of a conformal minimal surface in Rn , and we discuss the value distribution theory of the Gauss map of complete minimal surfaces of finite total curvature. The remainder of the book is devoted to results based on another classical complex analytic technique — the Riemann–Hilbert boundary value problem. In Chapter 6 we adapt it to minimal surfaces and holomorphic null curves. This technique, developed by the authors in a series of papers during 2015–2019 (some in collaboration with Barbara Drinovec Drnovˇsek), has proved very useful in the construction of complete bounded minimal surfaces and null curves. In particular, it led to major new contributions to the classical Calabi–Yau problem and Yang’s problem, which are presented in Chapter 7. We also mention a recent development of this technique which was used to establish the Calabi–Yau property for conformal superminimal surfaces in self-dual or anti-self-dual Einstein four-manifolds. In Chapter 8, the Riemann–Hilbert technique is applied to the construction of complete proper minimal surfaces from bordered Riemann surfaces into minimally convex domains in Rn for any n ≥ 3. This class contains all convex domains as well as many nonconvex ones. In dimension n = 3 it coincides with the class of meanconvex domains, and we show that it is the largest class for which our results hold. Combining these tools with other methods of complex and functional analysis, we initiate in Chapter 9 the study of minimal hulls of compact sets in Euclidean spaces. Since the theory of minimal surfaces is a huge subject, we are not trying to be comprehensive, which would be a hopeless task anyway. In particular, we do not treat several topics of interest, chief among them being the Plateau boundary value problem for minimal surfaces, which is well covered in a number of extant texts. We hope that the book will be useful not only to experts in the field, but also to graduate students and researchers in related fields. Hopefully it will convince the reader that this is a lively and vibrant field of geometry.

Granada and Ljubljana, December 2020

Antonio Alarc´on Franc Forstneriˇc Francisco J. L´opez

Acknowledgements

First and foremost, we wish to thank our colleagues who have contributed to this work through discussions and collaborations; in particular, Ildefonso CastroInfantes, Brett Chenoweth, Barbara Drinovec Drnovˇsek, Isabel Fern´andez, Jos´e A. G´alvez, Josip Globevnik, Finnur L´arusson, Joaqu´ın P´erez, Antonio Ros, and Francisco Urbano. We also thank John P. D’Angelo and Peter Landweber, who read parts of the text and offered advice on language and style. We wish to thank the funding agencies mentioned below for supporting our research which led to the preparation of this book. Antonio Alarc´on and Francisco J. L´opez were supported by the State Research Agency (AEI) and European Regional Development Fund (FEDER) via the grant no. MTM2017-89677-P, MICINN, the Junta de Andaluc´ıa grant no. P18-FR-4049, and the Junta de Andaluc´ıa - FEDER grant no. A-FQM-139-UGR18, Spain. Franc Forstneriˇc was supported by the research program P1-0291 and research grants J1-2152, J1-5432, J1-7256, J1-9104 from Slovenian Research Agency (ARRS), and by the Stefan Bergman Prize for the year 2019 awarded by the American Mathematical Society. We also thank our respective institutions, the University of Granada, Spain, the Faculty of Mathematics and Physics, University of Ljubljana and the Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia for providing a stimulating research environment and technical support, as well as for hosting our numerous mutual visits since 2011 and especially during the preparation of the book. Finally, we express our sincere thanks to the communicating editor, Remi Lodh, and the staff of Springer-Verlag for their professional work in the handling and technical preparation of the book.

ix

Contents

1

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation and Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Handlebody Structure of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Hermitian and K¨ahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Conformal Maps and Isothermal Coordinates on Surfaces . . . . . . . . . 1.9 The Beltrami Equation and J-Holomorphic Discs . . . . . . . . . . . . . . . . 1.10 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Divisors and the Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . . . . . 1.12 Holomorphic Approximation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Manifold-Valued Maps and the Oka Principle . . . . . . . . . . . . . . . . . . . 1.14 Holomorphic Sprays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Algebraic Sprays and Algebraic Approximation . . . . . . . . . . . . . . . . .

1 2 5 8 16 22 33 39 46 49 52 60 65 74 79 81

2

Basics on Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.1 Curvature of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.2 Variation of Area and Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 92 2.3 The Enneper–Weierstrass Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4 Nonorientable Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.5 The Gauss Map of a Minimal Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.6 Gaussian Curvature of a Minimal Surface . . . . . . . . . . . . . . . . . . . . . . 108 2.7 The Maximum Principle and the Isoperimetric Inequality . . . . . . . . . 113 2.8 A Gallery of Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3

Approximation and Interpolation Theorems for Minimal Surfaces . . . 131 3.1 Spaces of Conformal Minimal Immersions and Null Curves . . . . . . . 133

xi

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Contents

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12

Period Dominating Sprays of Maps into the Null Quadric . . . . . . . . . 136 A Semiglobal Approximation and Interpolation Theorem . . . . . . . . . 140 General Position Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Paths with Given Periods in Affine Algebraic Varieties . . . . . . . . . . . 151 The First Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 The Second Main Theorem: Fixed Component Functions . . . . . . . . . 160 Mittag-Leffler and Carleman Theorems for Minimal Surfaces . . . . . 164 Global Properties I: Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Global Properties II: Properness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Proof of Theorem 3.10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 The h-Principle for Minimal Surfaces and Null Curves . . . . . . . . . . . 185

4

Complete Minimal Surfaces of Finite Total Curvature . . . . . . . . . . . . . . 191 4.1 A Brief Survey of the Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . 193 4.2 Spaces of Functions and Conformal Minimal Immersions . . . . . . . . . 201 4.3 A Runge Theorem for Maps to CP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.4 Period Dominating Multiplicative Sprays . . . . . . . . . . . . . . . . . . . . . . . 206 4.5 Approximation and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 4.6 Intersections with Affine Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 4.7 An Effective Obstruction to Hitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

5

The Gauss Map of a Minimal Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.1 Period Dominating Sprays of Multipliers . . . . . . . . . . . . . . . . . . . . . . . 234 5.2 Paths With Given Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 5.3 Multipliers Providing Prescribed Periods . . . . . . . . . . . . . . . . . . . . . . . 241 5.4 Everybody is a Gauss Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 5.5 The Gaussian Image of Complete Minimal Surfaces I . . . . . . . . . . . . 249 5.6 The Gaussian Image of Complete Minimal Surfaces II . . . . . . . . . . . 255 5.7 Isotopies of Conformal Minimal Immersions . . . . . . . . . . . . . . . . . . . . 260

6

The Riemann–Hilbert Problem for Minimal Surfaces . . . . . . . . . . . . . . 265 6.1 The Complex Analytic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 6.2 The Riemann–Hilbert Problem for Null Discs in C3 . . . . . . . . . . . . . . 268 6.3 The Riemann–Hilbert Problem for Sprays of Null Discs in C3 . . . . . 275 6.4 The Riemann–Hilbert Problem for Minimal Surfaces in R3 and Null Curves in C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 6.5 The Riemann–Hilbert Problem for Null Discs in Cn for n > 3 . . . . . 283 6.6 The Riemann–Hilbert Problem for Null Curves in Cn and Minimal Surfaces in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.7 Exposing Boundary Points of Bordered Riemann Surfaces . . . . . . . . 292

7

The Calabi-Yau Problem for Minimal Surfaces . . . . . . . . . . . . . . . . . . . . 295 7.1 Examples by Jorge and Xavier and by Nadirashvili . . . . . . . . . . . . . . 297

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7.2 A Lower Bound on the Intrinsic Diameter . . . . . . . . . . . . . . . . . . . . . . 306 7.3 C 0 Small Perturbations Enlarging the Intrinsic Diameter . . . . . . . . . . 309 7.4 The Main Results on the Calabi-Yau Problem . . . . . . . . . . . . . . . . . . . 316 7.5 Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9 . . . . . . . . . . . . . . . . . . . . . . 325 7.6 Complete Dense Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8

Minimal Surfaces in Minimally Convex Domains . . . . . . . . . . . . . . . . . . 337 8.1 p-Plurisubharmonic Functions and p-Convex Domains . . . . . . . . . . . 339 8.2 Null Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.3 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.4 Lifting Boundaries of Conformal Minimal Surfaces . . . . . . . . . . . . . . 359 8.5 Proofs of Theorems 8.3.1, 8.3.3, 8.3.4, and 8.3.11 . . . . . . . . . . . . . . . 366 8.6 A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

9

Minimal Hulls, Null Hulls, and Currents . . . . . . . . . . . . . . . . . . . . . . . . . . 379 9.1 The Role of Hulls in Analysis and Geometry . . . . . . . . . . . . . . . . . . . . 380 9.2 Minimal Hulls and Null Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 9.3 Null Hulls of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 9.4 Rectifiable Sets, Varifolds, and Currents . . . . . . . . . . . . . . . . . . . . . . . . 394 9.5 Hulls Defined by Minimal Rectifiable Currents . . . . . . . . . . . . . . . . . . 399 9.6 Green Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 9.7 Currents Characterizing Minimal Hulls and Null Hulls . . . . . . . . . . . 404

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Chapter 1

Fundamentals

In this chapter we review the basic notions from the theory of real and complex manifolds that play an important role in this book. Proofs are for the most part omitted, but we supply references to standard works for further reading. After establishing the notation in Sect. 1.1, we discuss the basics on topological manifolds and surfaces in Sect. 1.2. We continue with a survey of differential calculus on smooth manifolds (Sects. 1.3 and 1.4), complex manifolds (Sect. 1.5), Riemannian manifolds (Sect. 1.6), and Hermitian and K¨ahler manifolds (Sect. 1.7). In Sect. 1.8 we discuss the notion of a conformal structure on a manifold, and we show that every Riemannian surface admits local isothermal coordinates. Another proof of this result via the Beltrami equation is indicated in Sect. 1.9. In Sections 1.10 and 1.11 we recall the fundamentals of Riemann surfaces, which play a central role in the theory of minimal surfaces. Finally, in Sections 1.12–1.15 we review the main results of holomorphic approximation theory that will be used in the book. The details for this part can be found in the monograph [140] and in the survey [128]. For further reading on calculus on smooth manifolds we recommend classics such as R. Abraham, J. E. Marsden, and T. Ratiu [2] and F. W. Warner [337]. For complex and algebraic manifolds we recommend the texts by J.-P. Demailly [105], P. Griffiths and J. Harris [167], L. H¨ormander [195], and R. O. Wells, Jr. [341], among many others. A comprehensive and highly readable introduction to Riemannian geometry can be found in the monographs by M. P. do Carmo [109, 110]. Finally, there are many excellent sources for the theory of Riemann surfaces, from classics such as L. V. Ahlfors and L. Sario [7], H. M. Farkas and I. Kra [123], and O. Forster [130], to the more recent texts by S. Donaldson [112] and D. Varolin [333].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_1

1

2

1 Fundamentals

1.1 Notation and Basics Notation. We denote by N = {1, 2, 3, . . .} the natural numbers, by Z the ring of integers, Z+ = {0, 1, 2, . . .}, and by R and C the fields of real and complex numbers, respectively. For any n ∈ N we denote by Rn the n-dimensional real Euclidean space with real coordinates (x1 , . . . , xn ), and by Cn the n-dimensional complex Euclidean space with √ complex coordinates z = (z1 , . . . , zn ), where zi = xi + iyi with xi , yi ∈ R and i = −1. We shall also write Cn∗ = Cn \ {0} and Rn∗ = Rn \ {0}. An exception to the above notation will be C∗ = C \ {0}, which is more customary than C∗ . The conjugate of a complex number z = x + iy is denoted z¯ = x − iy, and likewise for vectors. We write ℜ(x + iy) = x and ℑ(x + iy) = y. Given vectors v = (v1 , . . . , vn ) and w = (w1 , . . . , wn ) in Cn , we denote by v·w =

n

∑ vk wk ,

|v|2 = v · v =

k=1

n

∑ |vk |2 ,

k=1

their Hermitian scalar product and the squared norm, respectively. Their restrictions to the standard real subspace Rn ⊂ Cn are the Euclidean scalar product and the squared norm, respectively. Given a ∈ C and r > 0, set D(a, r) = {z ∈ C : |z − a| < r},

D = D(0, 1).

(1.1)

Given a point a = (a1 , . . . , an ) ∈ Cn and a number r > 0 we let   n Bn (a, r) = z = (z1 , . . . , zn ) ∈ Cn : |z − a|2 = ∑ |zi − ai |2 < r2

(1.2)

i=1

be the ball centred at a ∈ Cn of radius r. In particular, Bn = Bn (0, 1) stands for the unit ball of Cn . Thus, D(a, r) = B1 (a, r) and D = B1 . The corresponding balls in Rn are denoted BnR (a, r) and BnR . Note that B1R = (−1, +1). By a domain in Rn or Cn we mean a connected open subset. A closed domain is the closure of a domain, and a compact domain is a closed domain which is compact. We use the same terminology for sets in more general manifolds. Exterior products of vector spaces. In preparation for the discussion of differential forms on manifolds (see Sections 1.3 and 1.5) we recall the notion of exterior product or wedge product of vectors. This is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. Details can be found e.g. in [124, Chap. 1]. Let V be a finite-dimensional real or complex vector space, i.e., V ∼ = F n for some n ∈ N, where F ∈ {R, C} denotes the base field. For any k ∈ N we denote by ΛkV its k-th exterior power; this is a vector space of dimension nk whose elements are called k-vectors. Note that Λ1V = V , and we set Λ0V = F. The wedge product ΛkV ⊗ ΛmV (v, w) → v ∧ w ∈ Λk+mV

1.1 Notation and Basics

3

is linear in each factor and anticommutative, w ∧ v = (−1)km v ∧ w. If e1 , . . . , en is a basis of V then the k-vectors eI := ei1 ∧ ei2 ∧ · · · ∧ eik

(1.3)

for all multiindices I =  (i1 , . . . , ik ) with 1 ≤ i1 < i2 < · · ·k < ik ≤ n form a basis of ΛkV , so dim ΛkV = nk . In particular, ΛdimV V ∼ = F and Λ V = {0} for k > dimV . As an example, the wedge product of vectors v = ∑ni=1 vi ei and w = ∑ni=1 wi ei is v∧w =

∑

(vi w j − v j wi ) ei ∧ e j ∈ Λ2V.

(1.4)

1≤i< j≤n

For v, w ∈ R3 the vector v ∧ w ∈ Λ2 R3 ∼ = R3 is the usual vector product of v and w. A k-vector v ∈ ΛkV is said to be simple (or decomposable) if v = v1 ∧ v2 ∧ · · · ∧ vk for some linearly independent vectors v1 , . . . , vk ∈ V . Such a vector determines a k-dimensional linear subspace span{v1 , . . . , vk } of V , that is, an element of the Grassmann manifold Grk (V ) of all k-planes in V . Two simple k-vectors determine the same subspace if and only if they differ by a nonzero scalar multiple. Let V ∗ denote the dual space of V . Given a pair of vectors v ∈ V, φ ∈ V ∗ , we denote by φ , v the value of the linear functional φ : V → F ∈ {R, C} on the vector v. For every k ∈ {1, . . . , dimV } we have a nondegenerate bilinear pairing ΛkV ∗ ⊗ ΛkV → F given on decomposable vectors by φ1 ∧ φ2 ∧ · · · ∧ φk , v1 ∧ v2 ∧ · · · ∧ vk  = det (φi , v j )ki, j=1 .  ∗ This pairing induces an isomorphism Λk (V ∗ ) ∼ = ΛkV . Elements of Λk (V ∗ ) correspond to alternating k-linear functionals V k = V × · · · ×V → F ∈ {R, C}. Assume now for simplicity that F = R, and let V = Rn with its standard Euclidean scalar product · , ·  and norm |· |. The dual space V ∗ = (Rn )∗ , and the exterior powers ΛkV and ΛkV ∗ are then also equipped with dual Euclidean scalar products and norms. On decomposable k-vectors the scalar product is given by v1 ∧ v2 ∧ · · · ∧ vk , w1 ∧ w2 ∧ · · · ∧ wk  = det (vi , w j )ki, j=1 . If {e1 , . . . , en } is an orthonormal basis of Rn then the vectors eI (1.3) are an orthonormal basis of Λk Rn . We denote the Euclidean norm of a k-vector v ∈ Λk Rn by |v|E . In addition to the Euclidean norms on Λk Rn and Λk (Rn )∗ , there are the mass norm |v| and the comass norm |φ |, also dual to each other and defined by |φ | = sup{|φ , v| : v ∈ Λk Rn is a simple k-vector with |v|E = 1}, |v| = sup{|φ , v| : φ ∈ Λk (Rn )∗ , |φ | = 1}. Clearly,

(1.5)

4

1 Fundamentals

|φ | ≤ |φ |E = sup{|φ , v| : v ∈ Λk Rn , |v|E = 1} (since the supremum in |φ |E is over a larger set of vectors v ∈ Λk Rn ), and hence |v| ≥ |v|E . The Euclidean and mass norms agree for k = 1, but not for k > 1; however, they agree on decomposable vectors and covectors. Function spaces. Given an open set D ⊂ Rn and r ∈ Z+ ∪ {∞}, we denote by C r (D) the vector space of all r-times continuously differentiable functions on D. In particular, C 0 (D) = C (D) is the space of continuous functions on D. If x = (x1 , . . . , xn ) are coordinates on Rn and I = (i1 , i2 , . . . , in ) ∈ Zn+ is a multiindex, we denote the corresponding partial derivative of a function f (x1 , . . . , xn ) by ∂I f =

∂ |I| f ∂ |I| f = i . I 1 ∂x ∂ x1 · · · ∂ xnin

Here, |I| = i1 + · · · + in . If f = ( f1 , . . . , fm ) : D → Rm is a differentiable map on a domain D ⊂ Rn , we denote by   ∂ fi (p) (1.6) D f (p) = ∂xj i=1,...,m j=1,...,n

its Jacobian matrix of first order partial derivatives and, if m = n, by Jac( f ) its Jacobian determinant: Jac( f ) = det(D f ) = det(∂ fi /∂ x j ).

(1.7)

By C ω (D) we denote the space of real analytic functions on a domain D ⊂ i.e., functions f which admit a convergent power series expansion in a neighbourhood of any point a ∈ D: Rn ,

f (x) =

∑n cI (a)(x1 − a1 )i1 · · · (xn − an )in .

I∈Z+

Note that for any r ∈ Z+ ∪ {∞}, C r (D) is a Fr´echet algebra when endowed with the topology of uniform convergence of functions and their partial derivatives of order up to r on compact subsets of D. If r ∈ Z+ and 0 < α ≤ 1 then C r,α (D) denotes the space of functions of class C r (D) whose derivatives of top order r are H¨older continuous of order α. If K is a closed subset of Rn then C r (K) denotes the space of functions that are of class C r (U) on some open set U = U f ⊃ K, where we do not distinguish pairs of functions which agree on a neighbourhood of K. If r ∈ Z+ ∪{∞} then any f ∈ C r (K) extends to a function in C r (Rn ) supported in any given open set U ⊃ K. In particular, if K is compact, the extension can be chosen to have compact support: f ∈ C0r (Rn ). When K = D is the closure of an open set D ⊂ Rn with piecewise C 1 boundary, it follows from H. Whitney’s extension theorem [343] that f ∈ C r (D) if and only if f ∈ C r (D) and all its partial derivatives of order up to r extend continuously to D.

1.2 Topological Manifolds

5

For an open set D in Cn we denote by O(D) the Fr´echet algebra of holomorphic functions on D with the compact-open topology. By the Cauchy estimates, this coincides with the C r compact-open topology for any r ∈ N. Given a compact set K ⊂ Cn , we denote by O(K) the space of functions holomorphic on a variable neighbourhood of K, and by O(K) the uniform closure of { f |K : f ∈ O(K)} in the Banach algebra C (K) of all continuous functions on K. For a closed set K ⊂ Cn we ˚ the space of continuous functions K → C which denote by A (K) = C (K) ∩ O(K) are holomorphic in the interior K˚ of K. If K is compact, then A (K) is a Banach subalgebra of the Banach algebra C (K). The same notation is used for the corresponding function spaces on domains in real and complex manifolds.

1.2 Topological Manifolds A topological manifold of dimension dim M = n ∈ N is a second countable Hausdorff topological space M which is locally Euclidean of dimension n, in the sense that every point p ∈ M has an open neighbourhood U ⊂ M homeomorphic to an open set in Rn . A topological manifold of dimension 2 is a topological surface. i ⊂ Rn from open A collection U = {(Ui , φi )}i∈I of homeomorphisms φi : Ui → U n sets Ui ⊂ M onto open sets in R such that i∈I Ui = M is called an atlas on M, and elements (Ui , φi ) ∈ U are called local charts (or simply charts) on M. If we ask for the less restrictive condition that every point p ∈ M has an open neighbourhood homeomorphic to an open set in the closed halfspace Hn+ = {(x1 , . . . , xn ) ∈ Rn : xn ≥ 0},

(1.8)

then M is a topological manifold of dimension n with boundary. The boundary bM of M consists of all points p ∈ M for which there is a homeomorphism from an open neighbourhood of p in M onto an open set in Hn+ mapping p into the real hyperplane bHn+ = {(x1 , . . . , xn ) ∈ Rn : xn = 0}. Brouwer’s invariance of domain theorem implies that a point p ∈ bM does not admit an open neighbourhood homeomorphic to an open set in Rn , so the notion of boundary is consistent. The boundary bM is closed in M and is a topological manifold of dimension n − 1 without boundary, whereas the interior M˚ = M \ bM is a topological manifold of dimension n without boundary. We shall assume that manifolds have no boundary unless otherwise specified. A noncompact manifold without boundary is called an open manifold, and an open surface if it has dimension 2. A continuous map f : X → Y of topological spaces is said to be proper if for every compact K ⊂ Y the preimage f −1 (K) = {x ∈ X : f (x) ∈ K} is also compact. If X and Y are locally compact and paracompact Hausdorff spaces (for example, manifolds), then a map f : X → Y is proper if and only if for any closed discrete set E ⊂ X the image f (E) ⊂ Y is also closed and discrete. An important property of a

6

1 Fundamentals

proper map is that it is closed, i.e., the image of every closed set in X is a closed set in Y . Proper maps play a major role in geometry, and also in the present book. In particular, a continuous function f : X → R which is bounded from below is called an exhaustion function on X if it is a proper map, which means that for every number c ∈ R+ the closed sublevel set Xc = {x ∈ X : ρ(x) ≤ c} is compact. The following notion will be used primarily in surfaces; it plays a major role in holomorphic approximation theory (see Sect. 1.12). Definition 1.2.1. Let M be a manifold without boundary and K be a closed subset of M. A hole of K (in M) is a relatively compact connected component of M \ K, i.e., a connected component with compact closure in M. Let X be a topological space, and suppose that K1 ⊂ K2 ⊂ K3 ⊂ · · · is an increasing sequence of compact subsets of X whose interiors cover X. Then X has one end for every sequence U1 ⊃ U2 ⊃ U3 ⊃ · · · , where each Un is a connected component of X \ Kn . Roughly speaking, ends of X are the connected components of the ideal boundary of X, and each end represents a topologically distinct way to move to infinity within the space. If M is a compact manifold with boundary bM then the connected components of bM are the ends of its interior M \ bM. In the remainder of the section we will be dealing with topological surfaces. If M is a compact topological surface with boundary, then its boundary bM is a union of finitely many simple closed curves, homeomorphic copies of the circle S1 . A surface containing a topological subspace homeomorphic to an open M¨obius strip, i.e., the topological quotient of R2 \ {0} under the reflection about the origin 0 ∈ R2 , is said to be nonorientable; otherwise, the surface is orientable. A simple but important way of constructing new surfaces from old ones is the connected sum. Let M1 and M2 be connected disjoint surfaces. Choose closed discs D1 ⊂ M˚ 1 and D2 ⊂ M˚ 2 (homeomorphic images of the closed disc D ⊂ C). The sets M1 \ D˚ 1 and M2 \ D˚ 2 are topological surfaces containing bD1 and bD2 in their respective boundaries. For any homeomorphism f : bD2 → bD1 the adjunction space M1 # M2 = (M1 \ D˚ 1 ) ∪ f (M2 \ D˚ 2 ) is a topological surface called the connected sum of M1 and M2 . Its homeomorphism type does not depend on the choice of the discs D1 , D2 or the homeomorphism f . Furthermore, M1 # M2 is homeomorphic to M2 # M1 , and the operation is associative. (See for instance J. M. Lee [220, Chapter 6].) The most basic examples of compact surfaces are the sphere S2 and the torus T2 in the orientable framework, and the real projective plane RP2 in the nonorientable one. All compact surfaces are connected sums of these basic examples. Theorem 1.2.2. Let M be a connected compact topological surface without boundary. If M is orientable, then M is homeomorphic either to the sphere S2 or g to the connected sum T2 # · · · # T2 of g copies of the torus for a unique g ∈ N. If M g is nonorientable, then M is homeomorphic to the connected sum RP2 # · · · # RP2 of g copies of the real projective plane for a unique g ∈ N.

1.2 Topological Manifolds

7

Open surfaces also admit a classification in terms of that of compact surfaces and an ideal boundary (see I. Richards [297]). If M is a connected compact surface without boundary which is not homeomorphic to the sphere S2 , then the number gen(M) = g ∈ N in Theorem 1.2.2 is called the genus of M. Thus, T2 and RP2 have genus 1. We define the genus of the 2-sphere to be zero, gen(S2 ) = 0. The Euler number χ(M) of a connected compact surface M without boundary equals

2 − 2 gen(M) if M is orientable, (1.9) χ(M) = 2 − gen(M) if M is nonorientable. In particular, χ(S2 ) = 2. Theorem 1.2.2 says that a connected compact surface is determined by its orientability character and its genus (or its Euler number). Let M be a compact connected surface without boundary. Removing from M the interiors of pairwise disjoint closed discs D1 , . . . , Dm ⊂ M gives a compact connected surface M  with m boundary components bD1 , . . . , bDm having the same orientability character as M, the same genus as M, and the Euler number χ(M  ) = χ(M) − m.

(1.10)

Every connected compact surface with boundary is obtained in this way. We denote by H1 (M, Z) the first homology group of M with integer coefficients; this is the abelianization of the fundamental group π1 (M). If M is a compact orientable surface of genus g without boundary then H1 (M, Z) ∼ = Z2g ,

(1.11)

and if M has m ≥ 1 boundary components then H1 (M, Z) ∼ = Z2g+m−1 .

(1.12)

If M is a compact nonorientable surface of genus g then H1 (M, Z) ∼ = Zg−1 ⊕ Z2 .

(1.13)

It would be appropriate to discuss more carefully the topological structure of surfaces. This classical subject can be found in many texts. Since every topological surface admits a smooth structure which is unique up to diffemorphisms (see e.g. the survey by A. Hatcher [187]), it will be easier and closer to the spirit of our work to deal with this matter in Section 1.4 by using differential calculus.

8

1 Fundamentals

1.3 Differentiable Manifolds Let M be a topological manifold of dimension n ∈ N. Recall that an atlas on M is a collection U = {(U j , φ j )} of open sets U j ⊂ M and homeomorphisms φ j : U j → U j (local charts) onto open subsets U j of Rn such that j U j = M. Write Ui, j = Ui ∩U j . The atlas U is said to be of class C r for some r ∈ {1, 2, . . . , ∞, ω} (where ω stands for the real analytic class) if all transition maps φi, j = φi ◦ φ j−1 : φ j (Ui, j ) → φi (Ui, j )

(1.14)

∼ R2n , are C r diffeomorphisms. If dim M = 2n and the local charts take values in Cn = the atlas U is called a complex atlas if the transition maps φi, j are holomorphic. Two C r atlases U , V on M are said to be C r equivalent if their union U ∪ V is again a C r atlas; clearly this is an equivalence relation on the set of C r atlases. The union of all atlases in an equivalence class is a maximal C r atlas on M. Such an atlas determines on M the structure of a C r manifold, or simply a C r structure. A manifold of class C r is a topological manifold with a choice of a C r structure. A complex manifold of complex dimension n is a topological 2n-dimensional manifold with an equivalence class of complex atlases, i.e., a complex structure. A complex manifold of complex dimension 1 is called a Riemann surface. An atlas is said to be oriented if the transition maps φi, j preserve the orientation, i.e., their Jacobian determinants Jac(φi, j ) > 0 are positive. A manifold M is orientable if it admits an oriented atlas; an orientation on such M is determined by a choice of an oriented atlas. A connected orientable manifold has precisely two orientations; if one is determined by an atlas {(Ui , φi = (φi,1 , . . . , φi,n ))}, the opposite orientation is determined for instance by the atlas {(Ui , (−φi,1 , φi,2 , . . . , φi,n ))}. A holomorphic map f = ( f1 , . . . , fn ) between domains in Cn satisfies Jac( f ) = |Jac C ( f )|2 ≥ 0, where Jac( f ) is the Jacobian determinant of f considered as a map between domains in R2n and Jac C ( f ) = det(∂ fi /∂ z j ). In particular, a biholomorphic map has positive Jacobian. It follows that every complex manifold carries a natural orientation compatible with its complex structure. Let M and N be C r manifolds. A continuous map f : M → N is said to be smooth of class C r , or simply a C r map, if the composition ψ ◦ f ◦ φ −1 is of class C r for any pair of charts from the respective C r atlases on M and N. This notion is consistent precisely because the charts in a C r atlas are C r compatible. The analogous definition pertains to the case when M, N are complex manifolds, and we get the notion of a holomorphic map. A bijective map f : M → N such that both f and its inverse f −1 are of class C r is called a C r diffeomorphism (a biholomorphism in the holomorphic case). The set of all C r diffeomorphisms M → M is denoted Diff r (M) and is a group under composition. If M is a complex manifold then Aut(M) denotes the set of all

1.3 Differentiable Manifolds

9

holomorphic automorphisms of M, i.e., bijective holomorphic maps f : M → M with a holomorphic inverse f −1 . (The last condition is a consequence of the first two.) By a theorem of H. Whitney [344], every maximal atlas of class C r for some r ≥ 1 on a manifold M contains a C ω atlas. This means that every C r structure on M has a C r compatible structure of a real analytic manifold. However, there are topological manifolds of dimensions > 2 which do not admit any smooth manifold structure, and there are uncountably many non-diffeomorphic smooth structures on R4 (see R. Gompf [161]). On the other hand, every topological surface admits a smooth structure which is unique up to diffeomorphisms (see A. Hatcher [187]). In Sect. 1.8 we shall see that every orientable surface also admits the structure of a Riemann surface, i.e., a surface endowed with the structure of a one-dimensional complex manifold. The existence of complex structures on manifolds of even dimension > 2 is a considerably more delicate matter. Suppose that M is an n-dimensional manifold with boundary bM (see Sect. 1.2). i ⊂ Hn+ If U = {(Ui , φi )} is a C r atlas on M with φi = (φi,1 , . . . , φi,n ) : Ui → U   (see (1.8)) then, setting Ui = Ui ∩ bM and φi = (φi,1 , . . . , φi,n−1 ) we obtain a C r atlas U  = {(Ui , φi )} on bM which renders bM a C r manifold without boundary. Furthermore, if U is an oriented atlas then so is U  . The coherent orientation on bM is defined as follows. Considering the standard orientation on Hn+ as a domain in Rn , the coherent orientation on bHn+ = Rn−1 is defined to be (−1)n times its standard orientation; hence it agrees with the standard orientation on Rn−1 for even values of n and disagrees for odd values of n. By using charts in an oriented atlas on M, this notion carries over to any oriented manifold M with boundary bM and defines a coherent orientation on bM. The tangent bundle. A fundamental object associated to a C r manifold M (r ≥ 1) is its tangent bundle π : T M → M. This is a vector bundle of class C r−1 and rank n = dim M over M; real analytic if M is such, complex if M is a complex manifold. The fibre Tp M = π −1 (p) ∼ = Rn over p ∈ M is called the tangent space of M at p, and its elements are tangent vectors at p. The most intuitive geometric interpretation of a tangent vector at a point p ∈ M is the velocity vector at t = 0 of some differentiable path γ : (−ε, ε) → M with γ(0) = p. When M = Rn , the velocity vector is simply the derivative γ  (0). On an abstract manifold this does not work directly, but we can look at the velocity vector in any local chart around p and find the transformation rule. For computational purposes there is a more convenient interpretation of tangent vectors as linear functionals on C r (M) given by γ  (0)( f ) = ( f ◦ γ) (0),

f ∈ C r (M).

If M = Rn and γ  (0) = v = (v1 , . . . , vn ) ∈ Rn , this is simply the directional derivative n

v( f ) = ∑ vi i=1

∂f (p) = d f p (v), ∂ xi

p = γ(0).

Note that vi = v(xi ) where x = (x1 , . . . , xn ) are the coordinates on Rn . Thus, we may think of a tangent vector v as a first order linear partial differential operator at p:

10

1 Fundamentals

 n ∂  v = ∑ vi  , i=1 ∂ xi p

(v1 , . . . , vn ) ∈ Rn .

(1.15)

An operator of this type acting on the space of germs of smooth functions at p is characterized by the property that it is linear and satisfies the Leibniz rule: v( f1 f2 ) = f2 (p)v( f1 ) + f1 (p)v( f2 ). This allows us to identify Tp Rn with Rn , and hence T Rn = Rn × Rn with the projection onto the first factor. Given a C r map f : M → N, its differential d f p : Tp M → T f (p) N is defined by d f p (v)(g) = v(g ◦ f ),

v ∈ Tp M, g ∈ C r (N).

(1.16)

Note that d f p is a linear map which coincides with the classical differential when M = Rm and N = Rn . Indeed, in the standard bases ∂∂x , . . . , ∂ ∂xm on Tp Rm and 1

∂ ∂ ∂ y1 , . . . , ∂ yn 

on T f (p) Rn , the differential d f p is represented by the Jacobian matrix  ∂ fi D f (p) = ∂ x j (p) (1.6). In the special case when f : M → R is a function and we identify Tx R with R, it follows from (1.16) (taking g(x) = x) that v( f ) = d f p (v) ∈ R,

p ∈ M, v ∈ Tp M.

(1.17)

That is, a tangent vector v ∈ Tp M acts on a function f in the same way as the differential d f p : Tp M → R of f acts on the vector v. It is immediate from (1.16) that the chain rule holds: given C r maps f : M → N and h : N → Z we have that d(h ◦ f ) p = dh f (p) ◦ d f p ,

p ∈ M.

(1.18)

If φ = (φ1 , . . . , φm ) : U → Rm is a chart on M, the tangent vectors vi = ∂ /∂ φi ∈ Tp M satisfying (dφ ) p (vi ) = ∂ /∂ xi |φ (p) (i = 1, . . . , m) form a basis of Tp M for all p ∈ U. Given a C r map f : M → N and a point p ∈ M, we choose a pair of charts (U, φ ) on M and (V, ψ) on N with p ∈ U and f (p) ∈ V . It follows from (1.18) that the matrix of the differential d f p : Tp M → T f (p) N in the pair of bases {∂ /∂ φi }m i=1 on Tp M and n −1 {∂ /∂ ψ j } j=1 on T f (p) N is the Jacobian matrix D(ψ ◦ f ◦ φ )(φ (p)) (1.6). To every C r chart φ = (φ1 , . . . , φn ) : U → Rn on an open set U ⊂ M we associate the bijective fibrewise linear map ∼ =

Φ : T M|U = TU −→ U × Rn ,

Φ(v) = (p, (dφ ) p (v)), v ∈ TpU.

Such Φ is called a vector bundle chart on the tangent bundle T M. Given a C r atlas U = {(Ui , φi )} on M, we obtain the associated vector bundle atlas {(TUi , Φi )} on n n T M with transition maps Φi, j = Φi ◦ Φ −1 j : Ui, j × R → Ui, j × R given by Φi, j (p, v) = (p, Dφi, j (φ j (p))· v) ,

p ∈ Ui, j , v ∈ Rn .

In particular, pr2 ◦ Φi, j (p, · ) : Rn → Rn (where pr2 : Ui, j × Rn → Rn is the projection onto the second factor) is a linear isomorphism depending in a C r−1 manner on the

1.3 Differentiable Manifolds

11

base point p. Hence, the atlas {(TUi , Φi )} defines on T M the structure of a real vector bundle of rank n = dim M and of class C r−1 . Vector fields and their flows. Vector fields on a C r manifold M are sections of the tangent bundle π : T M → M, i.e., maps v : M → T M with π ◦ v = IdM . In local coordinates (x1 , . . . , xn ) on an open set U ⊂ M, we have n

v(p) = ∑ vi (p) i=1

∂   , ∂ xi p

p ∈ U.

(1.19)

The vector field v is of class C k for some k ≤ r − 1 if and only if its coefficient functions vi in any C r local chart are of class C k . A C 1 path γ : (a, b) → M is an integral curve of a vector field v if ˙ = v(γ(t)), γ(t)

t ∈ (a, b) ⊂ R.

∂ 

∂ ˙ = dγ t ∂t where ∂t Here, γ(t) is the coordinate vector field on R. In local coordinates, with v given by (1.19) and γ = (γ1 , . . . , γn ), this is equivalent to the system of ordinary differential equations

γ˙ j (t) = v j (γ1 (t), . . . , γn (t)),

j = 1, . . . , n.

(1.20)

Assume now that the vector field v is of class C k for some k ≥ 1. The local existence theorem says that any point p0 ∈ M has an open neighbourhood U ⊂ M for which there is a number δ > 0 such that for every p ∈ U there is a unique integral curve (−δ , +δ ) t → φt (p) ∈ M satisfying φ˙t (p) = v(φt (p)),

φ 0 (p) = p.

(1.21)

Furthermore, the maps φ and φ˙ = ∂ φ /∂t are of class C k in (t, p) ∈ (−δ , +δ ) ×U. The map φt satisfying (1.21) is called the (local) flow of the vector field v. Each trajectory t → φt (p) extends to the largest open interval I p = (α(p), ω(p)) ⊂ R,

−∞ ≤ α(p) < 0 < ω(p) ≤ +∞.

The open set Ω = {(t, p) ∈ R × M : t ∈ I p }

(1.22)

is called the fundamental domain of the vector field v. For every fixed t ∈ R, the time-t map φt is a diffeomorphism from its domain Mt = {p ∈ M : t ∈ Ip } onto its image in M, and we have the group property of the flow: φt+s = φt ◦ φs ,

φ 0 = IdM ,

φ−t = (φt )−1 .

(1.23)

A family {φt }t∈R with these properties is called a local 1-parameter group of diffeomorphisms of M. Conversely, every local 1-parameter group of diffeomorphisms is the flow of the vector field

12

1 Fundamentals

 ∂  v(p) =  ∂t

φt (p),

p ∈ M,

t=0

called the infinitesimal generator of {φt }t∈R . A vector field v of class C k is said to be complete (or completely integrable) if its fundamental domain equals R × M, that is, every trajectory φt (p) is defined for all t ∈ R. In such case, {φt }t∈R is a 1-parameter subgroup of the diffeomorphism group Diff k (M). Equivalently, the flow of v defines an action of the additive group (R, +) on M by diffeomorphisms. Conversely, every 1-parameter subgroup of Diff k (M) is the flow of a complete vector field, namely, its infinitesimal generator. Immersions, embeddings, submersions, local diffeomorphisms. Let M, N be C r manifolds for some r ≥ 1. A C r map f : M → N is said to be an immersion if its differential d f p : Tp M → T f (p) N is injective for all p ∈ M. If f is an immersion and a topological embedding (i.e., a homeomorphism onto its image f (M) ⊂ N in the relative topology on f (M) inherited from N), we say that f is a C r embedding. The image f (M) is then a C r submanifold of N. Every immersion f : M → N is an embedding on a neighbourhood of any point p ∈ M as can be seen from the implicit function theorem. More precisely, there are charts (U, φ ) on M, with p ∈ U and φ (p) = 0 ∈ Rm (m = dim M), and (V, ψ) on N, with f (p) ∈ V and ψ( f (p)) = 0 ∈ Rn (n = dim N ≥ m), such that f (U) ⊂ V and ψ ◦ f ◦ φ −1 (x1 , . . . , xm ) = (x1 , . . . , xm , 0, . . . , 0) ∈ Rn holds for all x = (x1 , . . . , xm ) ∈ φ (U) ⊂ Rm . If in addition f : M → N is a topological embedding then, after shrinking V around f (p) if necessary, we obtain   ψ( f (M) ∩V ) = ψ(V ) ∩ Rm × {0}n−m ⊂ Rn . This is precisely what it means for the subset f (M) to be a C r submanifold of N. The map f : M → N is said to be a submersion if d f p is surjective for all p ∈ M; in this case m = dim M ≥ n = dim N. In suitably chosen local coordinates around the points p ∈ M and f (p) ∈ N, a submersion corresponds to the projection x = (x1 , . . . , xn , . . . , xm ) −→ (x1 , . . . , xn ). A map f is a local diffeomorphism if d f p is bijective for all p ∈ M. A bijective local diffeomorphism is a diffeomorphism. A diffeomorphism f : M → N pushes forward a vector field v on M to the vector field f∗ v on N defined by ( f∗ v)( f (p)) = d f p (v p ), p ∈ M. If φt denotes the flow of v and ψt the flow of f∗V , then f ◦ φt = ψt ◦ f holds on the fundamental domain of v. The cotangent bundle and differential forms. The cotangent bundle T ∗ M of a smooth manifold M is the dual bundle of its tangent bundle T M. Elements of the cotangent space Tp∗ M = (Tp M)∗ at p ∈ M are linear functionals on the tangent space Tp M, called tangent covectors. Sections of T ∗ M are differential 1-forms on M. In local coordinates x = (x1 , . . . , xn ) on M and taking the vectors ∂∂xi (i = 1, . . . , n)

1.3 Differentiable Manifolds

13

as the standard basis of Tp M, the dual basis of Tp∗ M is given by the differentials dx1 , . . . , dxn . Indeed, in accordance with (1.17) we have that   ∂ ∂ xi = δi, j (the Kronecker delta). = dxi ∂xj ∂xj A 1-form is given in these coordinates by α(p) = ∑ni=1 ai (p)dxi where ai are functions. Its value on a vector field v = ∑nj=1 v j ∂∂x j is α(v)(p) = ∑ni=1 ai (p)vi (p).

More generally, a differential k-form on M for some k ∈ N is a correspondence which assigns to each p ∈ M an alternating k-linear map α p : (Tp M)k → R. We may view α p as an element of Λk Tp∗ M, the k-th exterior power of the cotangent space Tp∗ M. (See Sect. 1.1.) In local coordinates x = (x1 , . . . , xn ) on M, α has the form α(p) =

∑

1≤i1 0 if M is contractible. of cohomology rings; in particular, HdR ∗ (M) is naturally isomorphic This leads to the classical theorem of de Rham that HdR ∗ to the simplicial cohomology H (M, R) with coefficients in R; in particular, it is a topological invariant of M.

Orientation and volume forms. It is easily seen that a smooth manifold M is orientable if and only if it carries a volume form, that is, a nowhere vanishing form Ω of top degree n = dim M. The standard volume form on Rn is Ω 0 = dx1 ∧ dx2 ∧ · · · ∧ dxn .

(1.28)

Indeed, the choice of a volume form Ω on M determines an orientation on M according to the criterion that φ = (x1 , . . . , xn ) : U → Rn is a positively oriented chart on M if and only if Ω ( ∂∂x , . . . , ∂∂xn ) > 0. Conversely, if U = {(Ui , φi )} is an 1 oriented atlas on a locally finite covering of M, then by choosing a smooth partition of unity {χi } on M with suppχi ⊂ Ui for every i, the n-form Ω = ∑i χi φi∗ Ω 0 on M is a volume form determining the same orientation as the atlas U .

1.3 Differentiable Manifolds

15

If M is an oriented manifold with boundary bM and a volume form Ω , then the coherent orientation on bM (see p. 9) is determined by the volume form Ω (ν, · ) on bM, where ν is any outward pointing vector field on M along bM. This condition  ⊂ Hn+ around a point on ν means that for any local chart φ = (φ1 , . . . , φn ) : U → U p ∈ bM we have that ν(φn )(p) < 0. Integration and Stokes’s theorem. Differential n-forms on an orientable ndimensional manifold are natural objects for integration. An n-form on a domain U ⊂ Rn is given by α = f Ω 0 for some function f : U → R, where Ω 0 is the volume form (1.28). Assuming that f is Lebesgue measurable and f ∈ L1 (U), we define  U

f Ω0 =



f (x) dx1 . . . dxn .

U

(1.29)

If φ : U  → U is a C 1 diffeomorphism between domains in Rn , a simple calculation shows that φ ∗ Ω 0 = Jac(φ )Ω 0 . Hence,  U

φ ∗( f Ω 0) =

 U

( f ◦ φ ) Jac(φ )Ω 0 =

 U

( f ◦ φ ) Jac(φ ) dx1 . . . dxn .

If Jac(φ ) > 0, this equals (1.29) by the change of variables formula, so the integral of an n-form on Rn is independent of the choice of coordinates in the given orientation class. Let M be an orientable n-dimensional manifold. Choose an oriented locally finite atlas U = {(Ui , φi )} and a subordinate partition of unity {χi } on M. Given an n-form α on M with continuous coefficients and of compact support, we set  M

α =



∑ i

Rn

(φi−1 )∗ (χi α).

The n-form (φi−1 )∗ (χi α) is compactly supported in φi (Ui ) ⊂ Rn , and there are only finitely many nonzero terms since supp(α) is compact. The result is independent of the choices made as long as we are taking charts φi from the same maximal oriented atlas on M. Modulo convergence issues, the integral can be extended to forms with measurable coefficients and not necessarily compact support. If f : Z → M is a C 1 map from a k-dimensional oriented manifold Z to M, then   for every continuous k-form α on M we can define f α = Z f ∗ α, provided the integral converges. A special case is the integral over an oriented k-dimensional  submanifold Z → M with f the inclusion map; in this case we simply write Z α. We can now state one of the central results in this area. Theorem 1.3.1 (Stokes’s theorem). Assume that M is a compact oriented ndimensional manifold of class C 1 with coherently oriented boundary bM. For every (n − 1)-form α of class C 1 on M we have that  bM

In particular,



bM α

α =



dα. M

= 0 holds for every closed (n − 1)-form α on M.

(1.30)

16

1 Fundamentals

This result is a synthesis of many classical theorems, from the one-dimensional Leibniz–Newton formula ab f  (x)dx = f (b) − f (a), to Green’s formula in the plane (which also implies Cauchy’s integral formula for holomorphic functions), the classical Stokes’s formula relating the circulation of a vector field along the boundary of an oriented surface with the flux of its curl through the surface, the 3dimensional Gauss formula relating the flux of the vector field through the boundary of a domain in R3 to the integral of its divergence on the domain, etc. Stokes’s theorem extends to piecewise smooth oriented k-chains c in M, defining homology classes [c] ∈ Hk (M, Z), by 



dα = c

∂c

α

(1.31)

for any (k − 1)-form α on M. See e.g. F. W. Warner [337] for the details.

1.4 Handlebody Structure of Manifolds This section is a brief introduction to transversality and Morse theory. Among classical references, we mention the monographs by J. Milnor [260] (1963) and M. Goresky and R. MacPherson [163] (1988). We recall the Morse–Sard lemma, the Thom jet transversality theorem for maps between smooth manifolds, and we indicate how a Morse exhaustion function on a smooth manifold can be used to present the manifold as a smooth handlebody, that is, an object obtained by successive gluing of handles. Special attention is paid to the handlebody structure of surfaces. This approach, using strongly subharmonic Morse exhaustion functions on open Riemann surfaces, is employed in many constructions in the book. The Morse–Sard lemma and transversality. Let f : M → R be a differentiable function on a C 1 manifold M. A point p ∈ M is said to be a critical point of f if the differential d f p vanishes, and is a regular point otherwise. In local coordinates x = (x1 , . . . , xm ) on M, critical points are solutions of the system of equations ∂f = 0, ∂ xi

i = 1, . . . , m.

(1.32)

The set Crit( f ) = {p ∈ M : d f p = 0} of critical points of f is a closed subset of M. Conversely, every closed subset of M is the critical locus of some differentiable function on M. Its image f (Crit( f )) = {c ∈ R : there exists p ∈ Crit( f ) with f (p) = c}

(1.33)

is the set of critical values of f . A number c ∈ R which is not a critical value of f is called a regular value; this includes all numbers not in the range of f . Similarly one defines critical points and critical values of a C 1 map f : M → N between

1.4 Handlebody Structure of Manifolds

17

manifolds. Explicitly, Crit( f ) is the set of all points p ∈ M at which the differential d f p : Tp M → T f (p) N fails to be surjective. The following lemma due to A. P. Morse (1939) and A. Sard (1942) is one of the most important basic results in differential topology. It is a stepping stone in the proof of Thom’s transversality theorem described in the sequel. Lemma 1.4.1. The set of critical values of a smooth function f : M → R on a smooth manifold M has Lebesgue measure zero in R and is of the first category. In fact, this holds if M and f are of class C m with m = dim M. The analogous statement holds for maps f : M → Rn of class C r with r = max{1, m − n + 1}. Remark 1.4.2. Recall that a topological space T is a Baire space if any countable intersection of everywhere dense open subsets of T is everywhere dense; such a set is said to be of second category in T . The complement of a set of second category is a set of first category, a union of at most countably many closed nowhere dense subsets of T . A property is called generic if it holds for all elements in a set of second category. Every complete metric space is a Baire space.   Hence, Lemma 1.4.1 says that most values of a smooth function are regular values. A proof can be found in most texts on differential topology; see e.g. M. Golubitsky and V. Guillemin [159], M. Goresky and R. MacPherson [163], or Gromov [170, Sec. 1.3.2]. The Morse–Sard lemma has several major implications. An immediate one is that for most numbers c ∈ R the level set {p ∈ M : f (p) = c} of a smooth function f : M → R is either empty or a smooth hypersurface in M. Indeed, by the implicit function theorem the second alternative holds whenever c ∈ f (M) is a regular value of f . The Morse–Sard lemma is also the starting point of Thom’s transversality theorem [331] which we now describe. Let f : M → N be a smooth map between manifolds, and let Z be a smooth submanifold of N. (For the definitions, the class C 1 suffices.) We say that f is transverse to Z, denoted f  Z, if and only if d f p (Tp M) + T f (p) Z = T f (p) N holds for all p ∈ f −1 (Z).

(1.34)

For a point Z = {q} ⊂ N this is equivalent to q being a regular value of f . If f  Z, the implicit function theorem shows that the preimage f −1 (Z) ⊂ M, if nonempty, is a smooth submanifold of M of the same codimension as Z ⊂ N: dim M − dim f −1 (Z) = dim N − dim Z. In particular, if dim M + dim Z < dim N then a map f : M → N is transverse to Z if and only if f (M) ∩ Z = ∅. If Z is a closed submanifold of N without boundary then (1.34) is an open condition in the fine C 1 Whitney topology on the space C 1 (M, N). The basic transversality theorem of R. Thom [331] asserts that if M and N are smooth manifolds and Z ⊂ N is a closed smooth submanifold of N, then a generic smooth map f : M → N is transverse to Z. We are using the fine smooth Whitney

18

1 Fundamentals

topology on C ∞ (M, N) which renders it a Baire space. The precise amount of smoothness needed in this theorem depends on the dimensions of the manifolds. When N = R (or N = Rn ) this is just the Morse–Sard lemma, and the proof of the general case amounts to a reduction to this lemma. A particularly elegant proof of the transversality theorem was given by R. Abraham [1] in 1963; here is an outline. Given a smooth map f : M → N, we find a smooth map F : M × Rn → N for some n ≥ dim N such that F(· , 0) = f and ∂t F(p,t) : Rn → TF(p,t) N is surjective for every (p,t) ∈ M × Rn .

(1.35)

Such F is called a submersive family of maps. It can be obtained for example by composing f with flows φtii of suitably chosen complete smooth vector fields on N: F(p,t1 , . . . ,tn ) = φt11 ◦ · · · ◦ φtnn ( f (p)) ∈ N,

p ∈ M, t = (t1 , . . . ,tn ) ∈ Rn .

Set ft = F(· ,t) : M → N for t ∈ Rn and let π : M × Rn → Rn denote the projection π(p,t) = t. Assume now that Z is a closed smooth submanifold of N. Since F is a submersion, the preimage S = F −1 (Z) is a closed smooth submanifold of M × Rn . It is easily seen that the following conditions are equivalent (cf. [163, p. 40]): (a) (d ft ) p (Tp M) + T f (p) Z = T f (p) N. (b) The point (p,t) ∈ S is a regular point of the restricted projection π|S : S → Rn . By Sard’s lemma, almost every point t ∈ Rn satisfies condition (b). It follows that: Theorem 1.4.3 (Abraham [1]). If F : M ×Rn → N is a submersive family of smooth maps M → N (see (1.35)), then for any closed smooth submanifold Z ⊂ N the map ft = F(· ,t) : M → N is transverse to Z for every t ∈ Rn in the complement of a set of Lebesgue measure zero. By choosing t close to 0, the map ft is close to f0 = f , so we obtain an approximation of any given smooth map by maps transverse to Z. Theorem 1.4.3 holds for any smooth manifold in place of Rn as the parameter space. An important extension is Thom’s jet transversality theorem. If M and N are smooth manifolds and k ∈ Z+ , we denote by Jk (M, N) the manifold of k-jets of smooth maps M → N. Recall that a k-jet of a map f : M → N at a point p ∈ M is determined in any pair of local coordinates on the two manifolds by its k-th order Taylor polynomial at p. We refer to [163] or [170] for the precise definition and properties of jet manifolds. Given a smooth map f : M → N we denote by jk f : M → Jk (M, N) its k-jet extension. Theorem 1.4.4 (Jet transversality theorem). If M and N are smooth manifolds and Z is a smooth closed submanifold of Jk (M, N) for some k ∈ Z+ , then for a generic smooth map f : M → N its k-jet extension jk f : M → Jk (M, N) is transverse to Z. A proof and generalizations to the parametric and stratified cases can be found in [163] and in many other sources. The basic transversality theorem is a special case

1.4 Handlebody Structure of Manifolds

19

with k = 0. Analogous results hold for holomorphic maps between certain classes of complex manifolds; see [140, Sect. 8.8]. Hessian and Morse functions. Let x = (x1 , . . . , xn ) be coordinates on Rn . Given a domain U ⊂ Rn and a C 2 function ρ : U → R, the Hessian of ρ at a point x ∈ U is the symmetric quadratic form Hessρ (x) = Hessρ (x; · ) on Tx Rn ∼ = Rn given by Hessρ (x; ξ ) =

∂ 2ρ (x) ξ j ξk , j,k=1 ∂ x j ∂ xk n

∑

ξ = (ξ1 , . . . , ξn ) ∈ Rn .

(1.36)

2 The matrix Hρ (x) = ∂ x∂ j ∂ρx of the Hessian is the Jacobian matrix of the gradient

k map ∇ρ = ∂∂xρ , . . . , ∂∂xρn : U → Rn , and its trace is the Laplace operator on Rn : 1

∂ 2ρ (x). 2 i=1 ∂ xi n

tr Hρ (x) = Δ ρ(x) = ∑

(1.37)

The Hessian arises naturally as the second order term in the Taylor expansion of ρ: 1 ρ(x + ξ ) = ρ(x) + dρx (ξ ) + Hessρ (x; ξ ) + o(|ξ |2 ). 2 A change of coordinates x = φ (y) changes the Hessian by Hessρ◦φ (y; ξ ) = Hessρ (x; dφy (ξ )),

(1.38)

and hence the corresponding matrices change by Hρ◦φ (y) = Dφ (y)t Hρ (x)Dφ (y).

(1.39)

Recall that Dφ (y) denotes the Jacobian matrix (1.6) of φ at y. Assume now that x ∈ U is a critical point of ρ, i.e. dρx = 0. We say that x is a nondegenerate or Morse critical point of ρ if Hessρ (x) is nondegenerate, i.e., zero is not an eigenvalue of the matrix Hρ (x). We see from Hρ (x) = D(∇ρ)(x) that a critical point x is a Morse point if and only if the gradient map ∇ρ : U → Rn has maximal rank n at x. The number k ∈ {0, 1, . . . , n} of negative eigenvalues Hρ (x) is called the Morse index of the critical point x. Assume that x = 0 ∈ Rn is a Morse critical point of ρ. A theorem of Sylvester shows that for any nondegenerate symmetric n × n matrix H there is an invertible n × n matrix A such that At HA = Ik is the diagonal matrix whose first k diagonal entries equal −1 and the remaining n − k entries equal +1. In view of (1.39), the linear change of coordinates x → Ax therefore brings ρ to the following normal form, which was first described by M. Morse in 1934 (see e.g. J. Milnor [260]): 2 ρ(x) = ρ(0) − x12 − · · · − xk2 + xk+1 + · · · xn2 + o(|x|2 ).

(1.40)

20

1 Fundamentals

It follows that a Morse critical point is isolated, i.e., there are no other critical points in a small neighbourhood. Since critical points are solutions of ∇ρ(x) = 0 (1.32), this is also seen from Hρ (x) = D(∇ρ)(x) and the inverse function theorem. In view of the formula (1.38), the notions of a Morse critical point and its index are independent of coordinate changes of class C 2 , and hence they generalize to functions on smooth manifolds. Definition 1.4.5. A function ρ : M → R of class C 2 (M) is a Morse function if all its critical points are Morse points. Morse functions clearly form an open set in the fine C 2 Whitney topology on It follows from the jet transversality theorem (see Theorem 1.4.4), applied with k = 2 and N = R, that any C 2 function on M can be approximated arbitrarily closely in the fine C 2 topology by a Morse function. In particular: C 2 (M).

Proposition 1.4.6. Every C 2 exhaustion function M → R can be approximated arbitrarily closely in the fine C 2 topology by a smooth Morse exhaustion function whose critical points lie on distinct level sets and whose normal form (1.40) at any critical point is without the remainder term. The condition that distinct critical points lie on different level sets is easily achieved due to openness of the set of Morse functions in the fine C 2 topology. The last condition in the proposition is obtained by a simple modification of the function near any critical point; see [140, Lemma 3.10.3] for the details. Handlebody structure of a smooth manifold. Proposition 1.4.6 allows us to describe the handlebody structure of a smooth manifold. Let us first recall the meaning of this notion; see J. Milnor [260], R. Gompf and A. Stipsicz [162], or K. Cieliebak and Y. Eliashberg [89] for more details. Let Dk denote the closed unit ball in Rk . The boundary bDk = Sk−1 is the (k − 1)dimensional sphere. Note that D0 is a point which we denote by 0, and bD0 = ∅. The product H = H k,q = Dk × Dq is called the standard handle of index k and dimension m = k + q. The k-disc E = Dk × {0}q ⊂ H is the core, and the spherical cylinder bDk × Dq = Sk−1 × Dq is the attaching set of the handle. Assume now that M is a smooth manifold of dimension m ≥ 1 with boundary bM, k ∈ {1, . . . , m}, and φ : bDk × Dq = Sk−1 × Dq → bM,

k + q = m,

= is a smooth diffeomorphism onto its image in bM. The adjunction space M k q M ∪φ H is a handlebody obtained by gluing to M a handle H = D × D of index  is a k with the attaching map φ . After smoothing the corners at φ (Sk−1 × Sq−1 ), M smooth manifold with boundary  = (bM \ φ (Sk−1 × Dq )) ∪φ (Dk × Sq−1 ). bM  admits a deformation retraction onto M ∪φ E, where E ∼ Note that M = Dk is the core k−1 to bM via the map φ . disc of the handle H glued along its boundary sphere S

1.4 Handlebody Structure of Manifolds

21

This type of surgery arises naturally when considering the change of topology of sublevel sets of an exhaustion function at a Morse critical point. In fact, one of the basic results of Morse theory can be formulated as follows. Theorem 1.4.7. Let M be a smooth manifold of dimension m, ρ : M → R be a smooth Morse function, and a < b be regular values of ρ such that the set Ma,b := {x ∈ M : a ≤ ρ(x) ≤ b} is compact. Then the following hold. (a) If the set Ma,b does not contain any critical point of ρ, then the sublevel sets Mt = {x ∈ M : ρ(x) ≤ t} for t ∈ [a, b] are diffeomorphic to each other, and Mt is a smooth deformation retract of Ms for any a ≤ t < s ≤ b. (b) If Ma,b contains a unique critical point of ρ and this point has Morse index k, then Mb is diffeomorphic to Ma with an attached handle of index k. If k = 0 then Mb is diffeomorphic to the disjoint union of Ma and a compact m-ball. In case (a) one can find a diffeomorphism φ : Ma,b → {x ∈ M : ρ(x) = a} × [a, b] of the form φ (x) = (ψ(x), ρ(x)) by using the flow of a smooth vector field V with V (ρ) > 0 on Ma,b . Choosing the size of V in a suitable way, the flow φt of V maps the level set {ρ = a} diffeomorphically onto {ρ = a + t} for all t ∈ [0, b − a]. Case (b) can be seen quite explicitly by looking at the change of the sublevel set of the model quadratic Morse function (1.40) when passing the critical point at the origin. Since every smooth manifold M admits a Morse exhaustion function ρ : M → R (see Proposition 1.4.6), we infer that M is built by a sequence of surgeries, adding a new handle at every passage of a critical point of ρ. Local minima of ρ spawn new connected components, while at any local maximum of ρ one attaches a disc of dimension dim M and thereby compactifies the corresponding end. Handles attached at various stages of this procedure may cancel each other and the precise analysis is quite involved; see [162]. Handlebody structure of surfaces. Assume now that ρ : M → R is a smooth Morse exhaustion function on a surface M. For every regular value c of ρ the level set {ρ = c} is a union of finitely many smooth Jordan curves, and the sublevel set Mc = {ρ ≤ c} is a smooth compact surface with boundary bMc = {ρ = c}. If ρ has no critical values on an interval [a, b], then Ma,b = {x ∈ M : a ≤ ρ(x) ≤ b} = Mb \ M˚ a consists of finitely many pairwise disjoint compact annuli, one for each connected component of {ρ = a} (see Theorem 1.4.7 (a)). We have already observed above what happens at local minima or maxima of ρ. Suppose now that p ∈ M is a critical point of Morse index one, and assume that this is the only critical point on the level set {ρ = ρ(p)}. Choose numbers a < ρ(p) < b close enough to ρ(p) such that p is the only critical point of ρ in the compact set {a ≤ ρ ≤ b}. Theorem 1.4.7 (b) then says that Mb = {ρ ≤ b} is diffeomorphic to a handlebody obtained by attaching to Ma = {ρ ≤ a} the strip H = [−1, 1]2 (a handle of index 1) such that the sides {±1} × [−1, 1] of H are glued by a diffeomorphism onto disjoint arcs I± ⊂ bM. We have the following three distinct possibilities regarding the location of these arcs. (i) I+ and I− belong to the same connected component of bMa = {ρ = a}.

22

1 Fundamentals

(ii) I+ and I− belong to the same connected component of Ma but to different connected components of bMa . (iii) I+ and I− belong to different connected components of Ma . There are further distinctions with regard to the orientability character of the attachment. If Ma is orientable and the strip H is attached in an untwisted way then Mb is also orientable; otherwise it is nonorientable. Assume that we are in the orientable situation; we shall mainly consider this case in the book. In case (i) the surfaces Ma and Mb have the same genus, and the boundary bMb has one more connected component than bMa . In case (ii), gen(Mb ) = gen(Ma ) + 1 and the number of boundary components decreases by one. In this case, Mb \ M˚ a consists of a pair of pants (a compact surface with genus one and three boundary components) together with finitely many pairwise disjoint annuli; see Figure 1.1. In case (iii) the handle H connects two different connected components of Ma , the genus remains unchanged, and the number of boundary curves decreases by one. The Euler number decreases by one in cases (i) and (ii), and it increases by one in case (iii). bMb

Mb bMa

bMa

Ma

E bMa

bMa

Ma

Fig. 1.1 Adding a pair of pants

1.5 Complex Manifolds In this section we review basic differential calculus on complex manifolds. As we shall see, the complex structure provides another layer of structure for vector fields, differential forms, and other tensor fields. Basic examples of complex manifolds. We shall mainly be dealing with complex Euclidean spaces, complex projective spaces, their complex submanifolds and subvarieties, and holomorphic mappings between them.

1.5 Complex Manifolds

23

Example 1.5.1 (Affine complex manifolds). Let z = (z1 , . . . , zn ) be complex coordinates on Cn . A closed complex subvariety of Cn is a subset of the form X = {z ∈ Cn : f1 (z) = 0, . . . , fm (z) = 0}, where f1 , . . . , fm ∈ O(Cn ) are entire functions. If the defining functions f j are holomorphic polynomials then X is called an affine algebraic variety. If X has no singularities, it is called a complex submanifold of Cn . Such X is locally near any point p ∈ X the zero set of a collection of holomorphic functions g1 , . . . , gd in an open neighbourhood of p with C-linearly independent differentials; the number d is then called the codimension of X at p, and m = n − d is the dimension of X at p. (We assume that a submanifold has the same dimension at every point.) A closed complex subvariety X of Cn admits many holomorphic functions, obtained by restricting entire functions on Cn to X. In particular: (a) For every pair of distinct points p = q in X there is a holomorphic function f ∈ O(X) such that f (p) = f (q). (b) If K is a compact subset of X, then its O(X)-convex hull defined by =K O (X) = {z ∈ X : | f (z)| ≤ max | f | for all f ∈ O(X)} K K

(1.41)

is also compact. By considering functions f = eg , where g is an affine C-linear function on Cn , one  ⊂ Co(K) ∩ X where Co(K) denotes the closed convex hull of K in Cn . sees that K Definition 1.5.2. A complex manifold X is a Stein manifold if it satisfies the above axioms (a) and (b). This important class of complex manifolds was introduced by K. Stein [322] in 1951. Two years earlier, H. Behnke and K. Stein [58] proved that every open complex manifold of dimension one (an open Riemann surface) is a Stein manifold. Note that any closed complex submanifold of Cn is a Stein manifold. Conversely, it was shown by R. Remmert in 1956 that every Stein manifold is biholomorphic to a closed complex submanifold of a complex Euclidean space. More precise embedding theorems were obtained soon thereafter by E. Bishop (1961) and R. Narasimhan (1962). The optimal result providing embeddings of Stein manifolds of dimension n > 1 into Euclidean spaces of minimal possible dimension was established by Y. Eliashberg and M. Gromov (1992), with an improvement by J. Sch¨urmann (1997) for odd values of n. Here is a summary statement. Theorem 1.5.3. Every open Riemann surface (a Stein manifold of dimension 1) admits a proper holomorphic embedding into C3 . Every Stein manifold of complex   dimension n > 1 admits a proper holomorphic embedding into CN with N = 3n 2 +1 and this number is the smallest one with this property. It is conjectured that every open Riemann surface admits a proper holomorphic embedding into C2 ; however, few positive results are known on this problem. See

24

1 Fundamentals

also Theorem 1.10.3 and the discussion following it. We refer to [140, Chap. 9] for a more complete survey of this topic. A domain Ω in Cn (or in any Stein manifold) is a Stein manifold if and only if it is a domain of holomorphy, meaning that there exists a holomorphic function on Ω which does not admit a local holomorphic continuation across any boundary point. In particular, every convex domain in Cn is Stein. Another important geometric characterization of the class of Stein manifolds is by the existence of strongly plurisubharmonic exhaustion functions; see Definition 1.5.8 for this notion. Among the references for Stein manifold, we mention in particular the monographs by R. C. Gunning and H. Rossi [176], H. Grauert and R. Remmert [165], and L. H¨ormander [195]. The text [140] by the second named author is a suitable reference for the approximation and gluing techniques which will be used in this book. See Sections 1.12 and 1.13 for more on this subject.   Example 1.5.4 (Projective spaces and projective varieties). The n-dimensional complex projective space, CPn , consists of all complex lines through the origin in Cn+1 . Such a line λ ⊂ Cn+1 is determined by a point 0 = z = (z0 , . . . , zn ) ∈ λ ; we denote it by [z] = [z0 : z1 : · · · : zn ] and call these homogeneous coordinates on CPn . Clearly, [z] = [w] if and only if w = tz for some t ∈ C∗ . There is a unique → CPn , complex manifold structure on CPn for which the projection π : Cn+1 ∗ n n π(z) = [z] ∈ CP is holomorphic. A complex atlas on CP is given by the collection (U j , φ j ) ( j = 0, 1, . . . , n) where U j = {[z0 : z1 : · · · : zn ] ∈ CPn : z j = 0} and     z j−1 z j+1 z0 zn φ j [z0 : z1 : · · · : zn ] = ,..., , ,..., ∈ Cn . zj zj zj zj It is immediate that the transition maps φi ◦ φ j−1 are linear fractional. Note that CP1 = C = C ∪ {∞}

(the Riemann sphere).

Similarly one defines the real projective space RPn . It is a real submanifold of CPn consisting of all points [z0 : z1 : · · · : zn ] ∈ CPn with real coordinates. A nonzero holomorphic polynomial P(z0 , . . . , zn ) is said to be homogeneous of degree d ∈ N if P(tz0 , . . . ,tzn ) = t d P(z0 , . . . , zn ) for all t ∈ C. Its zero set   V (P) = [z0 : z1 : · · · : zn ] ∈ CPn : P(z0 , . . . , zn ) = 0 (1.42) is a well defined complex hypersurface in CPn . More generally, a collection of homogeneous polynomials P1 , . . . , Pm on Cn+1 determines an algebraic subvariety V (P1 , . . . , Pm ) = V (P1 ) ∩ · · · ∩ V (Pm ) ⊂ CPn . Subvarieties of CPn are called projective varieties, or projective manifolds when they are nonsingular. By Chow’s theorem [88, 167], every closed complex subvariety of CPn is algebraic and of the form V (P1 , . . . , Pm ) for some homogeneous polynomials in n + 1 variables. Complex projective spaces have very simple homology groups:

1.5 Complex Manifolds

25

 Hk (CPn , Z) =

Z 0

if k ≤ 2n is even, if k is odd or k > 2n.

(1.43)

The generator Lk of the group H2k (CPn , Z) ∼ = Z is the homology class of any projective linear subspace CPk → CPn . A complex subvariety X ⊂ CPn is said to be of pure dimension k if X has dimension k at every regular point. Such a variety determines a homology class [X] = d· Lk for some positive integer d = deg(X) ∈ N, called the degree of X. (Subvarieties with negative orientation determine negative multipliers of the generator Lk .) In particular, for every closed complex curve C ⊂ CPn we have [C] = deg(C)· L ∈ H2 (CPn , Z) ∼ = Z,

(1.44)

where L is the homology class of the projective line CP1 → CPn . For more in this subject, see for example the monographs by P. Griffiths and J. Harris [167] and R. Hartshorne [179].   Remark 1.5.5 (The degree of a projective variety and intersection numbers). The homological degree d = deg(X) of a projective variety has the following geometric interpretation. Assume that X ⊂ CPn is of pure complex dimension k. Choose a pair of projective linear subspaces E, L ⊂ CPn of dimensions n − k − 1 and k, respectively, such that (X ∪ L) ∩ E = ∅. (For dimension reasons this holds for a generic choice of E; note that E is a point when k = n − 1.) For every point p ∈ CPn \ E there exists a unique (n − k)-dimensional projective subspace Σ p of CPn such that E ∪ {p} ⊂ Σ p , and we get a holomorphic map τ : CPn \ E → L taking p to the unique point in L ∩ Σ p . The restriction τ|X : X → L is then a d-sheeted branched holomorphic covering, where d = deg(X) is the degree of X. The fibres of the projection τ : CPn \ E → L introduced above are (n − k)dimensional linear subspaces whose closures (obtained by adding E) are (n − k)dimensional projective subspaces. This also shows that the degree deg(X) equals the intersection number of X with any (n−k)-dimensional projective subspace Σ ⊂ CPn in general position with respect to X, meaning that X ∩ Σ is zero-dimensional. The intersections have to be counted with algebraic multiplicities, and they are simple intersections (of multiplicity one) for a generic choice of Σ . If X is a hypersurface in CPn defined as in (1.42) by a single irreducible homogeneous polynomial equation {P = 0}, then deg(X) equals the degree of P. In general, factoring P = P1d1 P2d2 · · · Pmdm where P1 , . . . , Pm are different irreducible homogeneous polynomials, we have that deg X = d1 + · · · + dm . Alternatively, such X can be understood as a chain X = ∑mj=1 d j X j with integer multiplicities, where X j = {Pj = 0}, in which case we again have deg(X) = deg P. Similarly one defines the degree of a holomorphic map F : X → CPn from a compact k-dimensional manifold into CPn as the intersection number of F with a generic (n − k)-dimensional projective subspace Σ ⊂ CPn , i.e., the number of solutions of the equation F(x) ∈ Σ taken with multiplicities. We refer to Fulton [154] for more information on intersection theory.  

26

1 Fundamentals

Holomorphic vector bundles. Vector bundles serve as a principal tool used to linearise problems in analysis and geometry. They are also a subject of intrinsic investigation with a profound impact on modern mathematics. We focus on holomorphic vector bundles, recalling those constructions that will be important to us. Similar constructions apply to other classes of vector bundles (topological, smooth, and with C replaced by another field such as R). A holomorphic vector bundle of rank n over a complex manifold M is a holomorphic fibre bundle π : E → M with fibre Y = Cn and structure group GLn (C). A vector bundle of rank n = 1 is called a line bundle. This means that E is a complex manifold, π : E → M is a surjective holomorphic map, and there exist an open cover ∼ = U = {Ui } of M and biholomorphic vector bundle charts θi : E|Ui → Ui × Cn such n that θi (Ex ) = {x} × C for each x ∈ Ui and   θi ◦ θ j−1 (x, v) = θi, j (x, v) = x, fi, j (x)v , (1.45) where x ∈ Ui, j = Ui ∩U j , v ∈ Cn , and fi, j : Ui, j → GLn (C) are holomorphic matrixvalued functions satisfying the conditions fi,i = 1,

fi, j f j,i = 1,

fi, j f j,k fk,i = 1.

(1.46)

A collection ( fi, j ) with these properties is called a holomorphic 1-cocycle on the cover U with values in GLn (C). Conversely, any such 1-cocycle determines a holomorphic vector bundle E → M, obtained by gluing the collection of products Ui × Cn with the maps fi, j according to the formula (1.45). It follows that every fibre Ex = π −1 (x) (x ∈ M) carries a natural structure of a complex vector space such that the fibre bundle charts θi : E|Ui → Ui × Cn are Cvector space isomorphisms on each fibre. A section of π : E → M is a map f : M → E such that π ◦ f = IdM . In particular, the zero section sends each point x ∈ M to the origin 0x ∈ Ex . Due to the existence of complex structures of fibres of E, the spaces ΓO (M, E) ⊂ Γ(M, E) of holomorphic and continuous sections, respectively, are complex vector spaces endowed with the compact-open topology. Given a holomorphic vector bundle atlas {(Ui , θi )} on E with transition maps fi, j (1.45), a section f : M → E is determined by a collection of maps fi : Ui → Cn satisfying the compatibility conditions fi = fi, j f j

on Ui, j .

(1.47)

The section is holomorphic if and only if the maps fi are such. An isomorphism of holomorphic vector bundles π : E → M, π  : E  → M is a biholomorphic map Φ : E → E  such that Φ : Ex → Ex is a C-linear isomorphism for any x ∈ M. Assume as we may that E and E  are presented by holomorphic vector bundle atlases over the same open cover U = {Ui }i∈I of M, and denote the respective transition maps (1.45) by fi, j and fi, j . In the vector bundle charts over Ui , Φ is given by a biholomorphic self-map of Ui × Cn of the form (x, v) → (x, φi (x)v), where φi : Ui → GLn (C) is a holomorphic map and we have that

1.5 Complex Manifolds

27

φi fi, j = fi, j φ j on Ui, j for all i, j ∈ I. Conversely, every collection of holomorphic maps φi : Ui → GLn (C) satisfying these compatibility conditions determines an isomorphism Φ : E → E  . It easily follows that the isomorphism classes of holomorphic vector bundles of rank n over M are in bijective correspondence with the elements of the first cohomology group H 1 (M, O GLn (C) ) with values in the sheaf of germs of holomorphic maps M → GLn (C). Note that this is a sheaf of nonabelian groups if n > 1. In the special case n = 1 we have GL1 (C) = C∗ , and O GL1 (C) = O ∗ is the sheaf of germs of nonvanishing holomorphic functions on M. We conclude that the isomorphism classes of holomorphic line bundles on a complex manifold M are in bijective correspondence with elements of the cohomology group H 1 (M, O ∗ ) = Pic(M),

(1.48)

called the Picard group of M. The group operation on Pic(M), which is induced by the multiplication of 1-cocycles (1.46) with values in O ∗ , corresponds to the tensor product E ⊗ E  of line bundles on M. In particular, each line bundle E has an inverse line bundle E −1 , with E ⊗ E −1 isomorphic to the trivial line bundle M × C. The Picard group can often be computed from the following exact sequence of sheaf homomorphisms (the exponential sheaf sequence), where σ ( f ) = e2πi f : σ

0 −→ Z −→ O −→ O ∗ −→ 1.

(1.49)

The relevant portion of the long exact cohomology sequence reads: c

1 H 2 (M, Z) −→ H 2 (M, O). (1.50) H 1 (M, Z) −→ H 1 (M, O) −→ H 1 (M, O ∗ ) −→

The homomorphism c1 : H 1 (M, O ∗ ) → H 2 (M, Z) is called the first Chern class map. We have the following obvious consequence. Theorem 1.5.6. If M is a complex manifold such that H 1 (M, O) = 0 = H 2 (M, O), then the map c1 : H 1 (M, O ∗ ) = Pic(M) → H 2 (M, Z) is an isomorphism. If M is a real manifold of class C r and E → M is a complex vector bundle with transition maps (1.45), then E is a complex vector bundle of class C r on M. For r = 0 we have a topological complex vector bundle. Replacing C by R we get (topological or smooth) real vector bundles. Cr

The holomorphic tangent bundle. Let us begin by considering the model space Cn with coordinates z = (z1 , . . . , zn ), where z j = x j +iy j for j = 1, . . . , n. A differentiable complex-valued function f (z1 , . . . , zn ) is holomorphic if and only if it satisfies the Cauchy–Riemann system of equations at every point: ∂f ∂f =i , ∂yj ∂xj

j = 1, . . . , n.

(1.51)

28

1 Fundamentals

Let us now conceptualize this condition. The tangent bundle T Cn = T R2n is spanned at each point by the values of the vector fields ∂∂x j , ∂∂y j for j = 1, . . . , n. We introduce the fibrewise linear operator J : T Cn → T Cn (a real endomorphism of T Cn ) by J

∂ ∂ = , ∂xj ∂yj

J

∂ ∂ =− . ∂yj ∂xj

(1.52)

Clearly, J 2 = −Id. This is the standard complex structure operator on Cn . It is immediate that a differentiable function f (z1 , . . . , zn ) is holomorphic if and only if for every point z in the domain of f , d fz (Jv) = i d fz (v),

v ∈ Tz Cn .

Indeed, applying this to the tangent vectors v j =

∂ ∂xj

gives the system (1.51).

Suppose now that M is a complex manifold of complex dimension n. The tangent bundle T M carries an R-linear endomorphism JM ∈ EndR T M, called the complex structure operator, which is uniquely determined by the condition that for every  ⊂ Cn on M we have holomorphic chart φ = (φ1 , . . . , φn ) : U → U dφ p (JM v) = J ◦ dφ p (v),

p ∈ U, v ∈ Tp M.

(1.53)

The definition of JM is consistent because charts are holomorphically compatible. The equation (1.53) means that the tangent map of the chart φ defines a holomorphic vector bundle chart on (T M, JM ). This is the holomorphic tangent bundle of M. Given another complex manifold (N, JN ), a differentiable map f : M → N is holomorphic (see p. 8) precisely when its differential d f p : Tp M → T f (p) N at any point p ∈ M intertwines the complex structure operators on T M and T N: d f p ◦ JM = JN ◦ d f p .

(1.54)

We extend the endomorphism J = JM : T M → T M to the complexified tangent bundle CT M = T M ⊗ C by setting J(v ⊗ c) = J(v) ⊗ c,

v ∈ T M, c ∈ C.

Since J 2 = −Id, the eigenvalues of J are ±i. Hence, we have a decomposition CT M = T 1,0 M ⊕ T 0,1 M

(1.55)

into the +i eigenspace T 1,0 M and the −i eigenspace T 0,1 M of J. In holomorphic coordinates z = (z1 , . . . , zn ) on a neighbourhood of a point p ∈ M we have



 ∂ ∂ ∂ ∂ ,..., ,..., , Tp0,1 M = spanC , Tp1,0 M = spanC ∂ z1 ∂ zn ∂ z¯1 ∂ z¯n where

1.5 Complex Manifolds

29

1 ∂ = ∂zj 2



∂ ∂ −i ∂xj ∂yj



1 ∂ = ∂ z¯ j 2

,



∂ ∂ +i ∂xj ∂yj

 .

The conjugation v ⊗ c → v ⊗ c¯ on CT M induces a C-antilinear isomorphism of T 1,0 M onto T 0,1 M. The Cauchy–Riemann equations (1.51) can be written as 2

∂f ∂f ∂f = +i = 0, ∂ z¯ j ∂xj ∂yj

j = 1, . . . , n,

(1.56)

so holomorphic functions are precisely the differentiable functions annihilated by the complex vector fields ∂∂z¯ j for j = 1, . . . , n. Note that T 1,0 M is a holomorphic vector bundle over M of rank n = dim M whose transition maps are given by the complex Jacobian matrices of the holomorphic transition maps between holomorphic charts on M. Sections of T 1,0 M are complex vector fields of type (1, 0); in local coordinates any such is of the form ∂

n

v=

∑ v j (z) ∂ z j .

(1.57)

j=1

A (1, 0)-vector field is holomorphic if it is a holomorphic section of T 1,0 M; equivalently, if its coefficiens v j in any holomorphic coordinate system are holomorphic functions. Sections of T 0,1 M are vector fields of type (0, 1); these are locally of the form w = ∑nj=1 w j (z) ∂∂z¯ j . The isomorphism (T M, J) ∼ = T 1,0 M. There is a vector bundle isomorphism ∼ =

Θ : (T M, J) −→ T 1,0 M,

v −→ Θ (v) =

1 (v − i Jv) , 2

(1.58)

with the inverse Θ −1 (w) = 2ℜ w = w + w ∈ T M,

w ∈ T 1,0 M,

such that Θ (Jv) = iΘ (v) for all v ∈ T M, i.e., the following diagram commutes: TM Θ



T 1,0 M

/ TM

J

i·

/



Θ

T 1,0 M.

In local holomorphic coordinates, the isomorphism Θ is given by  n  n ∂ ∂ ∂ Θ + b a ∑ j ∂ x j j ∂ y j −→ ∑ (a j + i b j ) ∂ z j . j=1 j=1

(1.59)

This isomorphism allows us to identify the holomorphic tangent bundle (T M, J) with the holomorphic (1, 0)-tangent bundle T 1,0 M, and real vector fields on M with

30

1 Fundamentals

vector fields of type (1, 0). A real vector field v (a section of T M) is holomorphic if and only if the associated (1, 0)-vector field Θ (v) = 12 (v − i Jv) is holomorphic. We see from (1.59) that the real vector field v = ∑nj=1 a j ∂∂x j + b j ∂∂y j is holomorphic if and only if the functions a j + i b j ( j = 1, . . . , n) are holomorphic. Let (M, JM ) and (N, JN ) be complex manifolds. Recall that a differentiable map f : M → N is holomorphic precisely when its differential d f p : Tp M → T f (p) N at any point p ∈ M intertwines the complex structure operators on M and N (see (1.54)). The complexified differential d f p : CTp M → CT f (p) N then respects the eigenvalue N is C-linear decompositions (1.55) on both sides, and the map d f p : Tp1,0 M → T f1,0 (p) N is C-antilinear for every p ∈ M. while d f p : Tp0,1 M → T f0,1 (p) Remark 1.5.7. We shall tacitly identify (T M, J) with T 1,0 M using the isomorphism (1.58) and will simply write T M for the holomorphic tangent bundle of a complex manifold.   The normal bundle of a complex submanifold. If M is a complex submanifold of a complex manifold N, then its tangent bundle T M is a holomorphic vector subbundle of T N|M , the restriction of the tangent bundle of N to M. The quotient holomorphic vector bundle νM/N = T N|M /T M is called the complex normal bundle of M in N. We have a short exact sequence β

0 −→ T M −→ T N|M −→ νM/N −→ 0,

(1.60)

where β is the quotient projection with the kernel T M. One can realise νM/N as a smooth complex vector subbundle of T N|M such that T N|M = T M ⊕ νM/N .

(1.61)

If M is Stein manifold (see Definition 1.5.2), then the sequence (1.60) also splits holomorphically and yields a holomorphic direct sum (1.61). For a survey and further references on this subject, we refer to [140, Chapters 2, 3]. Flows of holomorphic vector fields. Let v be a holomorphic vector field on a complex manifold M. The equation (1.21) for the flow of v can locally be solved for complex values of the time variable t. It yields a local flow φt (p) satisfying conditions (1.23) which is holomorphic in both the time and the space variables. A holomorphic vector field is said to be complete in complex time if its flow φt (p) is defined for all t ∈ C and p ∈ M; in such case the family {φt }t∈C is a holomorphic 1-parameter subgroup of the holomorphic automorphism group Aut(M). Recall that an entire vector field on Cn which is complete in real time (or even just in positive real time) is also complete in complex time; see [133, Corollary 2.2] and [3]. The holomorphic cotangent bundle. The complexified cotangent bundle CT ∗ M = T ∗ M ⊗ C of a complex manifold M decomposes as CT ∗ M = T ∗1,0 M ⊕ T ∗ 0,1 M,

1.5 Complex Manifolds

31

where in local holomorphic coordinates z = (z1 , . . . , zn ) we have Tp∗1,0 M = spanC {dz1 , . . . , dzn },

Tp∗ 0,1 M = spanC {d z¯1 , . . . , d z¯n }.

Tangent (1, 0)-covectors annihilate all tangent (0, 1)-vectors, and (0, 1)-covectors annihilate all tangent (1, 0)-vectors. T ∗1,0 M is a holomorphic vector bundle of rank n = dim M, the complex dual bundle of T 1,0 M. Its sections are (1, 0)-forms. In local holomorphic coordinates they have the form α = ∑ j=1 a j (z) dz j , and α is holomorphic if and only if its coefficient functions a j are such. Its value on a (1, 0)vector field v (1.57) is α(v) = ∑nj=1 a j v j . A (0, 1)-form is a section of T ∗ 0,1 M; it has a local form β = ∑ j=1 b j (z) d z¯ j . A (p, q)-form (p, q ∈ Z+ ) has a local expression α=

aI,K (z) dzI ∧ d z¯K ,

∑

(1.62)

#I=p, #K=q

where dzI = dzi1 ∧ · · · ∧ dzi p and d z¯K = d z¯k1 ∧ · · · ∧ d z¯kq . Every differential r-form on M admits a decomposition α = ∑ p+q=r α p,q into (p, q)-homogeneous parts. Let (M, JM ) and (N, JN ) be complex manifolds and f : M → N a holomorphic map. We have seen that the complexified differential d f p : CTp M → CT f (p) N N is a respects the ±i eigenspace decompositions (1.55), and d f p : Tp1,0 M → T f1,0 (p) C-linear map for every p ∈ M. It follows that the pullback of differential forms by a holomorphic map preserves their (p, q)-homogeneous parts, i.e., ( f ∗ α) p,q = f ∗ (α p,q ). In local coordinates z = (z1 , . . . , zm ) on M and w = (w1 , . . . , wn ) on N, the pullback of a typical term in (1.62) is given by f ∗ (a · dwI ∧ dwK )(z) = a( f (z)) d fI (z) ∧ d fK (z). Basic differential operators on complex manifolds. The exterior derivative (1.25) on a complex manifold splits as d = ∂ +∂, where in holomorphic coordinates z = (z1 , . . . , zn ) and for a function f we have ∂f =

∂f ∑ ∂ z j dz j , j=1 n

∂f =

n

∂f

∑ ∂ z¯ j d z¯ j .

(1.63)

j=1

In view of (1.56), the equation ∂ f = 0 characterizes holomorphic functions. The operators ∂ and ∂ extend to (p, q)-forms (1.62) by ∂ α = ∑ ∂ aI,K (z) ∧ dzI ∧ d z¯K ,

∂ α = ∑ ∂ aI,K (z) ∧ dzI ∧ d z¯K .

Thus, ∂ α is a (p + 1, q)-form while ∂ α is a (p, q + 1)-form. One easily verifies that ∂ 2 = ∂ ◦ ∂ = 0,

2

∂ = ∂ ◦ ∂ = 0,

∂ ∂ = −∂ ∂ ,

(1.64)

32

1 Fundamentals

where the last equation follows from the first two and 0 = d 2 = (∂ + ∂ )2 . Let d c = i (∂ − ∂ ) = 2 ℑ(∂ );

(1.65)

this is the conjugate differential. We have that d + i d c = 2∂ and dd c = i(∂ + ∂ )(∂ − ∂ ) = 2i ∂ ∂ . In one complex variable z = x + iy we have that  2  ∂ ∂2 ∂2 c · dz ∧ d z¯ = + dd = 2i dx ∧ dy = Δ · dx ∧ dy, ∂ z∂ z¯ ∂ x2 ∂ y2

(1.66)

(1.67)

where Δ is the Laplace operator; hence dd c f = 0 if and only if f is a harmonic function. (In some sources the Laplacian is defined with the opposite sign.) The operator dd c (1.66) is the pluricomplex Laplacian. It plays a major role in complex analysis. The restriction of dd c to any affine complex line in Cn is the Laplacian on that line. The equation dd c f = 0 characterizes pluriharmonic functions which locally coincide with real parts of holomorphic functions. It is also used to define the class of plurisubharmonic functions considered in the sequel. The Levi form and plurisubharmonic functions. A real-valued C 2 function ρ on a complex manifold (M, J) determines a Hermitian quadratic form on every tangent space Tp M, p ∈ M, by Lρ (p, v) =

 i  1 c dd ρ(p), v ∧ Jv = ∂ ∂ ρ(p), w ∧ w , 4 2

(1.68)

where v ∈ Tp M and w = Θ (v) = 21 (v − i Jv) ∈ Tp1,0 M (see (1.58)). This is called the Levi form of ρ at p. In local holomorphic coordinates z = (z1 , . . . , zn ) on M, with w = (w1 , . . . , wn ) ∈ Cn , the above expression equals n  ∂ 2 ρ(z) i ∂ ∂ ρ(z), w ∧ w = ∑ wi w j . 2 i, j=1 ∂ zi ∂ z¯ j

(1.69)

Definition 1.5.8. A real C 2 function ρ on a complex manifold (M, J) is strongly plurisubharmonic if it satisfies dd c ρ > 0, in the sense that dd c ρ, v ∧ Jv > 0 for every 0 = v ∈ T M.

(1.70)

The function ρ is (weakly) plurisubhamonic if dd c ρ ≥ 0. By (1.67), the notion of (strong) plurisubharmonicity in one complex variable coincides with (strong) subharmonicity, i.e., Δ ρ ≥ 0 (resp. Δ ρ > 0). If ρ is (strongly) plurisubharmonic on a complex manifold M then its restriction to any complex submanifold X ⊂ M is (strongly) plurisubharmonic on X. Conversely, if the restriction of ρ to every local smooth complex curve in M is (strongly) subharmonic, then ρ is (strongly) plurisubharmonic on M.

1.6 Riemannian Manifolds

33

The space Psh(M) of plurisubharmonic functions on M is closed under sums and products with nonnegative numbers. The standard definition of a plurisubharmonic function only assumes that ρ : M → [−∞, ∞) is upper semicontinuous and not identically equal to −∞, and dd c ρ ≥ 0 holds in the sense of currents. Here are some examples and a summary of the main properties of plurisubharmonic functions on a connected complex manifold M; see e.g. M. Klimek [207]. If u ∈ Psh(M) has a local maximum at some point, then u is constant. If f is holomorphic and not identically zero then log | f | is plurisubharmonic. If f is holomorphic then | f | p is plurisubharmonic for every p > 0. If f1 , . . . , fk ∈ O(M) then ∑ki=1 | fi |2 is plurisubharmonic. It is strongly plurisubharmonic at each point x ∈ M where the differentials d f1 , . . . , d fk span Tx∗1,0 M. 5. If u j ∈ Psh(M) for j = 1, . . . , k then max{u1 , . . . , uk } ∈ Psh(M). 6. If u1 ≤ u2 ≤ · · · ≤ u = lim j→∞ u j < +∞ and u j ∈ Psh(M) for every j, then u ∈ Psh(M) provided it is upper semicontinuous. If u is locally bounded from above then its upper regularization u∗ (z) = lim supζ →z u(ζ ) is plurisubharmonic. 7. If u ∈ Psh(M) and h : [−∞, ∞) → [−∞, ∞) is a convex increasing function then h ◦ u ∈ Psh(M). If u, h ∈ C 2 , u is strongly plurisubharmonic, and h is strongly convex and strongly increasing, then h ◦ u is strongly plurisubharmonic.

1. 2. 3. 4.

1.6 Riemannian Manifolds In this section we recall a few basic notions of Riemannian geometry. For a comprehensive introduction, see e.g. M. P. do Carmo [109, 110]. A Riemannian metric g on a smooth manifold M is a smoothly varying family of scalar products g p on the tangent spaces Tp M, p ∈ M. More precisely, g associates to each point p ∈ M a symmetric positive definite bilinear map g p : Tp M × Tp M → R depending smoothly on p ∈ M. (Here, smooth can mean any smoothness class C r provided M is of class C r+1 .) A smooth manifold endowed with a Riemannian metric is called a Riemannian manifold. In the model case when M is a domain in Rn with coordinates x = (x1 , . . . , xn ), a Riemannian metric g is given by n

gp =

∑

gi, j (p) dxi ⊗ dx j ,

p ∈ M,

(1.71)

i, j=1

where G(p) = (gi, j (p))ni, j=1 is a symmetric positive definite matrix for each p. (We shall often omit the tensor product sign ⊗ from the notation.) The value of g p on a pair of tangent vectors ξ = ∑i ξi ∂∂xi , η = ∑ ηi ∂∂xi ∈ Tp Rn is given by g p (ξ , η) =

n

∑

i, j=1

gi, j (p)ξi η j = G(p)ξ · η,

(1.72)

34

1 Fundamentals

where in the last expression we consider ξ = (ξ1 , . . . , ξn ) and η = (η1 , . . . , ηn ) as vectors in Rn and G(p)ξ is the product of the matrix G(p) with the vector ξ . A Riemannian metric g on a manifold M has the expression (1.71) in any local chart φ = (x1 , . . . , xn ) : U → φ (U) ⊂ Rn , with the associated matrix     ∂  ∂  n gi, j (p) = g p , G(p) = (gi, j (p))i, j=1 , , p ∈ U. (1.73) ∂ xi  p ∂ x j  p Example 1.6.1. The Euclidean metric ds2 on the Euclidean space Rn with coordinates x = (x1 , . . . , xn ) is the Riemannian metric represented by the identity matrix G = In of order n in the identity chart on Rn : ds2 = (dx1 )2 + (dx2 )2 + · · · + (dxn )2 .

(1.74)

 and a Riemannian metric g˜ on M,  the Given a smooth immersion x : M → M ∗ pullback metric g = x g˜ on M is defined on any pair of vectors ξ , η ∈ Tp M by g p (ξ , η) = g˜x(p) (dx p (ξ ), dx p (η)) .

(1.75)

If g˜ is of class C r and x is of class C r+1 , then g = x∗ g˜ is of class C r as well. Note  g) that x : (M, g) → (M, ˜ is an isometric immersion. In particular, a diffeomorphism  g) x : (M, g) → (M, ˜ satisfying x∗ g˜ = g is called an isometry. Let us find the relationship between the corresponding matrices. Fix a point  Assume that u1 , . . . , un is a basis of Tp M and v1 , . . . , vn p ∈ M and let q = x(p) ∈ M.  is a basis of Tq M. (These are the coordinate vectors in a suitable pair of local  at p and q = x(p), respectively.) Let G(p) be the matrix of coordinates on M and M  be the matrix of g˜q in the basis (v1 , . . . , vn ). g p in the basis (u1 , . . . , un ), and let G(q)  in this pair of bases. It Denote by P the matrix of the differential dx p : Tp M → Tq M then follows from (1.72) and (1.75) that for any pair of vectors ξ , η ∈ Rn ,   G(p)ξ · η = G(q)Pξ · Pη = P t G(q)Pξ · η, which is equivalent to

 G(p) = P t G(q)P.

(1.76)

The same formula pertains to the case when p = q and P is the change-of-basis matrix between two different bases on Tp M. Remark 1.6.2 (Existence of Riemannian metrics). Every manifold M of class C r+1 admits a Riemannian metric of class C r . Indeed, by Whitney’s theorem we can embed M as a closed C r+1 submanifold of the Euclidean space Rn with n = 2 dim M + 1; the restriction of the Euclidean metric ds2 on Rn to T M is then a Riemannian metric of class C r on M. Alternatively, take a locally finite atlas U = {(Ui , φi )} on M and let g = ∑i χi φi∗ (ds2 ), where ds2 is the Euclidean metric and {χi ≥ 0} is a smooth partition of unity subordinate to the cover {Ui } of M. (Note that a convex linear combination of metrics is again a metric.)  

1.6 Riemannian Manifolds

35

Example 1.6.3 (The first fundamental form of a surface). Let ds2 be the Euclidean metric (1.74) on Rn . Let us calculate the metric g = x∗ ds2 on a domain D ⊂ R2 , where   x(u1 , u2 ) = x1 (u1 , u2 ), . . . , xn (u1 , u2 ) , (u1 , u2 ) ∈ D is an immersion D → Rn . Denote its partial derivatives by xu1 , xu2 : D → Rn . The differential dx(u1 ,u2 ) : T(u1 ,u2 ) R2 → Tx(u1 ,u2 ) Rn at the point (u1 , u2 ) ∈ D applied to a tangent vector (ξ1 , ξ2 ) ∈ T(u1 ,u2 ) R2 ∼ = R2 is then given by dx(u1 ,u2 ) (ξ1 , ξ2 ) = ξ1 xu1 (u1 , u2 ) + ξ2 xu2 (u1 , u2 ).

(1.77)

Applying the formula (1.75) with g˜ = ds2 we see that g = x∗ ds2 = g1,1 du21 + g1,2 du1 du2 + g2,1 du2 du1 + g2,2 du22 ,

(1.78)

where the coefficients gi, j are given by g1,1 = |xu1 |2 ,

g1,2 = g2,1 = xu1 · xu2 ,

g2,2 = |xu2 |2 .

(1.79)

(On Rn we are using the Euclidean inner product and norm.) The matrix G(u1 , u2 ) of the metric g is given by G = (xu1 , xu2 )t (xu1 , xu2 ) = Jt J, where J = J(u1 , u2 ) is the Jacobian (n, 2)-matrix of the differential dx(u1 ,u2 ) . The Riemannian metric (1.78), (1.79) on D is traditionally called the first fundamental form of the immersed surface M = x(D) ⊂ Rn .   Example 1.6.4 (The spherical metric). The restriction of the Euclidean metric ds2 on Rn+1 (see (1.74)) to the unit sphere   Sn = (x0 , x1 , . . . , xn ) ∈ Rn+1 : x02 + x12 + · · · + xn2 = 1 is the spherical metric ds2Sn . It is most easily expressed in spherical coordinates. When n = 2 and using the coordinates (φ , θ ) ∈ [0, 2π) × [−π/2, +π/2] such that x 0 = cos θ cos φ ,

x1 = cos θ sin φ ,

x2 = sin θ ,

(1.80)

we have that ds2S2 = dθ 2 + cos2 θ · dφ 2 .

(1.81)

The area of the unit sphere S2 ⊂ R3 in this metric equals 4π. (The notion of volume of Riemannian manifolds is discussed below, see in particular (1.91).)   Example 1.6.5 (The Fubini–Study metric). The standard Riemannian metric on the complex projective space CPn (see Example 1.5.4) is the Fubini–Study metric 2 . In homogeneous coordinates ζ = (ζ , . . . , ζ ) on CPn it equals σ 2 = σCP n n 0

36

1 Fundamentals

σ2 = 4

|ζ ∧ dζ |2 . |ζ |4

(1.82)

The metric σ 2 is the real part of a K¨ahler metric on CPn ; see Section 1.7. In the affine coordinates z = (z1 , . . . , zn ) = (ζ1 /ζ0 , . . . , ζn /ζ0 ) on Cn = {ζ0 = 0} ⊂ CPn , σ2 = 4

(1 + |z|2 )|dz|2 − (¯z · dz)(z · d z¯) . (1 + |z|2 )2

(1.83)

In particular, in the affine coordinate z = x + iy ∈ C ⊂ CP1 we have that 2 σCP 1 =

4|dz|2 . (1 + |z|2 )2

(1.84)

Let P : S2 → C ∪ {∞} = CP1 be the stereographic projection from the point (0, 0, 1): z = P(x0 , x1 , x2 ) =

x 0 + i x1 cos θ = (cos φ + i sin φ ) ∈ CP1 , 1 − x2 1 − sin θ

(1.85)

where (φ , θ ) are the spherical coordinates (1.80) on S2 . A calculation shows that 2 σCP 1 =

4|dz|2 = dθ 2 + cos2 θ dφ 2 = ds2S2 , (1 + |z|2 )2

(1.86)

so the Fubini–Study metric σ 2 on CP1 agrees with the spherical metric ds2S2 on S2 via the stereographic projection. This agreement is our reason behind the choice of the normalizing constant 4 in (1.82); some texts use different choices.    The length of a path. Let |v| = g p (v, v) denote the norm of a vector v ∈ Tp M with respect to a given Riemannian metric g on M. The length of a piecewise C 1 path γ : [a, b] → M in the Riemannian manifold (M, g) is defined as (γ) =

 b a

˙ dt = |γ(t)|

 b a

˙ γ(t)) ˙ gγ(t) (γ(t), dt.

(1.87)

If γ is defined on [a, b) with a < b ≤ +∞, its length is given by (γ) = lim

 c

c→b a

˙ dt ∈ R+ ∪ {+∞}. |γ(t)|

Definition 1.6.6. A path γ : [0, b) → M (where b ∈ (0, +∞]) is divergent if the point γ(t) leaves any given compact set K ⊂ M as t approaches b. A Riemannian manifold (M, g) is complete if (γ) = +∞ for any divergent path γ : [0, b) → M of class C 1 . Consider a C 1 path γ : [a, b] → M. If t = t(s) is a monotone C 1 function of ˜ = γ(t(s)) has the same length as γ. The most natural s ∈ [α, β ] then the path γ(s) choice of parameter is the arc length:

1.6 Riemannian Manifolds

37

 t

s(t) = a

˙ dt, |γ(t)|

t ∈ [a, b].

(1.88)

Clearly, s : [a, b] → [0, (γ)] is an increasing surjective function satisfying s(t) ˙ = ˙ |γ(t)| for t ∈ [a, b]. If γ is an immersion then s(t) ˙ > 0 for all t ∈ [a, b], so we can express t = t(s) and consider the path [0, (γ)] s → γ(t(s)). We say that s is the natural parameter for γ, or that γ is parameterized by its arc length. The velocity vector γ  (s) with respect to the natural parameter has constant norm one: ˙ ˙ |t  (s)| = |γ(t)|/| s(t)| ˙ = 1. |γ  (s)| = |γ(t(s))|·

(1.89)

The volume of a Riemannian manifold. Let us see how the local expression (1.71) for a Riemannian metric g is affected by a change of coordinates. Let φ = (x1 , . . . , xn ) : U → φ (U) ⊂ Rn and φ˜ = (x˜1 , . . . , x˜n ) : U → φ˜ (U) ⊂ Rn be charts  = (g˜i, j ) denote the matrices on an open subset U ⊂ M. Let G = (gi, j ) and G (1.73) of the metric g on U in the coordinates x and x, ˜ respectively. Denote by ψ = φ ◦ φ˜ −1 : φ˜ (U) → φ (U) the transition map and by Dψ : φ˜ (U) → GLn (R) its Jacobian matrix. Then, P := Dψ ◦ φ˜ : U → GLn (C) is the change-of-basis matrix between { ∂∂x˜ , . . . , ∂∂x˜n } and { ∂∂x , . . . , ∂∂xn }. Hence, it follows from (1.76) that 1

1

 = P t GP, G

 = det G · (det P)2 . det G

The volume of U with respect to the metric g is defined as  √    d x˜1 · · · d x˜n . Vol(U) = det G dx1 · · · dxn = det G U

(1.90)

(1.91)

U

The last identity, which follows from dx1 · · · dxn = | det P| · d x˜1 · · · d x˜n and (1.90), shows that Vol(U) does not depend on the choice of parameterization. The formula  (1.91) allows us to define a Borel measure dV on M such that Vol(U) = U dV, and  to speak of the integral M f dV of a measurable function f : M → R with respect to dV . This measure is called the volume element associated to the Riemannian metric g. If M is orientable, the equation (1.91) shows that dV coincides with the volume form Ωg on M given in oriented local coordinates on U ⊂ M by √ (1.92) Ωg = det G dx1 ∧ · · · ∧ dxn . On a Riemannian surface (M, g) we use the term area instead of volume and write Area and dA, respectively. The volume form associated to g can also be expressed as follows. On any open subset U ⊂ M over which the tangent bundle T M|U is trivial the Gram–Schmidt method gives 1-forms φ1 , . . . , φn forming a g-orthonormal coframe for T ∗ M|U . The metric and the associated volume form are then given by n

g = ∑ φi ⊗ φi , i=1

Ωg = φ1 ∧ φ2 ∧ · · · ∧ φn .

(1.93)

38

1 Fundamentals

The forms φi can locally be chosen to be exact differentials, φi = dxi , if and only if g is locally isometric to the Euclidean metric ds2 on Rn given by (1.74). Divergence, gradient, and Laplacian. If (M, g) is a Riemannian manifold and X is a vector field of class C 1 on M, the divergence of X with respect to g is the function div(X) : M → R given in local coordinates (x1 , . . . , xn ) on M by   √ n 1 ∂ fi det G div(X) = ∑ √ , (1.94) ∂ xi i=1 det G where G = (gi, j ) is the matrix (1.73) of g and X = ∑ni=1 fi ∂∂xi . The divergence measures the rate of change of volume in the flow of X. More precisely, if D is a compact domain in M whose boundary bD has volume zero and φt denotes the flow of X, then  d   Vol(φt (D)) = div(X)dV. dt t=0 D The gradient of a C 1 function f : M → R with respect to the Riemannian metric g is the vector field ∇ f : M → T M defined for any p ∈ M and v ∈ Tp M by g p (∇ f (p), v) = d f p (v). In local coordinates the gradient is given by  n  ∂ f i, j ∂ g , ∇f = ∑ ∂xj i, j=1 ∂ xi

(1.95)

where (gi, j ) = G−1 . The Laplacian of f ∈ C 2 (M) is the function on M defined by Δ f = div(∇ f ). In local coordinates (x1 , . . . , xn ) the Laplacian equals   

n  ∂f ∂ √ 1 i, j ∂ i, j ∂ f Δf = ∑ g det G g +√ . ∂ xi ∂ x j ∂xj det G ∂ xi i, j=1

(1.96)

A function f on (M, g) is said to be harmonic if Δ f = 0. The divergence, gradient, and Laplacian on Rn with respect to the Euclidean metric ds2 (1.74) are occasionally denoted by div0 , ∇0 , and Δ 0 , especially when there is a need to emphasize that we are using the Euclidean metric. In particular, Δ0 f =

∂2 f ∂2 f + · · · + . ∂ xn2 ∂ x12

(1.97)

The classical divergence theorem, also known as the Gauss–Ostrogradsky theorem, asserts the following. This is a special case of Stokes’s Theorem 1.3.1.

1.7 Hermitian and K¨ahler Manifolds

39

Theorem 1.6.7. Let (M, g) be a compact Riemannian manifold with C 1 boundary bM. Let ι : bM → M be the inclusion map. For any C 1 vector field X on M we have 



div(X) dV = M

bM

g(X, ν) dA,

where dV and dA are the volume elements on (M, g) and (bM, ι ∗ g), respectively, and ν is the outward pointing unit normal vector field of bM.

1.7 Hermitian and K¨ahler Manifolds Let (M, J) be a complex manifold of dimension n. We have seen in Section 1.5 that the holomorphic tangent bundle (T M, J) is isomorphic to the (1, 0) tangent bundle T 1,0 M, with J corresponding to the multiplication by i on the latter (see (1.58)). Passing to duals, the cotangent bundle T ∗ M is isomorphic to T ∗1,0 M. Any tangent vector v ∈ T M is of the form v = w + w, where w ∈ T 1,0 M is a vector of type (1, 0) and w ∈ T 1,0 M = T 0,1 M is a vector of type (0, 1). Similar, a covector α ∈ T ∗ M decomposes as α = β + β¯ with β ∈ T ∗1,0 M and β¯ ∈ T ∗0,1 M. In local holomorphic coordinates (z1 , . . . , zn ) on M, with z j = x j + iy j , we have that ∂

n

v=

∂

j=1

and α=

∂

n

∂

∑ a j ∂ x j + b j ∂ y j = ∑ (a j + i b j ) ∂ z j + (a j − i b j ) ∂ z¯ j j=1

n

∑ aj

j=1

∂ ∂ +bj = β + β¯ , ∂xj ∂yj

β=

n

1

∑ 2 (a j + i b j )dz j .

j=1

Note that ∂

n

v=

∂

n

∑ v j ∂ z j + v j ∂ z¯ j

=⇒ Jv =

j=1

∂

∂

∑ i v j ∂ z j − iv j ∂ z¯ j .

j=1

A Hermitian metric h on a complex manifold (M, J) is a field of Hermitian inner products h p on the tangent spaces (Tp M, J) depending smoothly on p ∈ M. In local holomorphic coordinates (z1 , . . . , zn ) on U ⊂ M we have n

h=

∑

hi, j dzi ⊗ d z¯ j ,

(1.98)

i, j=1

where hi, j are functions on U. The value of h on a pair of tangent vectors at p, n

v=

∂

∂

∑ v j ∂ z j + v j ∂ z¯ j ,

j=1

n

w=

∂

∂

∑ w j ∂ z j + w j ∂ z¯ j

j=1

(1.99)

40

1 Fundamentals

is given by n

h p (v, w) =

∑

n

hi, j (p) dzi (v)d z¯ j (w) =

i, j=1

∑

hi, j (p)vi w j .

i, j=1

Note that h p (v, w) = h p (w, v) and hence hi, j = h¯ j,i . The matrix H(p) = (hi, j (p)) is positive definite Hermitian at every point p ∈ M. The real and imaginary parts of a Hermitian inner product on a complex vector space are an ordinary inner product and an alternating quadratic bilinear form on the underlying real vector space. This carries over to Hermitian vector bundles. In particular, given a Hermitian metric h on the tangent bundle T M, we write h = g − i ω,

ω = −ℑh.

g = ℜh,

(1.100)

The real part g = ℜh is an R-bilinear form on each tangent space Tp M which is symmetric (g p (v, w) = g p (w, v) for each v, w ∈ Tp M) and positive definite since g(v, v) = h(v, v) > 0 for any 0 = v ∈ T M. Hence, g is a Riemannian metric on M (see Section 1.6). Since h is Hermitian, we have for all p ∈ M and v, w ∈ Tp M that h p (Jv, w) = i h p (v, w),

h p (v, Jw) = −i h p (v, w),

h p (Jv, Jw) = h p (v, w). (1.101)

In particular, h p (Jv, v) = −h p (v, Jv) = i h p (v, v) is imaginary. It follows that g p (Jv, Jw) = g p (v, w),

g p (v, Jv) = 0.

This means that the Riemannian metric g = ℜh is J-invariant and the pair of vectors (v, Jv) is g-orthogonal for every v ∈ T M. Note that for each 0 = v ∈ Tp M, spanR {v, Jv} is the complex line in Tp M spanned by the vector v. The form ω = −ℑh is clearly R-bilinear on each tangent space Tp M, and in view of h p (v, w) = h p (w, v) it is alternating: ω(v, w) = −ω(w, v). Hence, ω is a real differential 2-form on M, called the fundamental form of the Hermitian metric h. If z = (z1 , . . . , zn ) are local holomorphic coordinates on M and v, w are real tangent vectors as in (1.99), a calculation gives ω(v, w) =

i n hi, j (p)(vi w j − v j wi ). 2 i,∑ j=1

This shows that the expression for ω in the coordinates (z1 , . . . , zn ) is ω=

i n hi, j dzi ∧ d z¯ j . 2 i,∑ j=1

(1.102)

In particular, ω is a (1, 1)-form (see (1.62)). Furthermore, properties (1.101) imply ω(Jv, Jw) = ω(v, w),

ω(v, Jv) = h(v, v) > 0 if v = 0.

(1.103)

1.7 Hermitian and K¨ahler Manifolds

41

The last property means that ω is a J-positive (1, 1)-form. The metrics h and g = ℜh can be expressed in terms of the fundamental form ω by g(v, w) = ω(v, Jw),

h(v, w) = ω(v, Jw) − i ω(v, w).

(1.104)

Definition 1.7.1. A Hermitian metric h on a complex manifold (M, J) is a K¨ahler metric if its fundamental form ω = −ℑh is closed, dω = 0. A closed J-positive (1, 1)-form ω on (M, J)) is called a K¨ahler form. A complex manifold M is K¨ahlerian if it admits a K¨ahler metric. A K¨ahler manifold is a pair (M, h) (or (M, ω)) where h is a K¨ahler metric and ω = −ℑh is a K¨ahler form on M. Example 1.7.2. On Cn with coordinates (z1 , . . . , zn ) the standard Hermitian metric is n

hst =

∑ dz j ⊗ d z¯ j .

(1.105)

j=1

The corresponding Riemannian metric is the Euclidean metric n

gst = ℜhst =

∑ dx j ⊗ dx j + dy j ⊗ dy j

j=1

where z j = x j + iy j , and the associated fundamental form is ω st =

i 2

n

1

i

∑ dz j ∧ d z¯ j = 2 ∂ ∂ |z|2 = 4 dd c |z|2 .

j=1

(See (1.66) for the definition of dd c .) Since dω st = 0, hst is a K¨ahler metric. Elementary linear algebra shows that at any point p in a Hermitian manifold (M, h, ω) there are local holomorphic coordinates z = (z1 , . . . , zn ) such that n

hp =

∑ (dz j ) p ⊗ (d z¯ j ) p ,

j=1

ωp =

i 2

n

∑ (dz j ) p ∧ (d z¯ j ) p .

(1.106)

j=1

One can characterize K¨ahler manifolds as the Hermitian manifolds (M, h) with the property that there is a holomorphic coordinate chart around any given point p ∈ M in which the metric h agrees with the standard metric hst (1.105) to order 2 at p (see [356, Proposition 7.14]).   Example 1.7.3. Let M be a complex manifold with a strongly plurisubharmonic function ρ : M → R, i.e., dd c ρ > 0 (see (1.70)). Then, ω = 14 dd c ρ is a K¨ahler form on M and ρ is called a K¨ahler potential of ω. Conversely, if (M, ω) is a K¨ahler manifold then for every point p ∈ M there are a neighbourhood U ⊂ M of p and a function ρ : U → R such that ω|U = 14 dd c ρ (see [266, Proposition 8.8]); any such ρ is called a local K¨ahler potential of ω. There is no comparable way of describing a general Riemannian metric in terms of a single function.  

42

1 Fundamentals

Example 1.7.4 (Fubini–Study form and metric on CPn ). The Fubini–Study metric has already been mentioned in Example 1.6.5. Here is a more complete description. given in coordinates ζ = (ζ0 , ζ1 , . . . , ζn ) by Consider the (1, 1)-form on Cn+1 ∗  = dd c log |ζ |2 . ω  is invariant under dilations ζ → tζ (t ∈ C∗ ), it passes down to a (1, 1)Since ω n ∗ form ω on Cn+1 ∗ /C = CP , called the Fubini–Study form. In the affine coordinates n (z1 , . . . , zn ) on C = {ζ0 = 0}, with z j = ζ j /ζ0 , we have ω = dd c log(1 + |z|2 ) = 2i ∂ ∂ log(1 + |z|2 )   ∂ |z|2 ∧ ∂ |z|2 ∂ ∂ |z|2 − = 2i 1 + |z|2 (1 + |z|2 )2 = 2i

(1 + |z|2 ) ∑ni=1 dzi ∧ d z¯i − ∑ni, j=1 zi z¯ j dzi ∧ d z¯ j (1 + |z|2 )2

.

If v ∈ T M is represented by the complex vector (v1 , . . . , vn ) ∈ Cn as in (1.99) and we write z · v = ∑ni=1 zi vi , a calculation shows that   4 (1 + |z|2 )|v|2 − |z · v|2 2 2 (1 + |z| )

4 4|v|2 2 2 + |z v − z v | . = |v| ≥ i j j i ∑ (1 + |z|2 )2 (1 + |z|2 )2 i< j

ω, v ∧ Jv =

Thus, ω is a positive (1, 1)-form and hence determines a K¨ahler metric on CPn defined by (1.104). In spite of its appearance, ω is not an exact form on CPn . In dimension n = 1 we have ω = 2i

dz ∧ d z¯ 4 dx ∧ dy = . (1 + |z|2 )2 (1 + |z|2 )2

Note that dz ⊗ d z¯ = (dx + i dy) ⊗ (dx − i dy) = (dx ⊗ dx + dy ⊗ dy) − i(dx ⊗ dy − dy ⊗ dx). Since dx ∧ dy = dx ⊗ dy − dy ⊗ dx, the Hermitian metric corresponding to ω is h=

4 |dz|2 4 dz ⊗ d z¯ 2 = − iω = σCP 1 − iω. (1 + |z|2 )2 (1 + |z|2 )2

2 Compare with (1.84) and recall that the Fubini–Study Riemannian metric σCP 1 2 2 3 coincides with the spherical metric dsS2 (1.86) on the unit sphere S ⊂ R under  the stereographic projection S2 → CP1 given by (1.85). 

1.7 Hermitian and K¨ahler Manifolds

43

˜ is a Hermitian manifold and f : M → M  h)  is a holomorphic immersion, If (M, ∗ ˜  are, then (M, h = f h) is also a Hermitian manifold. Furthermore, if g˜ and ω ˜ then g = f ∗ g˜ respectively, the Riemannian metric and the fundamental form of h,  are the corresponding quantities associated to h. If h˜ is a K¨ahler metric and ω = f ∗ ω  = 0.  = f ∗ (d ω) then so is h = f ∗ h˜ since dω = d( f ∗ ω) It follows that every complex submanifold M of CPn is a K¨ahler manifold with the K¨ahler metric induced by the Fubini–Study metric. Likewise, any complex submanifold of Cn is a K¨ahler manifold with the K¨ahler metric induced by the standard metric on Cn (see Example 1.7.2). Wirtinger’s theorem. Let (M, h) be a Hermitian manifold, and write h = g − iω as in (1.100). The volume of a complex submanifold (or a complex subvariety) of M is then given by the integral of a suitable exterior power of the fundamental form ω = −ℑh over the submanifold. Nothing of this sort can be done in the Riemannian case. Let us briefly explain this. On a neighbourhood of any given point in M we can find by Gram–Schmidt an h-unitary coframe of (1, 0)-forms φi = αi + i βi (i = 1, . . . , n = dim M) such that n

n

n

i=1

i=1

i=1

h = ∑ φi ⊗ φ i = ∑ (αi ⊗ αi + βi ⊗ βi ) − i ∑ αi ∧ βi = g − i ω. The collection {(αi , βi )}i=1,...,n is then an orthonormal coframe with respect to the Riemannian metric g = ℜh, and by (1.93) the volume form associated to g equals Ωg = α1 ∧ β1 ∧ · · · ∧ αn ∧ βn . On the other hand, we have that ω = ∑ni=1 αi ∧ βi and hence n

  ω n = ω ∧ · · · ∧ ω = n! Ωg . The restrictions of h, g, and ω to any complex d-dimensional submanifold X of M are the corresponding quantities on X, and hence the volume element dVX on X determined by g equals  1 dVX = ω d X . d! This implies the following theorem due to W. Wirtinger (see e.g. [88, p. 159]). Theorem 1.7.5. If (M, h) is a Hermitian manifold with the Riemannian metric g = ℜh and the fundamental form ω = −ℑh, then the g-volume of any d-dimensional complex analytic subvariety X of M equals Vol2d (X) =

1 d!

 X

ω d ∈ (0, +∞].

(1.107)

Remark 1.7.6. If X is a complex subvariety of dimension d in a complex manifold M and α is a 2d-form on M whose support intersects X in a compact set, then the

44

1 Fundamentals





integral X α = Xreg α is well defined since the singular locus Xsing = X \ Xreg is a negligible set for integration. The integral can be nonvanishing only for (d, d)forms, and it vanishes on any irreducible component of X of dimension less than d. Furthermore, Stokes’s theorem (see Theorem 1.3.1) extends to complex varieties with sufficiently regular boundaries; see e.g. [88, p. 176].   Wirtinger’s theorem is especially interesting on a K¨ahler manifold (M, h) due to the fact that ω = −ℑh is a closed (1, 1)-form. If X and X  is a pair of homologous compact complex subvarieties of M then, since dω = 0, Stokes’s theorem implies Vol(X) = Vol(X  ). That is, homologous compact complex subvarieties of a K¨ahler manifold have equal volume. Example 1.7.7. The Fubini–Study area of a complex curve C ⊂ CPn equals 

Area(C) = C

ω CPn ,

where ω CPn is the Fubini–Study form (see Example 1.6.5). The restriction of ω CPn to any projective line CP1 ⊂ CPn is the Fubini–Study form on that line, and its Riemannian metric agrees with the spherical metric on S2 ∼ = CP1 . In particular, 

Area(CP1 ) =

CP1

ω CP1 = 4π = the area of the unit sphere in R3 .

Recall (see Example 1.5.4 and in particular (1.44)) that every closed complex curve C ⊂ CPn is homologous to d· L, where L ∈ H2 (CPn , Z) ∼ = Z is the homology class of the projective line and d = deg(C) ∈ N is the degree of C. This gives the following. Corollary 1.7.8. The Fubini–Study area of a closed complex curve C ⊂ CPn equals Area(C) = 4π deg(C).

(1.108)

Volume minimizing properties of complex submanifolds. Let (M, h, ω) be a Hermitian manifold of dimension n, and let d ∈ {1, . . . , n}. An inequality due to 1 d W. Wirtinger [346] shows that the restriction of the form d! ω to a smooth 2ddimensional submanifold X of M is the volume form on X if and only if X is a complex submanifold; in all other cases it gives a smaller value. Since this is a pointwise result and for any point p ∈ M the metric h p on Tp M is C-linearly equivalent to the standard Hermitian metric hst (1.105) on Cn , it suffices to explain this particular case. Given vectors v1 , . . . , v2d ∈ Cn , we denote by |v1 ∧ · · · ∧ v2d | the Euclidean volume of the parallelepiped spanned by them. Proposition 1.7.9 (Wirtinger’s inequality). Let ω = 2i ∑nj=1 dz j ∧ d z¯ j . For any 2dvector v = v1 ∧ · · · ∧ v2d on Cn we have that −|v| ≤

1 d ω (v) ≤ |v|, d!

with an equality if and only if v is a positive (resp. a negative) 2d-vector, i.e., v = ± v1 ∧ iv1 ∧ · · · ∧ vd ∧ ivd for some v1 , . . . , vd ∈ Cn .

1.7 Hermitian and K¨ahler Manifolds

45

The proof amounts to a simple computation, see e.g. [88, p. 161]. As pointed out there, the analogous result holds on any Hermitian manifold (M, h, ω). In the language of calibrated geometry (see R. Harvey and H. B. Lawson [186]), the form 1 d d! ω is a calibration for real 2d-dimensional submanifolds of M. The following is an immediate consequence of Proposition 1.7.9. Corollary 1.7.10. If X is an oriented 2d-dimensional submanifold of a Hermitian manifold (M, ω), then − Vol2d (X) ≤

1 d!

 X

ω d ≤ Vol2d (X).

(1.109)

If Vol2d (X) is finite then the right or the left equality holds if and only if X is a complex submanifold with the positive resp. the negative orientation. If X is a compact complex subvariety of dimension d in a K¨ahler manifold (M, h, ω) and X  is a compact smooth submanifold of M of real dimension 2d which is homologous to X but fails to be complex, then Vol2d (X) =

1 d!

 X

ωd =

1 d!

 X

ω d < Vol2d (X  ),

where the last inequality holds by Proposition 1.7.9. Corollary 1.7.11. If (M, h, ω) is a K¨ahler manifold then any closed complex subvariety of M is an absolute volume minimizer in its homology class. The analogous conclusion holds when X ⊂ M is a compact complex subvariety with boundary to which Stokes’s theorem applies (see [88, p. 176]) and the K¨ahler form ω is exact, ω = dα for some 1-form α. This holds in particular for the standard K¨ahler form ω st on Cn . Here is a more precise result (see [88, p. 180]). Corollary 1.7.12. Let (M, ω) be a Hermitian manifold such that ω d = dα is exact for some d ∈ {1, . . . , n}. If X ⊂ M is a compact complex d-dimensional subvariety with boundary bX and X  ⊂ M is a smooth, compact, oriented 2d-dimensional submanifold with boundary bX  = bX (in the sense of Stokes’s theorem), then Vol2d (X) ≤ Vol2d (X  ). If (M, ω) = (Cn , ω st ) then we have strict inequality unless X = X  in the sense of currents. Proof. By Stokes’s theorem and Wirtinger’s inequality we have that 1 Vol2d (X) = d!



1 ω = d! X d



1 α= d! bX



1 α=  d! bX

 X

ω d ≤ Vol2d (X  ).

If the equality holds then X  is a complex submanifold in view of Proposition 1.7.9. If M = Cn , a theorem of R. Harvey and H. B. Lawson [180] shows that a complex subvariety with a given boundary is unique in the sense of currents.  

46

1 Fundamentals

1.8 Conformal Maps and Isothermal Coordinates on Surfaces Let (M, g) be a Riemannian manifold. We can measure the angle θ between any pair of nonzero tangent vectors u, v ∈ Tp M according to the formula cos θ =

g p (u, v) g p (u, v)  = . |u|g · |v|g g p (u, u) · g p (v, v)

(1.110)

Riemannian metrics g, g˜ on M are said to be conformally equivalent if they determine the same angle between any pair of nonzero tangent vectors: g˜ p (u, v) g p (u, v) = , |u|g · |v|g |u|g˜ · |v|g˜

0 = u, v ∈ Tp M, p ∈ M.

(1.111)

Lemma 1.8.1. Riemannian metrics g, g˜ on M are conformally equivalent if and only if there is a positive function μ > 0 on M such that g˜ = μg.

(1.112)

Proof. It is obvious that (1.112) implies (1.111). To prove the converse, fix a point p ∈ M and choose a g p -orthonormal basis v1 , . . . , vn of Tp M. These vectors are then pairwise g˜ p -orthogonal by (1.111). Let  μi = |vi |g˜ = g˜ p (vi , vi ) > 0. We must show that μi = μ j for all i, j ∈ {1, . . . , n}. Fix i = j and apply √ the hypothesis (1.111) to the pair of vectors (vi , vi + v j ). Note that |vi + v j |g = 2, and hence g˜ p (vi , vi + v j ) g p (vi , vi + v j ) μ2 μi 1 1 √ =  . = =  i = 2 |vi |g˜ · |vi + v j |g˜ μ 2 2 2 |vi |g · |vi + v j |g (μ /μ j i j) + 1 μi μi + μ j √ Since the√function h(t) = t/ t 2 + 1 is strictly increasing for t ≥ 0 and assumes the  value 1/ 2 at t = 1, it follows that μi = μ j as claimed.  A conformal structure of class C k on a smooth manifold M is determined by a choice of a conformal class of C k Riemannian metrics, where a pair of metrics g, g˜ determine the same conformal class precisely when (1.112) holds.  g)  An immersion Let (M, ˜ be another Riemannian manifold with dim M ≤ dim M.  g) x : (M, g) → (M, ˜ is said to be conformal if it preserves angles, i.e.,   g˜x(p) dx p (u), dx p (v) g p (u, v) = (1.113) |dx p (u)|g˜ · |dx p (v)|g˜ |u|g · |v|g holds for any pair of nonzero vectors u, v ∈ Tp M, p ∈ M. Clearly, this is equivalent to saying that the pullback metric x∗ g˜ is conformally equivalent to g, i.e., x∗ g˜ = μ g

1.8 Conformal Maps and Isothermal Coordinates on Surfaces

47

 g) holds for a positive function μ > 0 on M. Riemannian manifolds (M, g) and (M, ˜  are conformally equivalent if there is a conformal diffeomorphism x : M → M. Suppose now that M is a surface. Given a pair of conformally equivalent Riemannian metrics g and g˜ = μg of class C 1 on M, we see from (1.96) that their respective Laplacians satisfy (1.114) Δg = μ Δg˜ . Corollary 1.8.2. Conformally equivalent metrics on a surface determine the same harmonic functions. Example 1.8.3. It is classical that an immersion (x, y) : D → R2 on a domain D ⊂ R2(u,v) is conformal with respect to the Euclidean metric on R2 if and only if the map z = x + iy is holomorphic or antiholomorphic with respect to the complex variable ζ = u + iv on D, depending on whether it preserves or reverses the orientation.   The following observations will be used in the next chapter. Lemma 1.8.4. Given a domain D ⊂ R2 with coordinates (u, v) and an immersion x = (x1 , . . . , xn ) : D → Rn for some n ≥ 2, the following conditions are equivalent. (a) The immersion x is conformal in the Euclidean metrics on R2 and Rn . (b) The partial derivatives ∂∂ ux = xu = (x1,u , . . . , xn,u ) and ∂∂ xv = xv = (x1,v , . . . , xn,v ) have the same Euclidean length and are orthogonal to each other: |xu | = |xv | > 0,

xu · xv = 0.

(1.115)

(c) The vectors xu ± i xv ∈ Cn∗ are null vectors, i.e., they belong to the null quadric   A = An−1 = z = (z1 , z2 , . . . , zn ) ∈ Cn : z21 + z22 + · · · + z2n = 0 . (1.116) (d) The matrix of the Riemannian metric g = x∗ (ds2 ) on D (see (1.79)) equals G = (gi, j )2i, j=1 = μI,

(1.117)

where I is the identity matrix and μ = |xu |2 = |xv |2 > 0 on D. Proof. Condition (a) is equivalent to the metric g = x∗ (ds2Rn ) on D being conformally equivalent to the Euclidean metric ds2R2 , which by Lemma 1.8.1 means that g = μds2R2 for some positive function μ on D. In view of the formulas (1.78), (1.79) for the metric g, the latter condition is equivalent to (b) with μ = |xu |2 = |xv |2 . The equivalence (b) ⇔ (c) is seen from the identity n

∑ (xi,u ± i xi,v )2 = |xu |2 − |xv |2 ± 2 i xu · xv ,

i=1

and (b) ⇔ (d) is obvious.  

48

1 Fundamentals

We are interested in the simplest possible local form of a Riemannian metric up to conformal equivalence. There is an optimal answer to this question in dimension 2 which says that every sufficiently regular Riemannian metric is locally conformally equivalent to the Euclidean metric ds2 on R2 (see Theorem 1.8.6). We begin by introducing the relevant notion of an isothermal coordinate. Definition 1.8.5. A local chart φ = (x, y) : U → φ (U) ⊂ R2 on a Riemannian surface (M, g) is isothermal for the Riemannian metric g if g|U = μ(dx2 + dy2 ) = μ · φ ∗ ds2R2 holds for some positive function μ > 0 on U. Clearly, a chart φ is isothermal if and only if φ is a conformal diffeomorphism with respect to the pair of metrics g on U and ds2 on R2 . Note that a metric of the form μ|dz|2 with μ > 0 is a K¨ahler metric (see Definition 1.7.1). Theorem 1.8.6. A Riemannian metric g of H¨older class C k,α (k ∈ Z+ , 0 < α < 1) on a surface M admits an isothermal chart of class C k+1,α at any given point of M. Theorem 1.8.6 is a special case of [274, Theorem III] due to A. Nijenhuis and W. B. Woolf from 1963. This result has a long history going back to C. F. Gauss (1822) who proved the existence of isothermal coordinates on any surface with a real analytic metric, following an earlier result of J.-L. Lagrange (1779) for surfaces of revolution. Results for H¨older continuous metrics were obtained by A. Korn (1916) and L. Lichtenstein (1916). Subsequent accounts are due to C. B. Morrey (1938 and 1966, see [267, p. 366]), L. Bers (1952), L. Ahlfors (1955), and S. S. Chern (1955). A particularly simple proof for metrics of class C k,α (k ∈ N, 0 < α ≤ 1) was given by D. M. DeTurck and J. L. Kazdan [107] (1981) via the existence of harmonic coordinates; we recall it below. In the next section we outline another proof using the Beltrami equation which works for metrics of class C α with 0 < α ≤ 1. Proof. We give the proof of Theorem 1.8.6, due to DeTurck and Kazdan [107], for a metric of class C k,α with k ∈ N and 0 < α ≤ 1. Since the result is local, we may assume that M is a domain in R2 . The first part of the argument applies to any Riemannian metric g of class C k,α (k ∈ N, 0 < α ≤ 1) on a domain D ⊂ Rn (n ≥ 2), given in coordinates x = (x1 , . . . , xn ) by g = ∑ni, j=1 gi, j dxi ⊗ dx j . Set (gi, j ) = (gi, j )−1 . By standard results on elliptic partial differential equations, in a neighbourhood of any point p ∈ D there is a solution u ∈ C k+1,α of Δ u = 0 with prescribed value u(p) and differential du p , where Δ is the g-Laplacian (1.96) . Let y j (x) for j = 1, . . . , n be a solution ∂y

satisfying y j (p) = 0 and ∂ xij (p) = δi, j for i = 1, . . . , n. Then, y = (y1 , . . . , yn ) is a local coordinate system consisting of g-harmonic functions. Assume now that n = 2. Pick any harmonic solution u(x) with nonvanishing differential at p and choose v as the harmonic conjugate of u given by dv = ∗ du, where ∗ is the Hodge operator for the metric g. It is then easily verified that (u, v) is a local isothermal g-harmonic coordinate system near p ∈ D.  

1.9 The Beltrami Equation and J-Holomorphic Discs

49

Theorem 1.8.6 shows that a Riemannian surface (M, g), with g of some H¨older class C α (α > 0), carries an atlas U = {(Ui , φi )} consisting of isothermal charts for g. Note that the transition map φi, j = φi ◦ φ j−1 : φ j (Ui, j ) → φi (Ui, j ) between any pair of isothermal charts on (M, g) is a conformal map with respect to the Euclidean metric on R2 ∼ = C, hence it is a holomorphic or an antiholomorphic diffeomorphism (see Example 1.8.3). An atlas U with this property is called a conformal atlas. An oriented conformal atlas on a surface is the same thing as a complex atlas, and it determines on M the structure of a Riemann surface. Conversely, if U = {(Ui , φi )}i∈I is a conformal atlas on M, there is a Riemannian metric g on M such that every chart (Ui , φi ) is isothermal for g. Indeed, pick a locally finite refinement {V j } j∈J of the cover {Ui }i∈I with V j ⊂ Ui( j) for each j ∈ J. Let {χ j } j∈J be a smooth partition of unity on M with supp χ j ⊂ V j for every j. Then the metric g = ∑ j χ j φi(∗ j) (ds2 ) on M has the stated property. This gives: Corollary 1.8.7. Given a Riemannian metric g of some H¨older class on a smooth surface M, there exists a maximal conformal atlas on M (oriented if M is such) consisting of isothermal charts. Conversely, every conformal atlas on M is the atlas of isothermal charts for a Riemannian metric. Two Riemannian metrics determine the same maximal conformal atlas if and only if they are conformally equivalent. Hence, a conformal structure on a surface is determined by a choice of a conformal class of Riemannian metrics or, equivalently, by a choice of a conformal atlas. A surface endowed with a conformal structure is called a conformal surface. Since every topological surface admits a smooth structure and every smooth surface carries a smooth Riemannian metric, we have the following corollary. Corollary 1.8.8. Every surface carries a maximal conformal atlas. Every orientable surface carries a maximal oriented conformal atlas, that is, a complex atlas.

1.9 The Beltrami Equation and J-Holomorphic Discs In this section we present another proof of Theorem 1.8.6 on the existence of isothermal coordinates for a Riemannian metric on a surface. Its advantage is that it applies to metrics which are merely H¨older continuous. A Riemannian metric g on an oriented surface M determines a real endomorphism J : T M → T M of the tangent bundle of M by the condition that for any tangent vector 0 = ξ ∈ Tp M, the vector Jξ ∈ Tp M has the same g-length as ξ , it is g-orthogonal to ξ , and the pair (ξ , Jξ ) is a positive basis of the tangent space Tp M. These conditions uniquely determine J. Note that J 2 = −IdT M .

(1.118)

50

1 Fundamentals

An endomorphism J of the tangent bundle satisfying (1.118) is called an almost complex structure on M. A calculation gives the following local expression for J (in the positive orientation class) in terms of the metric g. Lemma 1.9.1. Assume that a Riemannian metric g on a domain in R2(x,y) is given by g = g1,1 dx2 + g1,2 dx dy + g2,1 dy dx + g2,2 dy2 , (1.119) √ √ with g1,2 = g2,1 . Set δ = g1,1 g2,2 − g21,2 > 0, b = g1,2 / δ , c = g2,2 / δ > 0. The associated almost complex structure J is then represented by the matrix function     −b −c 1 −g1,2 −g2,2 √ = b2 +1 . (1.120) g1,1 g1,2 b δ c In particular, J is of the same smoothness class as the metric g. By Lemma 1.8.1, Riemannian metrics g, g˜ are conformally equivalent if g˜ = μg for some function μ > 0; clearly, such metrics determine the same almost complex structure J. Conversely, a choice of J determines a conformal class of Riemannian metrics. The standard almost complex structure on R2(x,y) (cf. (1.52)) is given by Jst

∂ ∂ = , ∂x ∂y

Jst

∂ ∂ =− , ∂y ∂x

and it corresponds to the conformal class of the Euclidean metric ds2 = dx2 +dy2 on R2 ∼ R2 with the standard orientation. Identifying = C via (x, y) → x + iy, the operator √ Jst equals the multiplication by i = −1 on each tangent space Tp C ∼ = C. To find local isothermal coordinates, we associate to the metric g (1.119) the almost complex structure J (1.120) and look for a local diffeomorphism (u, v) = φ (x, y) = (φ1 (x, y), φ2 (x, y)) satisfying the equation dφ ◦ J = Jst ◦ dφ = i dφ ,

(1.121)

where in the last equality we use the complex notation φ = φ1 + iφ2 . A solution φ of (1.121) is said to be a (J, Jst )-holomorphic map. Note that a (Jst , Jst )-holomorphic map is a holomorphic map in the usual sense since in this case (1.121) is the standard Cauchy–Riemann system. The inverse map ψ = φ −1 then satisfies the equation dψ ◦ Jst = J(ψ) ◦ dψ,

(1.122)

i.e., it is (Jst , J)-holomorphic. (Here, J(ψ) denotes the operator J at the image point under ψ.) Applying (1.122) to the vector field ∂∂u gives ψv = dψ

∂ ∂ ∂ = dψ ◦ Jst = J(ψ) dψ = J(ψ) ψu . ∂v ∂u ∂u

1.9 The Beltrami Equation and J-Holomorphic Discs

51

By the definition of J, this means that the vectors ψu , ψv ∈ Tψ(u,v) R2 have the same g-length and are g-orthogonal, so the induced metric ψ ∗ (g) is of the form ψ ∗ (g) = (φ −1 )∗ (g) = μ(du2 + dv2 ) for a positive function μ > 0 (cf. (1.117)). Hence, if φ satisfies (1.121) or, equivalently, if ψ = φ −1 satisfies (1.122) then φ is an isothermal chart for g. Both equations (1.121), (1.122) have been studied extensively. Let us begin by showing how (1.121) reduces to the classical Beltrami equation in the theory of quasiconformal maps. Let z = x + iy be a complex coordinate on C. Assuming that J is given by the matrix in (1.120), we have that J

∂ ∂ ∂ = −c(x, y) + b(x, y) . ∂y ∂x ∂y

Write φ = φ1 + iφ2 . Applying the equation (1.121) to the vector field

∂ ∂y

gives

−cφx + bφy = iφy . Recall that φx = φz + φz¯ ,

φy = i(φz − φz¯ ).

Inserting these expressions into the previous equation shows that φ must satisfy the following Beltrami equation with the Beltrami coefficient μ: φz¯ = μφz ,

μ=

1 − c + ib . 1 + c + ib

(1.123)

Since b, c are real and c > 0, we have that |μ| < 1, and hence by continuity sup |μ(z)| < 1

(1.124)

z∈K

on any compact subset K in the domain of J. A homeomorphism φ satisfying the Beltrami equation (1.123) subject to the condition (1.124) is said to be quasiconformal. The Beltrami coefficient μ(z) is related to the distortion of φ (z), and the condition (1.124) is equivalent to the distortion being uniformly bounded. For the Beltrami equation and quasiconformal maps we refer to the monographs by L. Ahlfors [5], L. Ahlfors and L. Bers [6], K. Astala, T. Iwaniec and G. Martin [54], and O. Lehto and K. I. Virtanen [222], among others. We briefly recall the main result. Let μ : D → C be a measurable complex-valued function satisfying (1.124), with supremum replaced by the essential supremum. Assuming that the boundary bD has measure zero, we can extend the equation to the entire plane C by setting μ = 0 on C \ D. Then, the Beltrami equation (1.123) has a unique solution φ = φμ on C normalized by the conditions φ (0) = 0, φ (1) = 1, φ (∞) = ∞; this solution is a quasiconformal homeomorphism C → C which has generalized derivatives locally in L2 satisfying (1.123) almost everywhere. The general elliptic theory shows that φ is of class C r+1 on any domain on which the Beltrami coefficient μ is of class

52

1 Fundamentals

C r . This also holds if r is fractional. In particular, if the metric g and hence the almost complex structure operator J are H¨older continuous of order 0 < α < 1, then μ ∈ C α and the solution to (1.123) is of class C 1,α . (See for example [54, Theorem 5.3.4].) This gives a proof of Theorem 1.8.6 under the weakest regularity assumption g ∈ C α for the Riemannian metric. Replacing a domain in C by a smooth almost complex manifold (X, J) of higher dimension as the source, the equation (1.121) need not have any nontrivial local solutions. In other words, an almost complex manifold of dimension n > 1 need not admit any nonconstant local holomorphic functions. By a seminal result of A. Newlander and L. Nirenberg [273] from 1957, the existence of a local holomorphic chart at each point of (X, J) is equivalent to the vanishing of the Nijenhuis tensor. An almost complex structure J with this property is called integrable, and (X, J) is then an honest complex manifold. By A. Nijenhuis and W. B. Woolf [274, Theorem II] (1963) the same holds if J is of H¨older class C 1,α for some α > 0. On the other hand, the equation (1.122) for the inverse map ψ = φ −1 admits plenty of solutions when D is a disc in C and the codomain is an arbitrary almost complex manifold (X, J) of any dimension, with J of class C k,α for some k ∈ Z+ and 0 < α < 1; see Nijenhuis and Woolf [274, Theorem III]. These solutions, which are of class C k+1,α (one better than J), are called pseudoholomorphic curves, or J-holomorphic curves in the almost complex manifold (X, J). Almost complex analysis became popular after M. Gromov used it in his seminal work [169] from 1985 to establish the existence of global symplectic invariants. These techniques have found a plethora of applications in topology and geometry. In particular, pseudoholomorphic curves are used to define Floer homology and to distinguish smooth structures on manifolds. For more information we refer the reader to the surveys by J.-C. Sikorav [318] and D. McDuff and D. Salamon [247].

1.10 Riemann Surfaces A Riemann surface is a 1-dimensional complex manifold. More precisely, a Riemann surface structure on an oriented topological surface M without boundary is determined by a choice of a complex atlas U = {(U j , φ j )} on M, i.e., an atlas with biholomorphic transition maps φi, j = φi ◦ φ j−1 : φ j (Ui, j ) → φi (Ui, j ) ⊂ C. We have seen in Section 1.8 that a complex atlas on a surface is the same thing as an oriented conformal atlas. By Corollary 1.8.8, every orientable surface carries the structure of a Riemann surface. Another way of defining a Riemann surface structure on an oriented surface M is by choosing a smooth endomorphism J ∈ EndR T M with J 2 = −Id (an almost complex structure); every such structure J is integrable (see Section 1.9).

1.10 Riemann Surfaces

53

Since every Riemannian surface (M, g) admits local isothermal coordinates at any point of M (see Theorem 1.8.6) and g is a K¨ahler metric in such coordinates (see Definitions 1.7.1 and 1.8.5), we also infer the following. Corollary 1.10.1. Every oriented Riemannian surface (M, g) has a unique structure of a Riemann surface (up to biholomorphisms) in which g is a K¨ahler metric. Riemann surfaces, named after Bernhard Riemann, implicitly appeared already in the work of Carl F. Gauss who studied smooth surfaces conformally embedded  g) in a Euclidean space. Assume that M is a smooth oriented surface and (M, ˜ is a  let g = x∗ g˜ smooth Riemannian manifold. Given a smooth immersion x : M → M, be the induced metric on M. By Corollary 1.8.7 there exists an oriented conformal atlas on M consisting of g-isothermal charts. This endows M with the structure of a Riemann surface such that x is a conformal immersion. In particular, every oriented smoothly immersed surface M ⊂ Rn is a conformally immersed Riemann surface. Conversely, a theorem of R. R¨uedy [306] from 1971 says that every Riemann surface (either open or closed) admits a proper conformal embedding into R3 . It was B. Riemann who in his dissertation from 1851 first considered surfaces more general than planar domains as domains of holomorphic functions. The Riemann surface of a holomorphic function is a surface M lying over C (i.e., endowed with a local homeomorphism π : M → C), together with a function f : M → C such that for any open subset U ⊂ M for which π|U : U → π(U) ⊂ C is a homeomorphism, the function f ◦ (π|U )−1 : π(U) → C is holomorphic, and M is maximal in the sense that f does not extend to a holomorphic function on any larger surface over C. Any such surface M is open (noncompact). Conversely, R. Gunning and R. Narasimhan [175] proved in 1967 that every open Riemann surface M arises in this way (see Theorem 1.10.5). Riemann’s ideas were subsequently developed by many mathematicians, and Riemann surfaces started being considered as complex or algebraic curves in complex manifolds. For a history of the subject see the book by J. Stillwell [323]. Riemann surfaces arise naturally in diverse geometric contexts, which is one of the reasons for their versatility and ubiquity in mathematics, physics (in particular in string theory), and other fields of science. Indeed, it is fair to say that complex curves are everywhere. In particular, every regular minimal surface in Rn is parameterized by a conformal harmonic immersion from a Riemann surface (cf. Theorem 2.3.1 and Sect. 2.4). Most oriented surfaces admit many nonequivalent complex structures. Exceptions are some of the simplest surfaces one can think of, such as the plane C, the disc D = {z ∈ C : |z| < 1}, the Riemann sphere CP1 = C ∪ {∞} = C, and a few others. The three surfaces just named are the only simply connected Riemann surfaces up to biholomorphisms according to the Riemann–Koebe uniformization theorem from 1907 (see S. Donaldson [112] for discussion and references). All other Riemann surfaces arise as nonbranched holomorphic quotients of one of these special surfaces, where CP1 has no nontrivial quotients while the quotients of C are C∗ = C \ {0}

54

1 Fundamentals

and complex tori. All other Riemann surfaces are holomorphic quotients of the disc D. Classification of Riemann surface structures on an oriented topological surface, up to biholomorphisms in the connected component of the identity, is the subject of the theory of Teichm¨uller spaces; see O. Lehto [221]. In this and the following section we take a leisurely walk through some of the most important aspects of the theory of Riemann surfaces that will be used in the book. Details and references can be found in the monographs mentioned in the introduction to this chapter, and we provide additional ones along the way. Holomorphic and harmonic functions on Riemann surfaces. Basic differential operators on complex manifolds have already been introduced in Section 1.5. We recall them briefly in the simpler context of Riemann surfaces, also providing their expressions in any local holomorphic coordinate ζ = u + iv. The differential d splits into the sum (1.63) of C-linear and C-antilinear parts: d=

∂ ∂ ∂ ∂ du + dv = ∂ + ∂ = dζ + ¯ d ζ¯ . ∂u ∂v ∂ζ ∂ζ

(1.125)

A differentiable function f = x + iy : M → C is holomorphic if and only if it satisfies the Cauchy–Riemann equation in every local holomorphic coordinate: ∂ f = 0 ⇐⇒ xu = yv , xv = −yu . The algebra O(M) of all holomorphic functions on M is endowed with the compactopen topology. Note that O(M) = C if M is compact, connected and without boundary. The conjugate differential is defined as d c = 2ℑ(∂ ) = i(∂ − ∂ ) = −

∂ ∂ du + dv. ∂v ∂u

(1.126)

Equivalently, (d c f )(ξ ) = d f (−Jξ ),

ξ ∈ T M,

(1.127)

where J is the complex structure operator on M. We have that 2∂ = d + i d c ,

2∂ = d − i d c .

The Laplacian is given by (1.67):  2  ∂ ∂2 + dd c = 2i∂ ∂ = du ∧ dv = Δ · du ∧ dv. ∂ u2 ∂ v2

(1.128)

(1.129)

Here, Δ is the Laplacian in the Euclidean metric, which differs from the metric Laplacian (1.96) by the positive factor μ > 0 from the equation g = μ(du2 + dv2 ) in an isothermal coordinate u + iv. The Laplacian is a real operator, and it extends to complex-valued functions by dd c (x + iy) = dd c x + i dd c y. A function x ∈ C 2 (M) is harmonic if dd c x = 0. From (1.129) we infer that

1.10 Riemann Surfaces

55

x is harmonic ⇐⇒ dd c x = 0 ⇐⇒ ∂ ∂ x = 0 ⇐⇒ Δ x = 0.

(1.130)

In particular, any holomorphic function is harmonic. Since dd c is a real operator, the real and the imaginary components of a holomorphic function are harmonic functions. Conversely, a real function x ∈ C 2 (M) is harmonic (d(d c x) = 0) if and only if d c x is a closed 1-form on M. If this holds and M is simply connected or, more generally, if d c x is exact then by fixing a point p0 ∈ M and setting  p

y(p) = const +

p∈M

d c x,

p0

we get a function of class C 2 (M) satisfying dy = d c x. Hence, by (1.127) we have that dx(−Jξ ) = d c x(ξ ) = dy(ξ ) for every tangent vector ξ . Replacing ξ by −Jξ in this identity gives dx = −d c y. It then follows from (1.128) that 2∂ (x + iy) = (d − i d c )(x + iy) = (dx + d c y) + i(dy − d c x) = 0, so z = x + iy is a holomorphic function. Thus, the restriction of a harmonic function x : M → R to any simply connected domain U ⊂ M is the real part of a holomorphic function x + iy on U; we say that y is a harmonic conjugate of x. Meromorphic functions and 1-forms. Assume that M is a connected Riemann surface. We denote by M (M) the field of meromorphic functions on M, and by M∗ (M) = M (M) \ {0} the multiplicative abelian group of nonzero meromorphic functions. Note that M (M) ∪ {∞} coincides with the space of holomorphic maps M → CP1 = C ∪ {∞} into the Riemann sphere. A holomorphic or meromorphic 1-form on a Riemann surface M is a holomorphic or meromorphic section of the holomorphic cotangent bundle T ∗ M ∼ = T ∗1,0 M, also called the canonical bundle of M and denoted KM . In any local holomorphic coordinate ζ on M we have α = f dζ , where f = f (ζ ) is a holomorphic or a meromorphic function. We denote by M 1 (M) the vector space of meromorphic 1-forms, and M∗1 (M) = M 1 (M) \ {0}. Note that M 1 (M) is an M (M)-module. By Ω (M) = Ω 1 (M) we denote the complex vector subspace of M 1 (M) consisting of holomorphic 1-forms on M. If a meromorphic 1-form α = f dζ has a pole at a point p ∈ M then, choosing a compact domain D ⊂ M with smooth boundary bD such that p ∈ D˚ and α is holomorphic on D \ {p}, the residue of α at p equals 1 Res p α = 2πi

 bD

α.

(1.131)

Fundamentals on compact Riemann surfaces. Recall from Theorem 1.2.2 that every connected, compact, oriented, topological surface M without boundary is g homeomorphic either to the sphere S2 or to the connected sum T2 # · · · # T2 of g tori for a unique g ∈ N. This number g = gen(M) is the genus of M, where gen(S2 ) = 0. The Euler number of M equals χ(M) = 2 − 2gen(M) (cf. (1.9)).

56

1 Fundamentals

A nonconstant holomorphic map f : M → N between compact Riemann surfaces has at most finitely many critical points x1 , . . . , xm ∈ M, i.e., points at which the differential of f vanishes. In local coordinates z on M and w on N, centred at a point x0 ∈ M and its image point f (x0 ) ∈ N, respectively, f is of the form w = czk + O(zk+1 ) for some k ≥ 1 and c = 0. The integer k = ord f (x0 ) ∈ N is the order of f at x0 , and x0 is a critical point of f if and only if ord f (x0 ) > 1. By the argument principle for holomorphic functions it is easily seen that the number deg( f ) =

∑

ord f (x) ∈ N

(1.132)

f (x)=y

does not depend on the point y ∈ N. This number, called the degree of f , is the number of solutions of the equation f (x) = y counted with multiplicities. (This is in line with the notion of degree of a projective variety, see Remark 1.5.5.) Note that deg( f ) = 1 if and only if f is biholomorphic. When N = CP1 , this says in particular that a nonconstant meromorphic function f on a compact Riemann surface M has as many zeros as poles counted with multiplicities. The integer   b( f ) = ∑ ord f (x) − 1 ∈ Z+ (1.133) x∈M

is the branching number of the map f : M → N. We have the Riemann–Hurwitz formula: χ(M) = deg( f )χ(N) − b( f ). (1.134) (When N = CP1 , we have χ(N) = 2 and the formula reads χ(M) = 2deg( f )−b( f ).) This is seen by triangulating N so that every critical value of f is a vertex and all triangles are small enough so that they lift by f to triangles in M. Note that f is a covering map of degree deg( f ) over the complement of its finitely many critical values in N. Every edge and every cell in such a triangulation of N lifts to precisely deg( f ) distinct edges and cells in M, while the number of vertices in M is deg( f ) times the number of vertices in N minus b( f ), the loss of vertices occurring at the critical points of f . Adding up the number of vertices and triangles and subtracting the number of edges yields the formula (1.134). More precise information on the possible location and orders of zeros and poles of a holomorphic map M → CP1 is given by the Riemann–Roch theorem presented in Section 1.11. It shows that every compact Riemann surface M admits plenty of holomorphic maps M → CP1 of sufficiently large degree, depending on the genus of M. This and the Serre duality theorem (Theorem 1.11.7) are of key importance. Note that any collection f0 , f1 , . . . , fn ∈ M (M) of meromorphic functions, with at least one of them not identically zero, defines a holomorphic map M → CPn by M x −→ [ f0 (x) : f1 (x) : · · · : fn (x)] ∈ CPn . Any common zeros or poles of the fi ’s locally cancel out, so the map is well defined. By using also the general position theorem, one obtains the following classical embedding/immersion theorem; see P. Griffiths and J. Harris [167].

1.10 Riemann Surfaces

57

Theorem 1.10.2. Every compact Riemann surface admits a holomorphic embedding into CP3 and a holomorphic immersion into CP2 . R. Bryant [73] also showed that every compact Riemann surface embeds into CP3 as a holomorphic Legendrian curve. Most compact Riemann surfaces do not embed holomorphically into CP2 . An obstruction is shown by the following genus formula which holds for every smoothly embedded compact complex curve C ⊂ CP2 of degree d ∈ N: gen(C) =

(d − 1)(d − 2) . 2

(1.135)

(Recall that the degree of a complex curve in CPn equals the degree of its projection onto a generic projective line in CPn ; see Example 1.5.4.) Hence, a compact Riemann surface M whose genus is not of the form (d − 1)(d − 2)/2 with d ∈ N (e.g., gen(M) ∈ {2, 4, 5, 7, 8, 9, . . .}) does not embed holomorphically into CP2 . Open Riemann surfaces. A major development of function theory on open Riemann surfaces was made by H. Behnke and K. Stein [58] in 1949. Their paper was a prelude to Stein’s introduction in 1951 [322] of the class of manifolds which now bear his name. Of key importance in the theory of open Riemann surfaces are holomorphic approximations and interpolation theorems, extending the classical theorems of Runge and Weierstrass on the complex plane. This was done in [58] by considering an exhaustion of the surface M by an increasing sequence of smoothly bounded domains D  M. Attaching to D a disc along each of its finitely many boundary curves yields a compact Riemann surface. Applying the previously developed theory of compact Riemann surfaces, Behnke and Stein constructed Cauchy-type kernels on D and used them to prove approximation theorems for holomorphic functions of Runge subdomains of D, and then by induction on all of M. Some of these developments are indicated in Section 1.12. Here we survey a few of the highlights of the theory which were obtained by this approach. The following analogue of Theorem 1.10.2 is a special case of the embedding theorem for Stein manifolds; see [140, Theorem 2.4.1] and the references therein. Theorem 1.10.3. Every open Riemann surface admits a proper holomorphic embedding into C3 and a proper holomorphic immersion with simple double points into C2 . The question whether every open Riemann surface embeds as a smooth complex curve in the complex Euclidean plane C2 remains one of the most difficult open problems of complex analysis, known as the Forster–Bell–Narasimhan Conjecture [129, 59]. This conjecture concerns the existence of proper holomorphic embeddings into C2 , but even the existence of nonproper embeddings is an open problem. It is known that if C is a not necessarily proper embedded complex curve in C2 , then any smoothly bounded relatively compact domain in C admits a proper holomorphic embedding into C2 (see F. Forstneriˇc and E. F. Wold [147]). For a recent survey of this topic we refer to [140, Sects. 9.10–9.11]. This

58

1 Fundamentals

subject is discussed more carefully in Section 3.10 in connection to the related conjecture concerning the existence of proper conformal minimal embeddings of open Riemann surfaces into the Euclidean 4-space; see Conjecture 3.10.6. Recall that a continuous function ρ : M → [0, +∞) is an exhaustion function on M if for every c ∈ R+ the set {x ∈ M : ρ(x) ≤ c} is compact. A smooth function ρ is a Morse function if all its critical points are nondegenerate (see Definition 1.4.5). From Theorem 1.10.3 and Proposition 1.4.6 we infer the following corollary. Corollary 1.10.4. Every open Riemann surface admits a smooth strongly subharmonic Morse exhaustion function ρ : M → [0, +∞). Indeed, if f = ( f1 , f2 ) : M → C2 is a proper holomorphic immersion then for every a = (a1 , a2 ) ∈ C2 the function ρ(x) = | f1 (x)−a1 |2 +| f2 (x)−a2 |2 is a strongly subharmonic exhaustion function on M, and it is Morse for a generic choice of a. One can also construct such functions by using Runge’s approximation theorem. The following theorem of R. C. Gunning and R. Narasimhan [175] from 1967 says that every open Riemann surface is a Riemann domain spread over C. This result will be used frequently in the book. Theorem 1.10.5. Every open Riemann surface M admits a locally biholomorphic map M → C, and hence a nowhere vanishing exact holomorphic 1-form. Riemann surfaces of finite topological type. Topological invariants of a connected oriented surface M are its genus and the number of ends, either of which may be infinite. A surface M is said to have finite topological type if these numbers are finite; equivalently, if the homology group H1 (M, Z) = Zl is finitely generated. The following uniformization theorem is due to E. L. Stout [328, Theorem 8.1]. Theorem 1.10.6. Let S be an open Riemann surface with finitely generated first homology group H1 (S, Z). Then there exist a compact Riemann surface, R, and an analytic homeomorphism φ of S into R such that R \ φ (S) consists of finitely many closed discs with analytic boundaries together with finitely many isolated points. In other words, an open Riemann surface S with finitely generated first homology group is conformally equivalent to a domain of the form M = R\

m !

Di

(1.136)

i=1

in a compact Riemann surface R, where each D1 , . . . , Dm is either a point or a diffeomorphic image of the closed disc D = {z ∈ C : |z| ≤ 1} with real analytic boundary curve, and these sets are pairwise disjoint. It is easily seen that H1 (M, Z) ∼ = Zl ,

l = 2g + m − 1,

where g = gen(M) is the genus and m is the number of ends of M. Furthermore, we have the following well known result.

1.10 Riemann Surfaces

59

Lemma 1.10.7. On every connected open Riemann surface, M, of finite topological type there exist smooth Jordan curves C1 , . . . ,Cl representing a basis of H1 (M, Z) such that their union C = lj=1 C j has no holes in M (see Definition 1.2.1). An end (see p. 6) of a Riemann surface M which is conformally equivalent to the punctured disc D∗ = {z ∈ C : 0 < |z| < 1} is called a puncture of M. A surface M of the form (1.136) without punctures is bounded by finitely many Jordan curves; such M is called a bordered Riemann surface. Bordered Riemann surfaces and boundary regularity of conformal maps. Here is an intrinsic definition of a (compact) bordered conformal or Riemann surface.

Definition 1.10.8. Let S be a compact surface with nonempty boundary bS = m i=1 Γi consisting of finitely many Jordan curves. A conformal atlas on S is a topological atlas U = {(Ui , φi )} such that for every i, φi (Ui ) is an open subset of the upper halfplane H+ = {x+iy ∈ C : y ≥ 0} and the transition maps φi, j = φi ◦φ j−1 are conformal homeomorphisms. The pair (S, U ) is a compact bordered conformal surface, and a compact bordered Riemann surface if U is an oriented atlas. The interior S˚ = S \ bS is a bordered conformal surface, and a bordered Riemann surface if it is oriented. More precisely, a conformal or Riemann surface structure on a compact bordered surface is determined by an equivalence class of such atlases. The following observation is important for the uniformization theory. Assume that U is an oriented conformal atlas on S. Let (Ui , φi ), (U j , φ j ) ∈ U be a pair of charts around a boundary point p ∈ bS, with values in the upper halfplane H+ . Then, the points φi (p) and φ j (p) belong to bH+ = R = {x + iy : y = 0}, and each chart maps an arc I ⊂ bS around p onto an interval in R. By the Schwarz reflection principle, the transition map φi, j = φi ◦ φ j−1 extends across the interval φ j (I) ⊂ R to an injective holomorphic map on an open domain in C. These holomorphic extensions furnish a system of biholomorphic transition maps on an open Riemann  which contains S as a compact domain with real analytic boundary. surface, S, Furthermore, by gluing a disc onto S along every boundary curve in bS we obtain a compact Riemann surface R containing S (see Stout [328, Theorem 8.1]). This gives the following corollary, which is a special case of Theorem 1.10.6. Corollary 1.10.9. The interior S˚ = S \ bS of a compact bordered Riemann surface S is conformally equivalent to a domain M of the form (1.136) in a compact Riemann surface R such that each component Di of R \ M is a closed disc with real analytic boundary. Similarly, the interior of a nonorientable compact bordered conformal surface is conformally equivalent to a domain with real analytic boundary in a compact nonorientable conformal surface. A considerably more general uniformization theorem was proved by Z.-X. He and O. Schramm [188, Theorem 0.2] in 1993. Their result says that every open Riemann surface of finite genus and having at most countably many ends is conformally equivalent to a circle domain in a compact Riemann surface. The proof

60

1 Fundamentals

is substantially more involved than the proof of Corollary 1.10.9. Their result will be used in Section 7.4 where we study the Calabi–Yau problem for minimal surfaces. The following theorem summarizes the main results on boundary regularity of a conformal isomorphism between bordered conformal surfaces. We refer to G. M. Goluzin [160] for more complete information on this subject. Theorem 1.10.10. Let M and M  be compact bordered conformal surfaces with boundaries of class C k for some k ≥ 0, possibly noninteger. The following assertions hold for every conformal homeomorphism φ : M˚ → M˚  . (a) φ extends to a homeomorphism φ : M → M  . (C. Carath´eodory [76], 1913.) (b) If k = ∞ then the extension φ : M → M  is a diffeomorphism of class C ∞ (M). (P. Painlev´e [283, 284], 1887 and 1891.) (c) If k = m + α for some m ∈ Z+ and 0 < α < 1, then φ : M → M  is of the same class C k,α . (O. D. Kellogg [205], 1912, and S. E. Warschawski [338],1935.) Riemann surfaces of hyperbolic and parabolic type. The following classification of open Riemann surfaces has important implications in the theory of minimal surfaces; see H. M. Farkas and I. Kra [123, p. 179] and A. Grigor’yan [168]. Definition 1.10.11. A connected open Riemann surface is hyperbolic if it carries a nonconstant negative subharmonic function; otherwise it is parabolic. A connected complex manifold on which every negative plurisubharmonic function is constant is called a Liouville manifold; see [140, Definition 7.1.7(g)]. Thus, a Riemann surface is parabolic if and only if it is Liouville. The notion of hyperbolicity in Definition 1.10.11 differs from Kobayashi hyperbolicity; see S. Kobayashi [208] for the latter. For example, a compact Riemann surface with finitely many points removed is parabolic, the reason being that every bounded subharmonic function extends as a subharmonic function across a puncture and there are no nonconstant subharmonic functions on a compact Riemann surface due to the maximum principle. However, most such surfaces are Kobayashi hyperbolic, the simplest one being CP1 minus three points, which is Kobayashi hyperbolic by the classical Picard’s theorem.

1.11 Divisors and the Riemann–Roch Theorem We recall the notion of a divisor on a Riemann surface and the classical Riemann– Roch theorem. This will be used in Chapter 4 when considering minimal surfaces of finite total curvature. More complete expositions can be found in standard texts on the subject; see e.g. [7, 112, 123, 130, 333]. The spaces M (M), M∗ (M), M 1 (M), M∗1 (M), and Ω (M) have been introduced on p. 55. Definition 1.11.1. A divisor on a Riemann surface M is a function D : M → Z whose support supp(D) = {x ∈ M : D(x) = 0} is a closed discrete subset of M.

1.11 Divisors and the Riemann–Roch Theorem

61

Note that the support of a divisor on a compact Riemann surface is a finite set. It is customary to write D = ∑ D(x) · x = ∑ m j · x j , x∈M

j

where {x j } = supp(D) and m j = D(x j ) ∈ Z. The divisor with D(x) = 0 for all x ∈ M is called the zero divisor and denoted by 0. The set Div(M) of all divisors on M is an abelian group with the operation (D ± D )(x) = D(x) ± D (x) for x ∈ M. We have a partial ordering of divisors defined by D ≤ D ⇐⇒ D(x) ≤ D (x) for all x ∈ M. The divisors max{D, D } and min{D, D } are defined in an obvious way. A divisor D is called effective if D ≥ 0. Each divisor is the difference D = D − D of the effective divisors D = max{D, 0}, D = max{−D, 0}. The degree of the divisor D with finite support (which is always the case if M is compact) is the integer deg D =

∑ D(x) ∈ Z.

(1.137)

x∈M

Clearly, deg : Div(M) → Z is a group homomorphism. Example 1.11.2 (Principal divisors). A nonzero meromorphic function f ∈ M∗ (M) defines the divisor ( f ) whose support is the set of all zeros and poles of f , the value ( f )(x) at a zero x of f is the order of the zero, and the value at a pole is minus the order of the pole. Thus, if f has zeros at points x1 , . . . , xk of orders n1 , . . . , nk ∈ N and poles at points y1 , . . . , yl of orders m1 , . . . , ml ∈ N, then ( f ) = ∑ki=1 ni xi − ∑li=1 mi yi . A principal divisor is a divisor of the form ( f ) for some f ∈ M∗ (M). Every principal divisor on a compact Riemann surface has degree zero: deg( f ) = 0 for every f ∈ M∗ (M).

(1.138)

This says that f has as many zeros as poles when counting multiplicities; we have seen this in the discussion of the degree of a map (see (1.132)). On the other hand, every divisor on an open Riemann surface is a principal divisor; see Theorem 1.12.14.   Divisors D and D are said to be linearly equivalent, D ∼ D , if their difference D − D = ( f ) is a principal divisor. By (1.138), linearly equivalent divisors on a compact Riemann surface have the same degree. We denote by Div(M)/∼ the group of linear equivalence classes of divisors. Example 1.11.3 (Canonical divisors). A meromorphic 1-form ω ∈ M∗1 (M) determines a divisor (ω) ∈ Div(M) such that in any local holomorphic coordinate z on an open subset U ⊂ M, with ω = f dz for some f ∈ M∗ (U), we have (ω)(x) = ( f )(x) ∈ Z for all x ∈ U. For any pair of 1-forms ω, ω  ∈ M∗1 (M) their quotient ω/ω  = f ∈ M∗ (M) is a meromorphic function, so (ω) and (ω  ) are linearly equivalent and have the same degree if M is compact. This shows that for any nonzero

62

1 Fundamentals

1-form ω ∈ M∗1 (M), the map M (M) f → f ω ∈ M 1 (M) is a bijection. Any divisor of the form D = (ω) for some ω ∈ M∗1 (M) is called a canonical divisor, and the equivalence class of such divisors is the canonical class KM of M. By using the fact that every compact Riemann surface M admits a nonconstant holomorphic map f : M → CP1 (see Corollary 1.11.5), it is easily seen that deg KM = −χ(M) = 2gen(M) − 2.

(1.139)

Indeed, let deg( f ) and b( f ) denote, respectively, the degree (1.132) and the branching number (1.133) of f . We may assume that all critical values of f lie in C = CP1 \ {∞}. The 1-form α = dz has no zeros or poles on C. In the coordinate w = 1/z we have that dz = −dw/w2 , so α has a pole of order two at ∞ = {w = 0}. Thus, deg KCP1 = −2. Let ω = f ∗ α. On f −1 (C) ⊂ M we have ω = f ∗ (dz) = d f which has b( f ) zeros. Since ∞ is a regular value of f , f −1 (∞) consists of deg( f ) distinct points. As f ∗ α has a pole of order 2 at each of them, we infer that deg KM = deg( f ∗ α) = b( f ) − 2deg( f ) = −χ(M), where the last equality holds by the Riemann–Hurwitz formula (1.134).   Assume now that M is a compact Riemann surface. For any D ∈ Div(M) we denote by L(D) the complex vector subspace of M (M) defined by   L(D) = f ∈ M∗ (M) : ( f ) + D ≥ 0 ∪ {0}. (1.140) (The inclusion of the zero function is consistent it we treat its divisor as having the value +∞ at each point.) The space L(D) is finite-dimensional for every D. If D, D are linearly equivalent divisors with D = ( f0 ) + D for some f0 ∈ M∗ (M), then the ∼ = map f → f f0 for f ∈ M (M) induces an isomorphism L(D) → L(D ); in particular, dim L(D) = dim L(D ). For D = 0 we have L(0) = O(M) = C. Theorem 1.11.4 (The Riemann–Roch theorem). Let M be a compact Riemann surface with the canonical divisor class KM . For every divisor D on M, dim L(D) − dim L(KM − D) = 1 − gen(M) + deg D.

(1.141)

The formulas (1.138) and (1.139) give the following corollary. Corollary 1.11.5. Let D be a divisor on a compact Riemann surface M. (a) If deg D < 0 then L(D) = {0} and dim L(KM − D) = −1 + gen(M) − deg D. (b) If deg D > −χ(M) then L(KM − D) = {0} and dim L(D) = 1 − gen(M) + deg D. The corollary follows from the Riemann–Roch theorem as follows. If deg D < 0 then for every f ∈ M∗ (M) we have deg(( f ) + D) = deg( f ) + deg D = deg D < 0, so L(D) = {0} which proves (a). If deg D > −χ(M) then deg(KM − D) = deg(KM ) − deg D = −χ(M) − deg D < 0, so L(KM − D) = {0} by (a), which gives (b).

1.11 Divisors and the Riemann–Roch Theorem

63

When D = 0, we have L(0) = O(M) = C and deg D = 0, so (1.141) says that dim L(KM ) = gen(M). Choose a 1-form ω ∈ M∗1 (M); then (ω) represents KM . A function f ∈ M∗ (M) belongs to L((ω)) if and only if ( f ω) = ( f ) + (ω) ≥ 0, which means that f ω ∈ Ω (M) is a holomorphic 1-form. Thus, the Riemann–Roch formula (1.141) for the zero divisor D = 0 is equivalent to dim Ω (M) = gen(M).

(1.142)

This can be seen from the basic existence theory; see [123, Proposition III.2.7] which gives a more precise description of the basis of Ω (M) with respect to a canonical homology basis of H1 (M, Z). Similarly, the real vector space of real harmonic differentials on M has dimension 2gen(M); see [123, Corollary, p. 58]. This establishes the Riemann–Roch theorem for the zero divisor. The proof for any other divisor is obtained by a finite induction, changing the divisor by one point at a time and analysing what happens to the two dimensions on the left-hand side of the formula (1.141). A proof by sheaf-theoretic methods can be found in the book by O. Forster [130]. Another proof, using the existence of solutions to nonhomogeneous Laplace equations, is given in the book by S. Donaldson [112]. The Riemann–Roch theorem has an interpretation in terms of dimensions of spaces of sections of holomorphic line bundles. In order to explain this, we recall the correspondence between divisors and line bundles. Let π : E → M be a holomorphic line bundle (see Sect. 1.5). Let U = {(Ui , θi )}i∈I ∼ = with θi : E|Ui → Ui × C be a line bundle atlas on E with holomorphic transition maps θi, j (x, v) = (x, fi, j (x)v),

x ∈ Ui, j , v ∈ C.

A meromorphic section f of E is given by a collection of meromorphic functions fi ∈ M (Ui ) satisfying the compatibility conditions fi = fi, j f j on Ui, j (see (1.47)). Since fi, j ∈ O ∗ (Ui, j ), the divisors ( fi ) ∈ Div(Ui ) amalgamate into a divisor ( f ) ∈ Div(M). Its degree deg( f ) is called the Euler number or the Chern number of E, denoted deg(E) = c1 (E) ∈ Z. (If E → M is the fibre bundle with fibre CP1 , obtained by compactifying each fibre Ex ∼ = C ⊂ CP1 by adding the point at infinity, then deg(E) equals the difference of the intersection numbers of the graph of f with  The linear equivalence class the zero section E0 and the infinity section E∞ of E.) of the divisor ( f ) does not depend on f since the quotient of any two nontrivial meromorphic sections of E is a nonzero meromorphic function on M. Isomorphic line bundles give rise to linearly equivalent divisors and vice versa. It is a basic fact that every holomorphic line bundle E → M on a compact Riemann surface admits a nontrivial meromorphic section f , and hence the associated divisor ( f ) ∈ Div(M). This gives a group isomorphism ∼ =

Θ : Pic(M) −→ Div(M)/∼

(1.143)

from the Picard group Pic(M) = H 1 (M, O ∗ ) (1.48) of isomorphism classes of holomorphic line bundles on M onto the group Div(M)/∼ of linear equivalence

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classes of divisors. To see that Θ is surjective, let D = ∑ki=1 ni · xi ∈ Div(M) where supp(D) = {x1 , . . . , xk }. Choose pairwise disjoint coordinate discs xi ∈ Ui ⊂ M and functions fi ∈ M (Ui ) whose only zero or pole is at xi and ( fi )(xi ) = ni . Let U0 = M \ P and f0 = 1. The collection ( fi )ki=0 is then a meromorphic section f of the line bundle E → M given by the 1-cocycle fi, j = fi / f j ∈ O ∗ (Ui, j ), and ( f ) = D. Fix a nontrivial meromorphic section h of E, so E corresponds to the divisor D = (h) ∈ Div(M) under the isomorphism (1.143). Given f ∈ M∗ (M), the product f h is a holomorphic section of E if and only if ( f ) + D = ( f ) + (h) = ( f h) ≥ 0. Comparing with the definition of the vector space L(D) (1.140), we see that the map L(D) f −→ f h ∈ H 0 (M, E) is an isomorphism of L(D) onto the space H 0 (M, E) of holomorphic sections of the line bundle E → M. Note also that the divisor KM − D corresponds to the line bundle KM ⊗ E −1 where KM = T ∗ M is the canonical bundle of M. This gives the following equivalent formulation of Theorem 1.11.4. Theorem 1.11.6 (Riemann–Roch theorem for line bundles). For every holomorphic line bundle E on a compact Riemann surface M we have that dim H 0 (M, E) − dim H 0 (M, KM ⊗ E −1 ) = 1 − gen(M) + deg E.

(1.144)

There is yet another form of the Riemann–Roch formula. Let MM and MM1 denote the sheaves of germs of meromorphic functions and 1-forms on a Riemann surface M, respectively. A divisor D ∈ Div(M) clearly determines a germ of divisors at every point of M. We introduce the following sheaves: OD = { f ∈ M : ( f ) + D ≥ 0},

ΩD = {ω ∈ M 1 : (ω) + D ≥ 0}.

(We include the zero germs in both sheaves.) Obviously, H 0 (M, OD ) = L(D) (see (1.140)) and H 0 (X, ΩD ) = {ω ∈ M 1 (X) : (ω) + D ≥ 0}. Fix a 1-form ω 0 ∈ M∗1 (M). For every f ∈ M (U) on an open set U ⊂ M we have ( f ) + (ω 0 ) + D = ( f ω 0 ) + D. Hence, the following map is an isomorphism of sheaves: OD+(ω 0 ) −→ ΩD ,

f −→ f ω 0 .

(1.145)

The following is the second key result of this theory. Theorem 1.11.7 (Serre duality). For every divisor D on a compact Riemann surface M we have the isomorphisms H 1 (M, OD )∗ ∼ = H 0 (M, Ω−D ),

H 1 (M, ΩD ) ∼ = H 0 (M, O−D )∗ .

(1.146)

In particular, taking D = 0 in the above theorem gives the isomorphisms H 1 (M, O)∗ ∼ = H 0 (M, Ω ),

H 1 (M, Ω ) ∼ = H 0 (M, O)∗ ∼ = C.

From (1.142) and the first formula above in (1.147) it follows that

(1.147)

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dim H 1 (M, O) = dim H 0 (M, Ω ) = dim Ω (M) = gen(M).

(1.148)

Furthermore, from (1.145), (1.146), and L(KM − D) = H 0 (M, KM − D) we see that dim H 1 (M, OD ) = dim H 0 (M, Ω−D ) = dim H 0 (M, KM − D) = dim L(KM − D), and hence the Riemann–Roch formula (1.141) takes the form dim H 0 (X, OD ) − dim H 1 (X, OD ) = 1 − gen(M) + deg D.

(1.149)

The Picard Group of a Riemann surface. Recall (cf. (1.48)) that the Picard group Pic(M) = H 1 (M, O ∗ ) represents holomorphic isomorphism classes of holomorphic line bundles on M. Consider the exponential sheaf sequence (1.49), e2πi·

0 −→ Z −→ O −→ O ∗ −→ 1, and a part of the associated long exact cohomology sequence: c

1 H 2 (M, Z) −→ H 2 (M, O). H 1 (M, Z) −→ H 1 (M, O) −→ H 1 (M, O ∗ ) −→

If M is an open Riemann surface then H 1 (M, O) = 0 = H 2 (M, O). (This is a special case of the cohomology vanishing theorem on Stein manifolds.) Furthermore, H 2 (M, Z) = 0, and we conclude that H 1 (M, O ∗ ) = 0, i.e., every holomorphic line bundle on an open Riemann surface is trivial. More generally, every holomorphic vector bundle of arbitrary rank on an open Riemann surface is trivial according to the Oka–Grauert theorem, which says that, on any Stein manifold, the topological classification of complex vector bundles agrees with their holomorphic classification (see [140, Theorem 5.3.1]).

1.12 Holomorphic Approximation Theory Approximation of holomorphic maps plays a key role in global constructions in complex analysis and geometry. Since every minimal surface in Rn is the image of a conformal harmonic immersion from an open Riemann surface and the derivative of such a map is a holomorphic map into a certain quadric subvariety of Cn (see Theorems 2.3.1 and 2.3.4), it is not surprising that these techniques are also very useful in constructions of minimal surfaces in Euclidean spaces. In this and the following section we recall the main holomorphic approximation theorems that will be used and provide references. The recent survey [128] by J. E. Fornæss, F. Forstneriˇc, and E. F. Wold, and the monograph [140] by the second named author contain substantially more information and additional references.

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Holomorphic approximation theory on Riemann surfaces. We begin with the following version of the Runge approximation theorem which pertains both to compact and open Riemann surfaces. Theorem 1.12.1 (Runge’s theorem on Riemann surfaces). If K is a compact set in a Riemann surface M, then every holomorphic function f on a neighbourhood of K can be approximated uniformly on K by meromorphic functions F on M without poles on K, and by holomorphic functions on M if K has no holes (see Definition 1.2.1). Furthermore, the approximating function F can be chosen to agree with f to any given finite order on a given finite set of points in K and to have its poles in a given subset E ⊂ M \ K which contains a point in each hole of K. When M = C, this is the classical Runge’s theorem from 1885. The standard proof uses the Cauchy integral formula for holomorphic functions and the method of pushing the poles. The extension to all open Riemann surfaces was obtained by H. Behnke and K. Stein [58] in 1949. To this end, they constructed Cauchy type kernels, the so-called elementary differentials, on relatively compact domains in an arbitrary open Riemann surface; see [58, Theorem 3]. By using such kernels one can imitate the proof of Runge’s theorem on C; an outline can be found in [128, Theorems 2 and 4]. A functional analytic proof using Weyl’s lemma was given by B. Malgrange [237] in 1955; see also [130, Sect. 25]. The case when M is a compact Riemann surface is partly contained in the paper [58] by H. Behnke and K. Stein, with an improvement due to H. L. Royden [304, Theorem 10]. In the latter paper, f is allowed to have poles on K. Since meromorphic functions on a Riemann surface are the same thing as holomorphic maps to the Riemann sphere CP1 , Royden’s theorem can be stated as follows. Corollary 1.12.2 (Runge theorem for maps to CP1 ). Let M be a Riemann surface. If K is a compact set in M and E ⊂ M \ K is a set containing a point in each hole of K, then every holomorphic map f : U → CP1 from an open neighbourhood U ⊂ M of K can be approximated uniformly on K by holomorphic maps F : M → CP1 such that F −1 (∞) ⊂ ( f |K )−1 (∞) ∪ E. We may choose F to agree with f to a given finite order at any given finite set of points in K. A more precise recent result in the same direction is given by Theorem 4.3.1. Recall that the holomorphic hull of a compact set K in a complex manifold M is O (M) = {p ∈ M : | f (p)| ≤ max | f | for all f ∈ O(M)} K K

(1.150)

O (M) . An O(M)-convex (cf. (1.41)). The set K is said to be O(M)-convex if K = K =K O (Cn ) set in a Riemann surface M is also called a Runge set. When M = Cn , K  is the polynomial hull of K, and K is polynomially convex if K = K. O (M) is Lemma 1.12.3. For any compact set K in a Riemann surface M the hull K compact and equals the union of K and all its holes. In particular, K is O(M)-convex (a Runge set in M) if and only if it has no holes.

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67

 denote the union of K and all its holes. If M is compact and K = ∅, Proof. Let K =K O (M) = M. Assume now that M is open. Since the boundary of then clearly K  is a closed set without holes, and it is contained any hole of K is contained in K, K O (M) by the maximum principle for holomorphic functions. Choose a compact in K set K1 ⊂ M with smooth boundary such that K ⊂ K1 . Then, K1 has at most finitely many holes (at most one for each boundary curve), so the union of K1 with its holes  thus K  is compact. Assume now is a compact set without holes which contains K;  ∪ {p} is also a compact set without holes. Let f be  Then, K  = K that p ∈ M \ K.  and a holomorphic function on a neighbourhood of K  which equals zero near K f (p) = 1. By Theorem 1.12.1 there is a holomorphic function F ∈ O(M) such that O (M) . Thus, K O (M) = K.   |F(p)| > 1/2 > maxK |F|, and hence p ∈ /K  Proposition 1.12.4. Every open Riemann surface M admits a normal exhaustion by compact O(M)-convex sets Ki = Ki with smooth boundaries: K1 ⊂ K2 ⊂ · · · ⊂

∞ !

Ki = M,

Ki ⊂ K˚ i+1 for all i ∈ N.

(1.151)

i=1

Proof. Choose a normal exhaustion (1.151) of M by compact sets with smooth boundaries. For example, we can take Ki = {ρ ≤ ci } where ρ : M → R is a smooth exhaustion function and c1 < c2 < c3 < · · · with limi ci = +∞ are regular values of ρ. Since K1 has smooth boundary, it has at most finitely many holes, and the union 1 with smooth boundary which is O(M)of K1 with its holes is a compact set K 1 ⊂ K˚ i . Let K i be the convex by Lemma 1.12.3. Choose an index i2 > 1 such that K 2 2 union of Ki2 with its finitely many holes; this gives the next O(M)-convex set in the i ⊂ K i ⊂ · · · 1 ⊂ K sequence. Continuing inductively we find a normal exhaustion K 2 3 of M by compact O(M)-convex sets with smooth boundaries.   A convenient way of finding exhaustions satisfying Proposition 1.12.4 is by using (strongly) subharmonic exhaustion functions. If ρ : M → R is a nonconstant smooth subharmonic function (i.e., Δ ρ ≥ 0) then ρ has no local maxima, and hence a compact sublevel set {ρ ≤ c} has no holes. If in addition ρ is an exhaustion function on M (such exists by Corollary 1.10.4) then for any sequence c1 0 (with respect to some Riemannian distance function on M), then every function in A (K) is a uniform limit of meromorphic functions on M with poles in M \ K. An outline of proof of Theorem 1.12.7, along with references to sources where a complete proof can be found, is provided in [128, Theorem 6]. An important ingredient in the proof of Bishop’s theorem is the following localization theorem from [66]. Recall that O(K) denotes the space of functions that are holomorphic on neighbourhoods of a compact set K, and O(K) is the uniform closure of { f |K : f ∈ O(K)} in the Banach algebra C (K). Theorem 1.12.8. Let K be a compact set in a Riemann surface M and f ∈ C (K). If every point p ∈ K has a compact neighbourhood D p ⊂ M such that f |K∩D p ∈ O(K ∩ D p ), then f ∈ O(K). A simple proof of Theorem 1.12.8 using solutions of nonhomogeneous ∂ equations was given by A. Sakai [307] in 1972; see also [128, Theorem 6]. In light of Runge’s Theorem 1.12.1, the main point of Mergelyan’s theorem is to approximate a function f ∈ A (K) uniformly on K by functions holomorphic on open neighbourhoods of K. Examples show that this is not always possible if K has infinitely many holes (the so-called Swiss cheese phenomenon). This difficult analytic problem was solved for compact planar sets by A. G. Vitushkin [334] in

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1966. A compact set K in a Riemann surface M satisfying A (K) = O(K) is said to enjoy the Mergelyan property or the Vitushkin property. We refer to [128, Theorem 7] for a discussion and further references on this subject. Approximation on admissible sets in Riemann surfaces. We shall have to approximate functions of class A r (S) for any r ∈ Z+ on compact sets S of the following type in Riemann surfaces. Definition 1.12.9. Let M be a smooth surface. An admissible set in M is a compact set of the form S = K ∪ E, where K is a (possibly empty) finite union of pairwise disjoint compact domains with piecewise C 1 boundaries in M and E = S \ K˚ is a union of finitely many pairwise disjoint smooth Jordan arcs and closed Jordan curves meeting K only at their endpoints (if at all) and such that their intersections with the boundary bK of K are transverse. Admissible sets arise in handlebody decompositions of surfaces; see Section 1.4. Indeed, when passing a critical point p of index one of a Morse exhaustion function ρ : M → R on a surface M, the change of topology is described by attaching a smooth embedded arc E ⊂ M through p to a compact sublevel set Mc = {ρ ≤ c} for some c < ρ(p) close to ρ(p) so that all numbers in [c, ρ(p)) are regular values of ρ. For this reason, approximation on admissible sets is of particular importance and will be used frequently in the book. An admissible set has at most finitely many holes and hence Theorem 1.12.7 applies. It is also clear that for any Riemannian distance function dist on M and every small enough ε > 0, the set Sε = {p ∈ M : dist(p, S) < ε}

(1.152)

is an open neighbourhood of S which admits a deformation retraction onto S; we shall call such Sε a regular neighbourhood of S. Clearly, S has no holes and hence is Runge in any regular neighbourhood Sε . In our constructions of minimal surfaces it will be important not only to approximate maps on admissible sets, but also to control their integrals (periods) on closed curves forming a basis of the first homology group, as well as on some other arcs in order to satisfy interpolation conditions. We shall need special homology bases furnished by the following lemma taken from [132, Lemma 3.1]. Lemma 1.12.10 (Runge homology basis of an admissible set). A connected admissible set S has finitely generated first homology group H1 (S, Z) ∼ = Zl . Furthermore, there is a homology basis C = {C1 , . . . ,Cl } consisting of closed piecewise smooth Jordan curves in S such that C = li=1 Ci is connected and Runge in any regular neighbourhood Sε (1.152) of S, and every curve Ci ∈ C contains a nontrivial arc Ii disjoint from j=i C j . In the special case when S = K is a compact connected domain with piecewise C 1 boundary, the curves C1 , . . . ,Cl may be chosen such that any two meet only at a common point p0 ∈ K˚ and C = li=1 C j is a deformation retract of K.

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The second part of the lemma is classical; see e.g. H. M. Farkas and I. Kra [123]. One can also find a basis for H1 (S, Z) consisting of smooth Jordan curves, but piecewise smooth curves will suffice for our applications.

Proof. Let S = K ∪ E with K = m i=1 Ki , where K1 , . . . , Km are the connected components of K. If K = ∅, S is a single arc or closed curve and the result is trivial. Assume now that K = ∅. Since S is connected, E = nk=1 Ek is a union of finitely many smooth pairwise disjoint arcs Ek . In each component Ki of K we choose an interior point qi ∈ K˚ i which we shall call the vertex of Ki . The boundary i Γi, j consists of finitely many closed Jordan curves for some mi ≥ 1. bKi = mj=1 The standard construction (see [123]) gives for each i = 1, . . . , m a basis of H1 (Ki , Z) consisting of finitely many Jordan curves in K˚ i passing through the vertex qi and not intersecting elsewhere whose union is Runge in Ki ; we put all these curves in the family C under construction. For every i = 1, . . . , m and j = 1, . . . , mi we choose a pair of distinct points ai, j , bi, j ∈ Γi, j such that bi, j ∈ / E. We connect ai, j to the vertex qi by a smooth embedded arc Ai, j ⊂ K˚ i ∪ {ai, j }, chosen such that these arcs do not intersect each other, nor any of the chosen curves in the homology basis for Ki , except at qi . Recall that the arcs E1 , . . . , En are the connected components of E = S \ K. Every arc Ek in this collection is of one of the following three types: Case 1: Ek is attached to K with only one endpoint. Such arcs do not affect the homology. Let S0 denote the admissible set obtained by attaching all such arcs to K. Case 2: The endpoints of Ek lie in connected components Γi, j1 , Γi, j2 of bKi for some i ∈ {1, . . . , m}. (These components may be the same.) In this case, a new homologically essential closed curve in S is obtained by connecting the endpoints of Ek inside Ki as follows. Having traversed Ek to its endpoint in Γi, j2 , continue to the point ai, j2 along the (unique!) arc in Γi, j2 which does not contain bi, j2 , then go from ai, j2 to the vertex qi along the arc Ai, j2 , continue from qi to ai, j1 along Ai, j1 , and finally connect ai, j1 to the initial point of Ek by the arc in Γi, j1 not containing bi, j1 . We add all closed curves obtained in this way to the family C , and we denote by S1 the new admissible set obtained from S0 (see Case 1) in this way. Note that S1 still has the same number of connected components as K, namely m. Case 3: The endpoints of Ek belong to different connected components of K (and hence of S1 ). Let us call such an arc a bridge. Let S2 denote a connected admissible set obtained by attaching to S1 (see Case 2) a collection of bridges such that removing any one of them disconnects S2 . Paint these bridges black. (Such S2 need not be unique.) Clearly, the inclusion S1 → S2 induces an isomorphism of homology groups H1 (S1 , Z) ∼ = H1 (S2 , Z). We paint the remaining bridges red. For every red bridge Ek there are pairwise distinct bridges Ek = Ek1 , . . . , Eks , all but Ek black, and connected components Ki1 , . . . , Kis (islands) of K such that Ek1 connects Ki1 to Ki2 , Ek2 connects Ki2 to Ki3 , etc., until the cycle closes with the last bridge Eks connecting Kis back to Ki1 . We obtain a new closed curve in S by connecting the endpoint of each bridge Ek j in the above sequence within the domain Ki j to the initial point of the next bridge

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Ek j+1 , where Eks+1 = E1 . The connecting curves in domains Ki j are chosen as in Case 2 above. By attaching all red bridges to S2 we obtain the original admissible set S, and adding the corresponding closed curves to C furnishes a homology basis of S. See Figure 1.2.

Fig. 1.2 Two islands connected by three bridges

By the construction, every red bridge (see Case 3) is contained in precisely one curve in C . The same is true for each of the arcs in Case 2. Hence, every curve in the family C contains an arc which is disjoint from all other curves in C . Let |C | denote the union of all curves in C . By the construction, any point p ∈ Ki \ |C | (i = 1, . . . , m) can be connected by an arc in Ki \ |C | to the point bi, j ∈ Γi, j ⊂ bKi for some j ∈ {1, . . . , mi }. Hence, |C | has no holes in S, so it is Runge. We can make |C | connected by modifying each closed curve C ∈ C to pass through the vertex q1 ∈ K1 . Indeed, every such curve C passes through one of the vertices qi ∈ Ki , so it suffices to connect qi to q1 as described in Case 3 above, using only black bridges when passing between different connected components of K.   A function f : S = K ∪ E → C on an admissible set is of class C r (S) if it extends to a function in C r (M). By using Whitney’s jet-extension theorem (see [128, Appendix]) it is easily seen that every f ∈ A r (S) extends to a function f ∈ C r (M) which is ∂ -flat to order r on S, in the sense that lim Dr−1 x (∂ f ) = 0.

x→S

(1.153)

Here, Dkx denotes the k-jet at a point x, i.e., the value and all partial derivatives up to order k. Condition (1.153) is equivalent to asking that, in any local holomorphic coordinate z on an open set U ⊂ M around a point x0 ∈ S, the partial derivative ∂ f /∂ z¯ vanishes to order r − 1 on S ∩ U. We define the C r (S) norm of f as the maximum of derivatives of f up to order r at points x ∈ S (measured in a fixed finite system of holomorphic coordinate charts covering S), where for points z ∈ E \ K we consider only the derivatives tangential to the curves in E. The following Mergelyan type approximation theorem on admissible sets will be very important in the constructions in this book.

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Theorem 1.12.11 (Mergelyan approximation theorem on admissible sets). If S is an admissible set in a Riemann surface M, then every function f ∈ A r (S) (r ∈ Z+ ) can be approximated in C r (S) by meromorphic functions on M, and by holomorphic functions on M if S has no holes. Furthermore, the approximating functions can be chosen to agree with f on any given finite set A ⊂ S, and to agree with f to an ˚ arbitrary finite order at the points in A ∩ S. A proof of Theorem 1.12.11 by induction on r is given in [128, Theorem 16], reducing it to the basic case r = 0 covered by Theorem 1.12.7. The case r = 0 also follows from Theorem 1.12.8 and Mergelyan’s theorem in the plane. The analogous result holds if the function f has poles at finitely many interior points of S as in Corollary 1.12.2, and we can interpolate to any given finite order at such points. This generalization combines the special case of Theorem 1.13.1 (b) with X = CP1 and the Runge approximation theorem 1.12.1. An even more precise version of Theorem 1.12.11 on compact Riemann surfaces is furnished by Theorem 4.3.1, which is tailor-made for use in the constructions in Chapter 4. Remark 1.12.12 (On interpolation). In the Runge and Mergelyan theorems, interpolation at finitely many points p1 , . . . , pm in a given compact set K is a simple addition by the following argument. Assume for simplicity that K has no holes in M. Choose functions g1 , . . . , gm ∈ O(M) such that gi (p j ) = δi, j (the Kronecker delta). If F ∈ O(M) is uniformly close to f on K, then the function m

f˜ = F + ∑ ( f (pi ) − F(pi ))gi ∈ O(M) i=1

still approximates f on K, and it agrees with f at the points p1 , . . . , pm ∈ K. Similarly, we obtain approximation with jet interpolation to order N ∈ N at finitely many points p1 , . . . , pm ∈ K˚ by using a collection of functions gi ∈ O(M) which form a basis for the jet spaces of order N at the points p1 , . . . , pm . The analogous  argument applies with meromorphic functions gi without poles on K.  Weierstrass interpolation on open Riemann surfaces. The first general interpolation results for holomorphic functions were obtained by K. Weierstrass [339] in 1876. He proved that for any closed discrete set {ai }∞ i=1 in C and integers ki ∈ N there exists an entire function f ∈ O(C) with a zero of order ki at ai for every i ∈ N and no other zeros. He found such a function in the form of an infinite product ∞ z ki gi (z) e , f (z) = zk ∏ 1 − ai i=1

where k ≥ 0 is the order of zero at 0, ai = 0 for all i, and the holomorphic polynomials gi are chosen such that the product converges. By using the Runge approximation theorem of H. Behnke and K. Stein [58] (see Theorem 1.12.1), H. Florack [127] in 1949 extended Weierstrass’s theorem to all open Riemann surfaces as follows.

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Theorem 1.12.13. Let M be an open Riemann surface and A = {ai }∞ i=1 be a closed discrete subset of M. Given integers ki ∈ N there exists a function f ∈ O(M) which vanishes to order ki at the point ai for every i ∈ N and has no other zeros. An immediate consequence is that every meromorphic function on an open Riemann surface is the quotient of two holomorphic functions, and every divisor on such a surface is a principal divisor. A more precise result in the same direction is the following Weierstrass–Florack jet interpolation theorem (see [127]). Theorem 1.12.14. Let M be an open Riemann surface, K be a compact O(M)convex subset of M, A = {ai }∞ i=1 be a closed discrete subset of M, U ⊂ M be an open set containing A ∪ K, and f ∈ M (U) be a meromorphic function whose only zeros and poles are at the points of A. Given integers ki ∈ N and a number ε > 0, there exists a function F ∈ M (M) such that (a) |F(z) − f (z)| < ε for all z ∈ K, (b) F − f vanishes to order ki at the point ai ∈ A for every i ∈ N, and (c) F has no zeros or poles on M \ A. This follows by combining Theorem 1.12.13 and the Runge approximation theorem for nowhere vanishing holomorphic functions. The latter is a special case of approximation of manifold-valued maps considered in the following section, but this particular case is classical and can be treated by elementary methods. In the function-theoretic sense, Stein manifolds (see Definition 1.5.2) are higherdimensional analogues of open Riemann surfaces. The following extension of Runge’s Theorem 1.12.1 to Stein manifolds is due to K. Oka [277] (1936) and A. Weil [340] (1935); interpolation was added by K. Oka and H. Cartan. Theorem 1.12.15 (Cartan–Oka–Weil theorem). If K is a compact O(M)-convex subset of a Stein manifold M, then every function f ∈ O(K) can be approximated uniformly on K by functions F ∈ O(M). If in addition A is a closed complex subvariety of M and f |A∩K extends to a holomorphic function on A, then the approximating functions F can be chosen to agree with f on A, and to agree with f to any given finite order on A ∩ K. If furthermore f is holomorphic on a neighbourhood of A ∪ K then the approximating functions F can be chosen to agree with f to any given finite order on A. The most efficient proof of the Cartan–Oka–Weil theorem uses L. H¨ormander’s solutions to nonhomogeneous ∂ -equations with L2 -estimates [194, 195]. A proof of Theorem 1.12.15 by H¨ormander’s method, including a parametric version and approximation with interpolation for sections of holomorphic vector bundles over Stein manifolds, can be found in [140, Sect. 2.8]. A survey of L2 -techniques in complex geometry is given in T. Ohsawa’s book [276].

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1.13 Manifold-Valued Maps and the Oka Principle In this section we present a few key results of holomorphic approximation theory for manifold-valued maps. A comprehensive account can be found in the monograph [140]; see also the surveys [145] and [128, Sect. 7]. Given a pair of complex manifolds M and X, we denote by O(M, X) the space of holomorphic maps M → X endowed with the compact-open topology. If K is a compact subset of M and r ∈ Z+ , we denote by A r (K, X) the space of maps K → X of class C r (meaning that they admit C r extension to some open neighbourhood of ˚ Similarly, O(K, X) stands for the K in M) which are holomorphic in the interior K. space of holomorphic maps f : U = U f → X from open neighbourhoods of K in M, and O(K, X) is the uniform closure of { f |K : f ∈ O(K, X)} in C (K, X). We begin with the following Mergelyan theorem for manifold-valued maps from Riemann surfaces. Theorem 1.13.1 (Mergelyan theorem for manifold-valued maps). Let M be a Riemann surface and X be an arbitrary complex manifold. (a) If K is a compact set in M such that A (K) = O(K), then A (K, X) = O(K, X). This holds in particular if K has at most finitely many holes in M. (b) If S = K ∪ E is an admissible set in M (see Definition 1.12.9) and r ∈ Z+ then every map in A r (S, X) can be approximated in C r (S, X) by maps U → X holomorphic on open neighbourhoods U of S in M. Furthermore, the approximating maps can be chosen to agree with the given map at any finite set of points p1 , . . . , pm in K (in case (a)) or in S (in case (b)); at the ˚ we can interpolate to any given finite order. points p j ∈ K˚ (or p j ∈ S) Part (a) of Theorem 1.13.1 is [141, Theorem 1.4] (see also [128, Corollary 7]). Part (b) follows by the same proof and using Theorem 1.12.11 (the Mergelyan theorem for functions on admissible sets). A compact set K in a Riemann surface M satisfying A (K) = O(K) (see p. 68) is said to enjoy the Mergelyan property or the Vitushkin property. The first part of Theorem 1.13.1 says that if a compact set K in a Riemann surfaces has the Mergelyan–Vitushkin property for functions, then it has the same property for maps to an arbitrary complex manifold. The main point of the proof is that, assuming A (K) = O(K), the graph G f = {(p, f (p)) : p ∈ K} ⊂ M × X

(1.154)

of any map f ∈ A (K, X) has a basis of open Stein neighbourhoods in M × X. (A compact set with this property is called a Stein compact.) This is a special case of the important result of E. M. Poletsky (see [128, Theorem 32]) to the effect that, if the set K is a Stein compact in a complex manifold M and f : K → X is a continuous map which is locally uniformly approximable by holomorphic maps, then the graph G f ⊂ M × X (1.154) is a Stein compact. The existence of a Stein neighbourhood

1.13 Manifold-Valued Maps and the Oka Principle

75

of G f , along with standard techniques from Stein manifold theory, easily reduce the approximation theorem for maps f ∈ A (K, X) (and for maps of class A r (K, X) in case (b)) to the corresponding result for functions (see [128, Lemma 3]). The interpolation condition in Theorem 1.13.1 follows in view of Remark 1.12.12. The class of complex manifolds X for which the Runge–Oka–Weil theorem holds for maps M → X from any Stein source manifold M is by now fairly well understood; they are called Oka manifolds in honour of K. Oka’s pioneering work [278] on the second Cousin problem. This class of manifolds was introduced by F. Forstneriˇc [138] in 2009, and the subject is treated in [140, Chapter 5]. The following is [140, Definition 5.4.1]. Definition 1.13.2. A complex manifold X is an Oka manifold if every holomorphic map K → X from a neighbourhood of any compact convex set K ⊂ Cn (n ∈ N) can be approximated uniformly on K by entire maps Cn → X. Several other characterizations of Oka manifolds can be found in [140, Sect. 5.15] and [210]. The following result is a special case of [140, Theorem 5.4.4]. Theorem 1.13.3 (Runge theorem for maps to Oka manifolds). Assume that M is a Stein manifold (in particular, M may be an open Riemann surface), M  ⊂ M is a (possibly empty) closed complex subvariety, and X is an Oka manifold endowed with a Riemannian distance function dist. Given a compact O(M)-convex set K ⊂ M and a continuous map f : M → X which is holomorphic on a neighbourhood of K ∪ M  , there exist for every ε > 0 a neighbourhood U of K ∪ M  and a homotopy ft : M → X (t ∈ [0, 1]) such that f0 = f and the following conditions hold for every t ∈ [0, 1]. • • • •

The map ft is holomorphic on U. We have that sup p∈K dist( ft (p), f (p)) < ε. The map ft agrees with f to any given finite order r ∈ N along M  . The map f1 is holomorphic on M.

The analogous result holds for families of maps fq : M → X depending continuously on a parameter q ∈ Q in a compact Hausdorff space. The analogue of Theorem 1.13.3 also holds for sections f : M → Z of any holomorphic fibre bundle π : Z → M with an Oka fibre Z p = π −1 (p) over a Stein manifold M. Furthermore, ramifications of π are allowed provided the initial section is holomorphic near the set of critical values of π as in the following theorem. Theorem 1.13.4 (The basic Oka principle for sections of ramified maps). Let π : Z → M be a surjective holomorphic map of a reduced complex space Z onto a Stein manifold M. Assume that M  is a closed complex subvariety of M such that π : Z \ π −1 (M  ) → M \ M  is a holomorphic fibre bundle with Oka fibre, K is a compact O(M)-convex subset of M, and f : M → Z is a continuous section of π which is holomorphic on an open neighbourhood of K ∪ M  . Let dist be a distance function on Z. Given r ∈ N and ε > 0, there exist an open neighbourhood U ⊂ M of K ∪ M  and a homotopy of continuous sections ft : M → Z (t ∈ [0, 1]) satisfying the conclusion of Theorem 1.13.3.

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The main use in this book of Oka manifolds and the above two theorems is indicated in Corollary 1.13.9, which says that the complex hyperquadrics that arise in the theory of minimal surfaces are Oka manifolds. Theorem 1.13.4 was proved in [135, Theorem 2.1]; see also [140, Theorem 6.14.6]. We shall use it in the case when M is an open Riemann surface and M  is a closed discrete set in M. The hypothesis in both theorems that the initial map f be defined and continuous on all of M is clearly necessary, as there may exist maps K → X which do not extend to continuous maps M → X. For X = C, Theorem 1.13.3 is equivalent to the Cartan–Oka–Weil Theorem 1.12.15. The conclusion of Theorem 1.13.3 is called the basic Oka principle with approximation and interpolation for the natural inclusion map ι : O(M, X) −→ C (M, X)

(1.155)

from the space of all holomorphic maps into the space of all continuous maps M → X, where both spaces are endowed with the compact-open topology. The parametric Oka principle for the inclusion ι refers to the analogous statement for continuous families of maps depending on a parameter in a Euclidean compact P, with the homotopy being fixed on a compact subset Q ⊂ P. Neglecting approximation and interpolation, this is illustrated by the following diagram: f0

/ O(M, X) _ o7 o o o o o ι ooo   ooooo f0 / C (M, X) P

Q _

f1

It says that any continuous map f0 : P → C (M, X) with f0 (Q) ⊂ O(M, X) can be deformed to a continuous map f1 : P → O(M, X) by a homotopy ft : P → C (M, X) (t ∈ [0, 1)) that is fixed on Q (see [140, Theorem 5.4.4]). More generally, any continuous map ι : A → B between topological spaces for which the above statement holds with respect to the commuting diagram /A qq8 q q f1 q ι qqq q q   qqq f0 /B P

Q _

is said to satisfy the parametric h-principle. This terminology was introduced by M. Gromov and we refer to his works [172, 173, 170]. The following is a well known consequence of the parametric h-principle. Proposition 1.13.5. A continuous map ι : A → B which satisfies the parametric h-principle is a weak homotopy equivalence, i.e., it induces a bijection of the path components of the two spaces and an isomorphism of their homotopy groups ∼ = πk (ι) : πk (A, a) −→ πk (B, ι(a)) for every a ∈ A and k = 1, 2, . . ..

1.13 Manifold-Valued Maps and the Oka Principle

77

Proof. Assume that ι : A → B satisfies the parametric h-principle. By taking P to be a singleton and Q = ∅ we see that every element in B is connected by a path to an element in ι(A). Applying the same result with P = [0, 1] and Q = {0, 1} ⊂ P we see that every path f0 : [0, 1] → B connecting a pair of elements in ι(A) can be deformed with fixed ends to a path f1 = ι ◦ g where g : [0, 1] → A. This shows that ι induces a bijection of the path components of the two spaces. Similarly one proves the statement concerning the k-th homotopy groups by applying the parametric hk+1 principle with the pairs P = Sk , Q = ∅ and P = BR ⊂ Rk+1 , Q = bP = Sk .   We have the following corollary (see [140, Corollary 5.5.6] for the first part and F. L´arusson [213] for the second one). Recall that a continuous map f : A → B is a homotopy equivalence if there is a continuous map g : B → A such that g ◦ f is homotopic to the identity on A and f ◦ g is homotopic to the identity on B. Corollary 1.13.6. If M is a Stein manifold and X is an Oka manifold then the inclusion (1.155) is a weak homotopy equivalence. It is a genuine homotopy equivalence when M admits a strongly plurisubharmonic exhaustion function with finitely many critical points; this holds for any affine algebraic manifold, and also for any open Riemann surface of finite topological type. L´arusson’s result in [213] (the second part of the above corollary) relies on the theory of absolute neighbourhood retracts, abbreviated ANRs, and the fact that a weak homotopy equivalence between CW complexes is a homotopy equivalence according to a theorem of J. H. C. Whitehead [246, p. 74]. Since every ANR is homotopy equivalent to a CW complex (see R. Palais [285, Theorem 15]), the same conclusion holds for maps between a pair of ANRs. The nontrivial part is to show that certain mapping spaces, such as those involved here, are ANRs. Theorem 1.13.3 has a long history going back to seminal works of K. Oka [278] (1939) and H. Grauert [164] (1958); see the introduction to [140, Chapter 5] and the survey [145]. The classical Oka–Grauert principle pertains to the case when X is a complex Lie group or a complex homogeneous manifold, i.e., a complex manifold with a transitive holomorphic action of a complex Lie group by holomorphic automorphisms of X. It has major consequences for the classification theory of holomorphic vector bundles and principal fibre bundles on Stein manifolds; see [140, Chapters 5 and 8]. Modern extensions of Oka theory were initiated by M. Gromov [171] in 1989. He found weaker sufficient conditions on the manifold X such that Theorem 1.13.3 holds. We recall the following condition. Definition 1.13.7. A complex manifold X is flexible if there exist finitely many Ccomplete holomorphic vector fields V1 , . . . ,Vm on X whose values at any point x ∈ X span the tangent space Tx X. Every flexible manifold is an Oka manifold (see [140, Proposition 5.6.22]). Furthermore, every complex homogeneous manifold X is flexible. Indeed, if a complex Lie group G acts holomorphically on X on the left, then every vector v ∈ g in the Lie algebra g of G determines a complete holomorphic vector field on X with

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the flow φt (x) = etv · x for t ∈ C and x ∈ X. Assuming that the action is transitive, these vector fields span the tangent space of X at every point. Example 1.13.8 (Complete holomorphic flows on the null quadric). If P(z) is a homogeneous quadratic polynomial on Cn for some n ≥ 2 such that the hyperquadric A = {P = 0} is smooth away from the origin, then the manifold X = A \ {0} is flexible. Indeed, the holomorphic vector fields V j,k =

∂P ∂ ∂P ∂ − , ∂ z j ∂ zk ∂ zk ∂ z j

1 ≤ j = k ≤ n,

(1.156)

are tangential to A, they span the tangent space Tz A at each point 0 = z ∈ A, they are linear and hence C-complete, and A \ {0} is an invariant set for their flows. Hence, X = A \ {0} is an Oka manifold to which Theorem 1.13.3 applies. Likewise, the level sets {P = c} for c ∈ C∗ are Oka manifolds. In particular, the punctured null quadric A∗ := A \ {0} ⊂ Cn (1.116) is an Oka manifold. In this case, P(z) = ∑nj=1 z2j and the vector fields (1.156) equal V j,k = z j

∂ ∂ − zk . ∂ zk ∂zj

(1.157)

A calculation shows that the flow φt (z) of V j,k for t ∈ C is given by φt (z) j = z j cost − zk sint, φt (z)k = z j sint + zk cost, φt (z)i = zi for i = j, k.

(1.158)

This flow consists of complex rotations in the coordinate ( j, k)-plane and is the identity on the complementary subspace. Every complex hyperquadric   Σc = (z1 , . . . , zn ) ∈ Cn : z21 + · · · + z2n = −c (1.159) for c ∈ C is an invariant set of these flows. In fact, A∗ = Σ0 \ {0} and Σc for c = 0 are homogeneous manifolds of the complex orthogonal group On (C) = {A ∈ GLn (C) : AAt = I}, and also of its index 2 subgroup SOn (C). Any pair of hyperquadrics Σc for c ∈ C∗ are biholomorphic to each other by a dilation.   Corollary 1.13.9. Theorem 1.13.3 and Corollary 1.13.6 hold for maps from any Stein manifold M (in particular, from any open Riemann surface) to every flexible complex manifold X, in particular, to each of the following manifolds: 1. The punctured plane X = C∗ . 2. The punctured null quadric X = A∗ ⊂ Cn given by (1.116) for any n ≥ 3. 3. Any hyperquadric X = Σc in (1.159) with c ∈ C∗ .

1.14 Holomorphic Sprays

79

1.14 Holomorphic Sprays In holomorphic approximation theory for manifold-valued maps it is necessary to consider families of maps depending holomorphically on complex parameters. Such families will also be used in many proofs in this book. Definition 1.14.1. Let M and X be complex manifolds. A holomorphic spray of maps M → X is a holomorphic map F : M ×V → X, where V ⊂ CN is a connected open neighbourhood of the origin in a complex Euclidean space. The map F0 = F(· , 0) : M → X is called the core map, or simply the core of F. The spray F is dominating if the partial differential  ∂ F(p,t)  : CN −→ TF(p,0) X is surjective for every p ∈ M, (1.160) ∂t t=0 and is strongly dominating if ∂ F(p,t)/∂t is surjective for every (p,t) ∈ M ×V . We shall often omit the word “holomorphic” and simply talk of sprays of maps. A spray in the above definition is sometimes called a local spray, indicating that the parameter set V is a possibly small open neighbourhood of the origin in a Euclidean space. Local dominating sprays are ubiquitous; in particular we have the following result (cf. [140, Corollary 8.10.4]). Lemma 1.14.2. Assume that K is a Stein compact in a complex manifold M (i.e., K has a basis of open Stein neighbourhoods in M) and X is a complex manifold. Given a holomorphic map f : U → X from an open neighbourhood U ⊂ M of K, there are an open neighbourhood U0 ⊂ U of K and a dominating spray F : U0 ×V → X with the core F(· , 0) = f |U0 . Such a spray F is obtained by composing flows of (not necessarily complete) holomorphic vector fields on X defined on a neighbourhood Ω ⊂ M × X of the graph G f (U) = {(p, f (p)) : p ∈ U} ⊂ M × X of f |U ; see the formula (1.161) below. Note that G f (U) admits an open Stein neighbourhood in M × X by Siu’s theorem [140, Theorem 3.1.1], and the rest follows from Cartan’s Theorem A. An important result which lies at the heart of Oka theory is that one can glue a pair of local holomorphic sprays F : UA × V → X, G : UB × V → X, defined on open neighbourhoods UA ,UB of a suitable pair of compact sets A, B ⊂ M and such that F and G are sufficiently uniformly close to each other on (UA ∩ UB ) × V , into a holomorphic spray f˜ : U  × V  → X over an open neighbourhood U  ⊂ X of A ∪ B and with 0 ∈ V   V a smaller neighbourhood of the origin in CN , such that f˜ approximates the initial sprays F, G on the respective domains. A sufficient condition for this gluing is that the sets C = A ∩ B and D = A ∪ B are Stein compacts and A \ B ∩ B \ A = ∅. A pair of compacts (A, B) with these properties is called a Cartan pair. We refer to [140, Sections 5.7–5.9] for the details. A spray F in Definition 1.14.1 is called a global spray if the parameter set V equals CN . In general, there is no global dominating spray with a given core

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map f : M → X, the reason being that X need not admit any nonconstant entire holomorphic maps from Euclidean spaces. However, such sprays exist if M is a Stein manifold and X is a flexible manifold (see Definition 1.13.7), and in particular if X is a complex homogeneous manifold. Indeed, assume that V1 , . . . ,VN are C-complete holomorphic vector fields on X which span the tangent space of X at each point. Let φt j denote the flow of V j for time t ∈ C. Given a holomorphic map f : M → X, we obtain a dominating spray F : M × CN → X with the core F(· , 0) = f by taking F(p,t1 , . . . ,tN ) = φt11 ◦ · · · ◦ φtNN ( f (p)) ∈ X

(1.161)

for p ∈ M and t = (t1 , . . . ,tN ) ∈ CN . (A spray of this type is called a flow-spray.) Indeed, by the definition of the flow of a vector field we have that  ∂ F(p,t)  = Vi ( f (p)), p ∈ M, (1.162) ∂ti t=0 and since these vectors span the tangent space T f (p) X, the spray is dominating. An important example is a global spray σ : X ×CN → X whose core is the identity map on X: σ (x, 0) = x for all x ∈ X. Such σ is called a spray on X. More generally, we may take as the domain of the spray the total space E of any holomorphic vector bundle π : E → X. A complex manifold X which admits a global dominating spray is called elliptic (see Gromov [171] and [140, Chap. 6]). In particular, any flexible manifold X is elliptic, with a dominating spray σ : X × CN → X defined by σ (x,t1 , . . . ,tN ) = φt11 ◦ · · · ◦ φtNN (x) ∈ X

(1.163)

for all x ∈ X and t = (t1 , . . . ,tN ) ∈ CN , using the same notation as in (1.161). A seminal result due to M. Gromov [171] is that every elliptic manifold is an Oka manifold; the converse holds on Stein manifolds but fails in general (see Y. Kusakabe [211]). For more information we refer to [140, Chaps. 5, 6]. It follows from Theorem 1.13.3 that any local holomorphic spray over a holomorphic map f : M → X (see Definition 1.14.1) from a Stein manifold M to an Oka manifold X can be interpolated along M × {0} by a global spray M × CN → X. That the converse also holds is a nontrivial recent result of Y. Kusakabe [210] which gives the following characterization of the class of Oka manifolds. Theorem 1.14.3. A complex manifold X is an Oka manifold if and only if every holomorphic map f : M → X from a Stein manifold M is the core of a dominating holomorphic spray F : M × CN → X. In our applications of the technique of sprays in minimal surface theory, the crucial role is played by sprays of maps from open Riemann surfaces satisfying certain period domination conditions on finitely many Jordan curves in M. Such period dominating sprays are first introduced in Lemma 3.2.1 and are then used in many instances throughout the book.

1.15 Algebraic Sprays and Algebraic Approximation

81

1.15 Algebraic Sprays and Algebraic Approximation The punctured null quadric A∗ , and the hyperquadrics Σc in (1.159) for c ∈ C∗ , also satisfy the following approximation theorem concerning algebraic maps from affine algebraic varieties. This result is a special case of [140, Theorem 6.15.1]. Theorem 1.15.1. Assume that X ⊂ Cn (n ≥ 3) is the punctured null quadric A∗ (1.116) or a hyperquadric Σc (1.159), c ∈ C∗ . If A is an affine algebraic variety, K is a compact O(A)-convex subset of A, and f : K → X is a holomorphic map which is homotopic to the restriction f0 |K of an algebraic map f0 : A → X, then f can be approximated uniformly on K by algebraic maps f˜ : A → X. In particular, a holomorphic map A → X which is homotopic to an algebraic map A → X can be approximated uniformly on compacts in A by algebraic maps A → X. This holds for any null homotopic holomorphic map A → X. In view of [140, Example 6.15.7], the condition that the given map f be homotopic to an algebraic map A → X cannot be dispensed with in general. Remark 1.15.2. The reason for mentioning Theorem 1.15.1 is our hope that the method used in its proof might be useful in the construction of complete conformal minimal surfaces M → Rn of finite total curvature; see Chapter 4 for this topic. In this case, M is a finitely punctured compact Riemann surface M \ {p1 , . . . , pm }, and the Weierstrass data of the minimal surface must have an effective pole at each puncture pi to ensure completeness. The methods used in Chapter 4 employ the classical theory of Riemann surfaces, but the proofs are technically fairly involved. Unfortunately, Theorem 1.15.1 by itself does not suffice for this application, but we hope that it could be developed in a suitable way.   Theorem 1.15.1 actually holds for maps from affine algebraic varieties into any algebraically subelliptic manifold, that is, an algebraic manifold X which admits a finite dominating family of algebraic sprays defined on algebraic vector bundles over X; see [140, Definition 5.6.13 (e) and Theorem 6.15.1]. The cited result, which was proved in [136, Theorem 3.1], assumes the existence of a homotopy of holomorphic maps ft : K → X (t ∈ [0, 1]) connecting f to f0 |K , and the conclusion is that one can approximate the homotopy by the restrictions to K × [0, 1] of algebraic maps A × C → X. The conditions in Theorem 1.15.1 imply the existence of such a homotopy. Indeed, since X is an Oka manifold and K has a basis of open Stein neighbourhoods, every homotopy K × [0, 1] → X connecting a pair of holomorphic maps can be deformed with fixed ends to a homotopy of holomorphic maps in view of [140, Theorem 5.4.4]. The fact that the manifolds A∗ and Σc are algebraically elliptic (a stronger condition than algebraic subellipticity) is a special case of the results of K. Kaliman and M. Zaidenberg in [204, Sect. 5]. They showed that on any affine variety of the form X = {uv = P(x)} ⊂ C2+k , where P is a nonconstant holomorphic polynomial on Ck , k ∈ N, the tangent space of X at any regular point is spanned by finitely many algebraic vector fields with algebraic flows φti , t ∈ C. Such vector fields are called

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locally nilpotent derivations. The algebraic spray σ : X × CN → X (1.163) obtained by composing these flows is then dominating. The hyperquadrics in (1.15.1) can be 2 by a linear change of coordinates on written in the form uv = P(x) = x12 + · · · + xn−2 n C (see (4.18), (4.19)), so the results of Kaliman and Zaidenberg apply. However, it is easy to find explicit algebraic vector fields with algebraic flows on Cn (i.e., locally nilpotent derivations) which dominate the regular locus of every fibre of the quadratic polynomial h(z1 , . . . , zn ) = z21 + z22 + · · · + z2n .

(1.164)

The level sets of h are the hyperquadrics Σc (c ∈ C) in (1.159), with Σ0 being the null quadric A in (1.116). Proposition 1.15.3. Let n ≥ 3. There exist finitely many linear vector fields V i on Cn , i = 1, . . . , N, with quadratic polynomial flows φ i : Cn × C → Cn such that V i (h) = 0 and for each z ∈ Cn \ {0} the vectors V i (z) for i = 1, . . . , N span the tangent space of the level set {h = h(z)} at z. In particular, the manifolds h−1 (c) = Σc (c ∈ C∗ ) and h−1 (0) \ {0} = A∗ are algebraically elliptic. The first statement implies that the algebraic spray σ : Cn × CN → Cn in (1.163) obtained by composing the flows φ i is dominating on each fibre h−1 (c) = Σc (c = 0) and on h−1 (0) \ {0} = A∗ . Hence, each of these manifolds is algebraically elliptic. Proof. We begin with the case n = 3. Write z = (z1 , z2 , z3 ) ∈ C3 . It is easy to verify that the following three vector fields on C3 satisfy the stated conditions: V 1 (z) = (−z3 , iz3 , z1 − iz2 ), V 2 (z) = (z3 , iz3 , −z1 − iz2 ), V 3 (z) = (z3 − iz2 , iz1 , −z1 ). Their respective flows are given by

t2 t2 φ 1 (z,t) = z1 − tz3 + (iz2 − z1 ), z2 + itz3 + (iz1 + z2 ), z3 + t(z1 − iz2 ) , 2 2

2 2 t t φ 2 (z,t) = z1 + tz3 − (z1 + iz2 ), z2 + itz3 + (z2 − iz1 ), z3 − t(z1 + iz2 ) , 2 2

t2 3 2 2 φ (z,t) = z1 + (t + t )(z3 − iz2 ), z2 + itz1 + t (z2 + iz3 ), z3 − tz1 + (iz2 − z3 ) . 2 If n > 3, we can use such vector fields in any set of three variables, keeping the other variables fixed. Furthermore, one can do the same in any linear coordinate system on Cn obtained by a complex orthogonal rotation in On (C); such rotations preserve the bilinear form (z, w) → ∑nj=1 z j w j , and hence the polynomial h in (1.164). By adding more vector fields to the collection, we can inductively reduce the dimension of the algebraic set in Cn where a given family fails to span the tangent space of the fibre of h, thereby showing that finitely many vector fields of this type satisfy the proposition. We leave the details to the reader.  

Chapter 2

Basics on Minimal Surfaces

In this chapter we present the basic notions of the theory of minimal surfaces in Euclidean spaces Rn and holomorphic null curves in Cn . We focus on those parts of the theory which are most relevant to this book. In Section 2.1 we recall the notion of curvature of an immersed surface in Rn . In Section 2.2 we derive Lagrange’s formulas for the first and the second variation of the area. These formulas provide a characterization of minimal surfaces as immersed surfaces with vanishing mean curvature vector field. In Section 2.3 we show that a smooth conformal immersion x : M → Rn of a Riemann surface M into Rn parameterizes a minimal surface in Rn if and only if x is a harmonic map. This leads to the classical Enneper–Weierstrass representation formula, which provides the key connection between conformal minimal immersions M → Rn and holomorphic maps with vanishing real periods from M into the punctured null quadric in Cn . This brings complex analysis into the picture in a natural way, and is the basis for subsequent developments. In Section 2.4 we show how to treat nonorientable minimal surfaces in Rn by complex analytic methods applied to their orientable double covers. In Section 2.5 we introduce the generalized Gauss map of a minimal surface. This topic of major interest is treated in more detail in Chapter 5 where we show in particular that every meromorphic function on an open Riemann surface M is the Gauss map of a conformal minimal immersion M → R3 . In Section 2.6 we consider the Gaussian curvature of a minimal surface, and we begin the discussion of complete minimal surfaces of finite total curvature. These are among the simplest minimal surfaces whose key feature is that their Weierstrass data are given by meromorphic functions on a finitely punctured compact Riemann surface. This topic is continued and expanded in Chapter 4. In Section 2.7 we recall some basic instances of the maximum principles and briefly discuss the isoperimetric inequality for minimal surfaces in Rn . The former topic is expanded in Chapters 8 and 9; see in particular Sections 8.6 and 9.2. We conclude the chapter with a survey of some classical and a few recent examples of minimal surfaces in Section 2.8.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_2

83

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2.1 Curvature of Surfaces In this section we recall the notion of curvature of a smoothly immersed surface in Rn for any n ≥ 3, following the classical approach as in the books by M. P. do Carmo [109] and R. Osserman [282]. The geometric meaning of curvature is first explained in a coordinate-free manner on an embedded surface S ⊂ Rn , followed by computations in local coordinates. We also explain the connection with the Gauss map of a surface and its derivative, the Weingarten map. We begin by recalling the notion of curvature of a curve. Assume that α(s) is an immersed C 2 path in Rn parameterized by arc length (1.88). Its velocity vector T (s) = α  (s) has unit length for each s. Differentiating the identity T (s) · T (s) = 1 gives T  (s) · T (s) = 0, which means that the curvature vector T  (s) = α  (s) is orthogonal to the tangent vector T (s) for each value of s. The number κ(s) := |T  (s)| = |α  (s)| ∈ [0, ∞) is the curvature of α at s. When κ(s) = 0, we can write α  (s) = κ(s)ν(s)

(2.1)

where ν(s) ∈ Rn is a unit vector, called the principal normal vector of α at s. The reciprocal value 1/κ(s) ∈ (0, +∞] is the radius of curvature of α at s. This is the radius of the circle in the affine 2-plane Σ ⊂ Rn through the point α(s), spanned by the vectors T (s) and ν(s), which provides the best quadratic approximation of α at the point α(s). If the curvature κ(s) vanishes identically, then by integrating the equation T  (s) = 0 we see that α is linear and its image lies in an affine line. Assume now that x : M → Rn (n ≥ 3) is an immersed surface of class C 2 . All calculations will be of local nature, and we shall work on a contractible chart U ⊂ M with coordinates u = (u1 , u2 ) such that the restriction x|U : U → Rn is an embedding onto the embedded oriented surface S = x(U) ⊂ Rn . The tangent plane Tp S is a subspace of Tp Rn ∼ = Rn . Its orthogonal complement N p S = (Tp S)⊥ ⊂ Tp Rn

(2.2)

is a vector subspace of dimension n − 2 in Tp Rn , called the normal space of S at p; its elements are normal vectors to S at p. A normal vector field along S is a section N : S → T Rn |S of the restricted tangent bundle T Rn |S := p∈S Tp Rn ∼ = S × Rn such that N(p) ∈ N p S for all p ∈ S. Such N is a unit normal vector field if in addition |N(p)| = 1 for all p ∈ S. By using the parameterization p = x(u1 , u2 ) and writing N(p) = N(x(u1 , u2 )) = N(u1 , u2 ),

(2.3)

we may consider N as a map N : U → Rn such that N(q) is normal to Tx(q) S = dxq (Tq M) for each point q = (u1 , u2 ) ∈ U. (Strictly speaking this is a different map, but the obvious abuse of notation should not cause any difficulties.)

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Fix a point p = x(q) ∈ S with q ∈ U and consider curvatures of embedded curves of class C 2 in S passing through p. Every such curve is of the form α(t) = x(u1 (t), u2 (t)), where t → (u1 (t), u2 (t)) is an embedded C 2 curve in U with (u1 (t0 ), u2 (t0 )) = q for some t0 . Let s = s(t) denote the arc length on α (see (1.88)). The chain rule gives ∂ x dui , i=1 ∂ ui ds 2

α  (s) = ∑

2 ∂ x d 2 ui ∂ 2 x dui du j . + ∑ 2 i=1 ∂ ui ds i, j=1 ∂ ui ∂ u j ds ds 2

α  (s) = ∑

(2.4)

Assume that α(s0 ) = p ∈ S and write v = v(s0 ) = α  (s0 ) ∈ Tp S. Since the vector fields ∂∂uxi = xui (i = 1, 2) are tangent to S, taking the scalar product of the second equation in (2.4) with a unit normal vector N ∈ N p S gives κ N (p, v) := α  (s0 ) · N =

2

∑

i, j=1

hNi,j (p)

du j dui (s0 ) (s0 ), ds ds

(2.5)

where p = x(q) and the coefficients hNi,j (p) are given by hNi,j (p) =

∂ 2x (q) · N = xui u j (q) · N. ∂ ui ∂ u j

(2.6)

The number κ N (p, v) ∈ R is called the normal curvature of the surface S at the point p in the tangent direction v = α  (s0 ) ∈ Tp S and the unit normal direction N. We see from (2.5) and (2.6) that κ N (p, v) only depends on the tangent vector v of the curve and the normal vector N, so the notation is justified. If κ(s0 ) = |α  (s0 )| = 0 then κ N (p, v) = α  (s0 ) · N = κ(s0 ) ν(s0 ) · N = κ(s0 ) cos θ , where θ ∈ [0, π] is the angle between the curvature vector α  (s0 ) = κ(s0 )ν(s0 ) (see (2.1)) and the unit normal vector N ∈ N p S. In particular, if θ = 0 or θ = π (that is to say, ν(s0 ) = ± N) then κ N (p, v) = ±|α  (s0 )|. We say that the curve α is N-normal if this equality holds, including the case α  (s0 ) = 0. If n = 3 then such a curve is obtained for instance by intersecting S with the affine 2-plane Σ at p spanned by the vectors v and N. Let us express the normal curvature κ N (p, v) (2.5) in terms of the derivatives of u(t) = (u1 (t), u2 (t)) with respect to the original variable t; these will be denoted by a dot over the letter. We have dui /ds = u˙i /s˙ and s˙2 = |x| ˙ 2 = g1,1 u˙21 + 2g1,2 u˙1 u˙2 + g2,2 u˙22 , where G = (gi, j ) is the matrix of g = x∗ ds2 at q. Inserting into (2.5) gives at t = t0 :

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κ N (p, v) =

∑2i, j=1 hNi,j (p)u˙i u˙ j ∑2i, j=1 gi, j u˙i u˙ j

=

∑2i, j=1 xui u j u˙i u˙ j · N ∑2i, j=1 gi, j u˙i u˙ j

.

(2.7)

The numerator on the right-hand side of (2.7) is called the second fundamental form of S at p in the normal direction N ∈ N p S. Since the expression (2.7) is clearly independent of the norm of the tangent vector 0 = v ∈ Tp S, we can extend the definition of the curvature to all nonzero vectors by v κ N (p, v) = κ N p, , 0 = v ∈ Tp S. |v| For a fixed point p ∈ S, the continuous function v → κ N (p, v) ∈ R achieves its minimum and maximum on the unit circle {v ∈ Tp S : |v| = 1}: κ1N (p) := min κ N (p, v), |v|=1

κ2N (p) := max κ N (p, v). |v|=1

(2.8)

These numbers are the principal curvatures of the surface S at the point p in the normal direction N ∈ N p S. Their average H N (p) =

κ1N (p) + κ2N (p) 2

(2.9)

is the mean curvature of S at p in the normal direction N, and their product K N (p) = κ1N (p)κ2N (p) is the Gaussian curvature of S at p in the normal direction N. Let   F N (p) = hNi,j (p)

(2.10)

(2.11)

denote the matrix (2.6) of the second fundamental form at the point p = x(q) ∈ S in the normal direction N ∈ Np S, and let G = (gi, j ) be the matrix (1.73) of the Riemannian metric g = x∗ ds2 at the point q ∈ U in the basis ∂∂u , ∂∂u . Since the 1 2 normal curvature κ N (p, v) (2.7) is the quotient of the quadratic forms determined by F N (p) and G, its extremal values (2.8) are roots of the equation   det F N (p) − λ G = 0. Expanding the determinant gives   (det G)λ 2 − g2,2 hN1,1 + g1,1 hN2,2 − 2g1,2 hN1,2 λ + det F N (p) = 0. From the Vieta formulas for the sum and the product of roots we find that   H N (p) = (2 det G)−1 g2,2 hN1,1 + g1,1 hN2,2 − 2g1,2 hN1,2 = (2 det G)−1 (g2,2 xu1 u1 + g1,1 xu2 u2 − 2g1,2 xu1 u2 ) · N

(2.12)

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and K N (p) =

N N N 2 det F N (p) h1,1 h2,2 − (h1,2 ) = . det G g1,1 g2,2 − (g1,2 )2

(2.13)

Note that the vector (2 det G)−1 (g2,2 xu1 u1 + g1,1 xu2 u2 − 2g1,2 xu1 u2 ) in (2.12) does not depend on the choice of orientation on U. Its projection H ∈ N p S to the normal space N p S in the direction of the tangent space Tp S is called the mean curvature vector of the surface S at the point p. The formula (2.12) then says that H N (p) = H · N

for all N ∈ N p S.

(2.14)

By inserting p = x(q) for q ∈ U ⊂ M, these quantities become functions on U depending on the choice of a unit normal vector field N : U → Rn to x|U . Remark 2.1.1. In dimension n = 3 there is locally only one unit normal vector field N to the immersion x : U → R3 which is compatible with the given orientation on U ⊂ M. This normal vector field is called the Gauss map of x; see (2.18) and (2.19). Since the mean curvature vector field H is orthogonal to x, we have that H = λ N for a scalar-valued function λ , and we see from (2.14) that λ = H N . Thus, H = H· N

when n = 3,

where H = H N is the mean curvature function and N is the Gauss map of x. In this case, the function K N = K given by (2.10) is the classical Gaussian curvature of the Riemannian surface (M, g = x∗ ds2 ). Note that the sign of the mean curvature H N depends on the sign of N, but the Gaussian curvature K does not.   The formula (2.12) for the mean curvature simplifies in isothermal coordinates (u1 , u2 ) for the metric g = x∗ ds2 (see Theorem 1.8.6). In this case, G = (gi, j ) = μI and det G = μ 2 . By (2.12) we have that   hN1,1 (p) + hN2,2 (p) xu1 u1 + xu2 u2 · N Δ0 x · N 1 N = = = Δ x · N, (2.15) H (p) = 2μ 2μ 2μ 2 where Δ x is the g-Laplacian of x (1.96) evaluated at the point q ∈ U with x(q) = p and the last equality holds by (1.114). Lemma 2.1.2. If x : M → Rn is an immersion of class C 2 then Δ x = 2H,

(2.16)

where Δ is the intrinsic Laplacian (1.96) of the metric g = x∗ ds2 at a point q ∈ M and H is the mean curvature vector (2.14) of the surface S = x(M) at the point p = x(q). In particular, (Δ x)(q) is orthogonal to the plane dxq (Tq M) for every q ∈ M. Proof. It suffices to prove the statement in last sentence of the lemma. Indeed, if this holds, we see from (2.15) that the vector 12 Δ x fits the definition (2.14) of the mean curvature vector H, so by uniqueness the formula (2.16) follows.

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Let (u1 , u2 ) be isothermal coordinate for the metric g = x∗ ds2 on an open set U ⊂ M. In these coordinates the immersion x is conformal, so (1.115) implies xu1 · xu1 = xu2 · xu2 ,

xu1 · xu2 = 0.

Differentiating the first identity on u1 and the second one on u2 yields xu1 u1 · xu1 = xu1 u2 · xu2 = −xu2 u2 · xu1 , whence Δ0 x · xu1 = 0. Likewise, differentiating the first identity on u2 and the second one on u1 gives Δ0 x · xu2 = 0. Since the vectors xu1 (q) and xu2 (q) span  dxq (Tq M) = Tp S, we see that Δ x = μ1 Δ0 x is orthogonal to S.  Remark 2.1.3. In an isothermal coordinate ζ = u1 + iu2 the formula (2.16) is equivalent to the following one, as seen from (2.15): Δ 0 x = xu1 u1 + xu2 u2 = 2μH,

μ = |xu1 |2 = |xu2 |2 > 0.

(2.17)

See also [109, Proposition 2, p. 201] and [282, Lemma 4.1].   The Gauss map and the Weingarten map. We give another point of view on the curvature of an immersed surface in Rn , based on the variation of its Gauss map. We begin by considering the case n = 3. Let S ⊂ R3 be an embedded orientable surface. The normal bundle to S is a trivial line bundle, and S admits precisely two unit normal vector fields ± N; a choice of N determines an orientation on S. We may consider N as a map   (2.18) N : S −→ S2 = (x1 , x2 , x3 ) ∈ R3 : x12 + x22 + x32 = 1 into the unit 2-sphere. This is called the Gauss map of the surface S ⊂ R3 . In terms of a cooriented local parameterization x = x(u1 , u2 ) we have that N(u1 , u2 ) =

xu1 × xu2 , |xu1 × xu2 |

(2.19)

where v × w denotes the vector product in R3 . We are interested in the rate of change of the tangent plane Tp S as the point p ∈ S varies. Equivalently, we consider the variation of the Gauss map N : S → S2 (2.18). The differential of N at p is a linear map dN p : Tp S → TN(p) S2 . Since the 2-plane TN(p) S2 ⊂ R3 is the orthogonal complement of the vector N(p), it is parallel to Tp S, so we may consider the differential of N at p as a linear map dN p : Tp S −→ Tp S,

p ∈ S.

(2.20)

Its negative −dN p is called the Weingarten map or the shape operator of the surface S at the point p ∈ S. We now show that the Weingarten map is self-adjoint and its eigenvalues are the principal curvatures of the surface in the given normal direction N.

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89

Lemma 2.1.4. The map dN p is self-adjoint, i.e., for any pair v1 , v2 ∈ Tp S we have dN p (v1 ) · v2 = v1 · dN p (v2 ).

(2.21)

Furthermore, for every unit vector v ∈ Tp S the curvature κ N (p, v) (2.5) equals κ N (p, v) = −dN p (v) · v.

(2.22)

In particular, the principal curvatures (2.8) are the eigenvalues of −dN p and the corresponding eigenvectors are orthogonal. Proof. We may assume that the vectors v1 , v2 in (2.21) are linearly independent. Pick a local parameterization x(u1 , u2 ) of S near p such that x(0, 0) = p and

∂x (0, 0) = vi for i = 1, 2. ∂ ui

(2.23)

Using the notation N(u1 , u2 ) as in (2.3), we write xui =

∂x , ∂ ui

Nui =

∂N , ∂ ui

xui u j =

∂ 2x , ∂ ui ∂ u j

i, j = 1, 2.

(2.24)

The chain rule gives dN p (vi ) = dN p (xui (0, 0)) = Nui (0, 0),

i = 1, 2.

(2.25)

Since the vector field N is normal to S while xui is tangential to S, we have N · xu1 = 0 and N · xu2 = 0. Differentiation gives the equations −Nu1 · xu1 −Nu1 · xu2 −Nu2 · xu1 −Nu2 · xu2

= = = =

N · xu1 u1 N · xu2 u1 N · xu1 u2 N · xu2 u2

= h1,1 , = h2,1 , = h1,2 = h2,1 , = h2,2 ,

(2.26)

where (hi, j ) is the matrix of the second fundamental form (2.6) in the normal direction N. In particular, Nu1 · xu2 = Nu2 · xu1 . At the origin (u1 , u2 ) = (0, 0) we get in view of (2.23) and (2.25) the equation (2.21). Given a tangent vector 0 = v ∈ Tp S, write v = ξ1 v1 + ξ2 v2 , where v1 , v2 are given by (2.23) and ξ1 , ξ2 ∈ R. In view of (2.26) we obtain the following equation: −dN p (v) · v = − (ξ1 Nu1 + ξ2 Nu2 ) · (ξ1 xu1 + ξ2 xu2 ) = h1,1 ξ12 + 2h1,2 ξ1 ξ2 + h2,2 ξ22 . Let (gi, j ) be the matrix of the metric g = x∗ ds2 at the point (0, 0). Dividing the above equation by |v|2 = g1,1 ξ12 + 2g1,2 ξ1 ξ2 + g2,2 ξ22 and taking into account (2.7) gives −dN p (v) · v h1,1 ξ12 + 2h1,2 ξ1 ξ2 + h2,2 ξ22 = = κ N (p, v). |v|2 g1,1 ξ12 + 2g1,2 ξ1 ξ2 + g2,2 ξ22

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This implies (2.22) for every unit vector v ∈ Tp S.   This can be extended to any dimension n ≥ 3. Let π p : Rn = Tp Rn → Tp S denote the orthogonal projection whose kernel is the normal space to S at p. Pick a unit normal vector field N along S. Let dN pT = π p ◦ dN p : Tp S → Tp S

(2.27)

denote the orthogonal projection of the differential dN p to Tp S. We see as before that dN pT is self-adjoint. Let x : M → S ⊂ Rn be a smooth immersion parameterizing S, and let q ∈ M be a point with x(q) = p. Then there is a unique (self-adjoint) linear map ANq : Tq M → Tq M such that dxq ◦ ANq = −dN pT ◦ dxq : Tq M → Tp S.

(2.28)

As in the case n = 3, ANq (and −dN pT ) is called the Weingarten map associated to the normal vector field N. For any g-unit vector w ∈ Tq M (where g = x∗ ds2 ) the vector v = dxq (w) ∈ Tp S is a unit vector in the Euclidean norm, and by (2.28)   gq ANq w, w = (dxq ◦ ANq w) · dxq (w) = −dNpT v · v = κ N (p, v), (2.29) where κ N (p, v) is the normal curvature defined by (2.6). If F N (p) = (hNi,j (p)) is the matrix (2.11) of the second fundamental form and (gi, j ), (ai, j ) are the matrices of the metric gq and of the Weingarten map ANq , respectively, in the basis ∂∂u , ∂∂u , it follows from (2.29) that 1

2

F N (p) = (hNi,j (p)) = (gi, j ) · (ai, j ).

(2.30)

Gaussian curvature of a surface. It is a remarkable discovery of C. F. Gauss that the Gaussian curvature of a surface can be expressed solely in terms of its first fundamental form g. This is Gauss’s Theorema Egregium (the Remarkable Theorem). The most general formula of this type is the Brioschi formula. We recall the following special case in which the metric g is orthogonal, g = Adu2 + Bdv2 :      ∂ B A 1 ∂ √u √v Kg = − √ + . (2.31) ∂v 2 AB ∂ u AB AB (See [294, Corollary 10.2, p. 234].) By Theorem 1.8.6, any smooth Riemannian metric g on a surface admits local isothermal coordinates z = x + iy in which g = μ(dx2 + dy2 ) = μ|dz|2 . (Such a metric g is a K¨ahler metric, see Definition 1.7.1.) By (2.31) the Gaussian curvature of g equals

2.1 Curvature of Surfaces

Kg = − where Δ 0 =



1 2μ ∂2 ∂ u2

91

∂ μu ∂ μv + ∂u μ ∂v μ



=−

1 1 Δ 0 log μ = − Δ log μ, 2μ 2

(2.32)

+ ∂∂v2 and Δ = μ1 Δ 0 is the g-Laplacian (1.96). 2

Note that any K¨ahler metric of the form g = | f (z)|2 |dz|2 , where f is a nonvanishing holomorphic function, has vanishing Gaussian curvature. Example 2.1.5. The Poincar´e metric on the disc D = {z ∈ C : |z| < 1} is given by gP =

4|dz|2 . (1 − |z|2 )2

(2.33)

It is a complete K¨ahler metric with constant Gaussian curvature K = −1. The latter 4 is seen by the following calculation. Set μ(z) = (1−|z| 2 )2 . Then:  ∂2  log 4 − 2 log(1 − |z|2 ) ∂ z∂ z¯ 2(1 − |z|2 ) − 2z(−¯z) 2z ∂ = 4 =4 ∂ z 1 − |z|2 (1 − |z|2 )2 8 = (1 − |z|2 )2 = 2μ(z).

Δ0 log μ(z) = 4

Hence by (2.32) we have K=−

Δ0 log μ = −1. 2μ

The classical Schwarz–Pick lemma (see S. Kobayashi [208, p. 2]) says that any holomorphic map f : D → D satisfies the inequality | f  (z)| 1 ≤ , 1 − | f (z)|2 1 − |z|2

z ∈ D.

(2.34)

If the equality holds at some point of D, then it holds at all points and f is a holomorphic automorphism of the disc. Recall that the group Aut(D) consists of M¨obius transformations:   z−a : a ∈ D, t ∈ R . Aut(D) = z → eit 1 − az ¯ An equivalent interpretation of the Schwarz–Pick lemma is that holomorphic maps D → D are distance-decreasing in the Poincar´e metric, and orientation-preserving isometries are precisely holomorphic automorphisms of the disc. The Poincar´e disc (D, gP ) of constant Gaussian curvature −1 is the model space for the theory of Kobayashi hyperbolicity of complex manifolds; see S. Kobayashi [208]. The Schwarz–Pick type lemmas play an important role in the study of the Gauss map of minimal surfaces; see Sections 5.5 and 5.6.  

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2.2 Variation of Area and Minimal Surfaces Assume that M is a smooth compact surface with boundary and x : M → Rn (n ≥ 3) is a C 2 immersion. A smooth 1-parameter family of C 2 maps x t : M → Rn ,

t ∈ (−ε, ε) ⊂ R

(2.35)

is called a variation of x with fixed boundary if x0 = x and x t = x on bM for all t. (It will suffice to assume that x t is twice differentiable in t and its first and second order t-derivatives have two continuous spatial derivatives.) The vector field E(p,t) = ∂ x t (p)/∂t ∈ Rn ,

p∈M

(2.36)

is called the variational vector field of x t . For a fixed p ∈ M this is simply the velocity vector field of the path t → x t (p) along its trajectory. Note that E = 0 on bM × (−ε, ε). For each t close enough to 0 the map x t is an immersion. We are interested in the variation of area of the surfaces S t = x t (M) ⊂ Rn with fixed boundaries bS t = x(bM). (Our assumptions imply that the area is twice continuously differentiable on t.) We recall the following classical definition. Definition 2.2.1 (Minimal surfaces). A C 2 immersed surface x : M → Rn is a minimal surface if for every compact domain D ⊂ M with smooth boundary and any smooth variation x t of x fixing bD, the first variation of area at t = 0 vanishes:  d  Area(x t (D)) = 0. (2.37) dt t=0 In other words, minimal surfaces are stationary points of the area functional. The same definition applies to immersed submanifolds of any dimension m in an ¯ replacing the area by m-dimensional arbitrary smooth Riemannian manifold (M, g), volume associated to the metric g¯ (see e.g. H. B. Lawson [216]). We wish to relate the first variation of area (2.37) to geometric quantities related to the immersed surface x : M → Rn . The answer is given by the following classical first variational formula for the area, due to Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia, also known as Giuseppe Ludovico De la Grange Tournier). Theorem 2.2.2 (First variation of area). Assume that M is a smooth compact surface with boundary bM and x : M → Rn (n ≥ 3) is a C 2 immersion. For any smooth variation x t : M → Rn of x0 = x which is fixed on bM we have that   d  t Area(x (M)) = −2 E· H dA, (2.38) dt t=0 M where E = ∂ x t /∂t|t=0 is the variational vector field (2.36) at t = 0, H is the mean curvature vector field (2.14) of the immersion x, and dA is the element of the surface area with respect to the Riemannian metric x∗ ds2 on M.

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We also obtain a formula for the second variation of area; see (2.50). Similar formulas hold for the variation of volume of an immersed compact submanifold with ¯ (cf. boundary of any dimension in an arbitrary smooth Riemannian manifold (M, g) (9.26)); see e.g. H. B. Lawson [216, Theorem 4, p. 7]. The proof given here for the special case of surfaces in Euclidean spaces is simpler and does not require the full machinery of Riemannian geometry. Before proving Theorem 2.2.2 we discuss some consequences. Assume for simplicity that x is a smooth immersion (of class at least C 4 ), so that H is a vector field of class C 2 . By taking a smooth function f : M → R+ vanishing on bM and considering the C 2 variation x t = x + t f H we get   d  t Area(x (M)) = −2 f |H|2 dA ≤ 0. dt t=0 M If H is not identically zero, then by choosing f to be positive at some point where H = 0 yields a negative number. Hence, deforming the surface x(M) ⊂ Rn in the direction of the mean curvature vector field H, the area strictly decreases for small t > 0. More precisely (cf. Lawson [216, p. 5] and note that his definition of the mean curvature vector field differs from ours by a factor of 2): The mean curvature vector field is the (negative) gradient of the area function on the space of immersions M → Rn with fixed boundary bM → Rn . The formula (2.38) implies the following result of Meusnier from 1776. Theorem 2.2.3 (Meusnier’s theorem). A C 2 immersed surface x : M → Rn is a minimal surface if and only if its mean curvature vector field H vanishes identically. Proof. If H = 0 on M then x is minimal, as seen directly from (2.38). Conversely, assume that H(p0 ) = 0 for some p0 ∈ M. Let D ⊂ M be a small compact disc around p0 and E : M → Rn be a smooth vector field supported on D such that E· H ≥ 0 and (E· H)(p0 ) > 0. Then, (2.38) shows that the variation x t = x + tE strictly decreases  the area of x t (M) for small t > 0, so x is not minimal.  Corollary 2.2.4. A C 2 immersed surface x : M → Rn (n ≥ 3) is a minimal surface if and only if x is harmonic in the Riemannian metric g = x∗ ds2 . In particular, a smooth conformal immersion x : M → Rn from a Riemann surface M is minimal if and only if dd c x = 0. Proof. By (2.16) we have Δ x = 2H, where Δ is the g-Laplacian. The first conclusion then follows from Theorem 2.2.3, and the second one is seen from (2.17), which relates the metric Laplacian to dd c in isothermal coordinates.   Proof of Theorem 2.2.2. We begin by making a conceptual simplification which reduces the proof to the case of normal variations. Assume for simplicity that x is an embedding; the same proof will apply to immersions. At every point q in the embedded surface S = x(M) ⊂ Rn the tangent

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space is the orthogonal direct sum Tq Rn = Tq S ⊕ Nq S of the tangent and the normal space. We identify Nq S with a linear subspace of Rn and set Λq = q+Nq S ⊂ Rn . Pick a small neighbourhood U ⊂ Rn of S such that the sets in the family {Λq ∩U : q ∈ S} are pairwise disjoint. Let V = q∈S Λq ∩U and denote by π : V → S the projection with fibres π −1 (q) = Λq ∩U for q ∈ S. Consider a smooth variation x t : M → Rn of x which is fixed on bM. For small t we have x t (M) ⊂ V and π(x t (M)) ⊂ S. We split x t as x t (p) = π(x t (p)) + y t (p) ∈ Λπ(x t (p)) , p ∈ M, (2.39) where y t (p) ∈ Nπ(x t (p)) . Note that π(x t (p)) = x(p) for all p ∈ bM and all t. For small t, the map h t = x−1 ◦ π ◦ x t : M → M is a small perturbation of the identity map h0 = IdM which is fixed on bM, hence a diffeomorphism. Thus, π ◦ x t = x ◦ h t . (If x is an immersion, it is locally an embedding, and the same argument applied locally still gives a family of diffeomorphisms h t : M → M satisfying π ◦ x t = x ◦ h t .) Precomposing (2.39) by the inverse (h t )−1 : M → M gives a normal deformation X t := x t ◦ (h t )−1 = x + y t ◦ (h t )−1 = x +Y t of X 0 = x such that x t (M) = X t (M) and X t (p) ∈ Λx(p) ∩ U for each p ∈ M and all small values of t ∈ R. In particular, all t-derivatives of X t at t = 0 are orthogonal to x. We call such X t a normal variation. The variation X t is called normal to order k ∈ N if the variational vector field ∂ X t /∂t vanishes to order k at t = 0. It follows that S = x(M) is a minimal surface if and only if the variation of area (2.37) vanishes for all normal variations. Furthermore, since the variational vector field ∂ X t /∂t|t=0 = ∂Y t /∂t|t=0 of X t is the normal component of the variational vector field E = ∂ x t /∂t|t=0 and the mean curvature vector field H is orthogonal to S, the variational formula (2.38) holds for x t if and only if it holds for X t (since the tangential component of E disappears when multiplied by H). Assume now that x t : M → Rn is a normal variation of x 0 = x : M → Rn with the variational vector field ∂ x t /∂t (2.36). Set E=

∂ x t   , ∂t t=0

E2 =

∂ 2 x t   . ∂t 2 t=0

(2.40)

We have the following second order Taylor expansion of x t with respect to t: 1 x t = x + tE + t 2 E 2 + o(t 2 ). 2

(2.41)

Let (u1 , u2 ) be isothermal coordinates for x on a local chart U ⊂ M (see Theorem 1.8.6). Denote by G t = (gi,t j ) the matrix of the Riemannian metric g t = (x t )∗ ds2 , with g0 = g := x∗ ds2 and G 0 = G = μ(δi, j ), where μ = |xu1 |2 = |xu2 |2 > 0 on U and δi, j is the Kronecker delta. Differentiating (2.41) on ui gives 1 xut i = xui + tEui + t 2 Eu2i + o(t 2 ), 2

i = 1, 2.

2.2 Variation of Area and Minimal Surfaces

95

Noting that gi,t j = xut i · xut j and gi, j = xui · xu j = μδi, j we obtain    1 gi,t j = μδi, j + t Eui · xu j + Eu j · xui + t 2 Eui · Eu j + Eu2i · xu j + Eu2j · xui + o(t 2 ). 2 Therefore, det G t = μ 2 + 2tμ (Eu1 · xu1 + Eu2 · xu2 ) + c2t 2 + o(t 2 )

(2.42)

where 2   c2 = − (Eu1 · xu2 + Eu2 · xu1 )2 + 4(Eu1 · xu1 )(Eu2 · xu2 ) + μ ∑ |Eui |2 + Eu2i · xui . i=1

(2.43) The coefficient c2 will be used in the proof of the second variational formula. Consider the first order term in (2.42). Since the vector field E is normal to x, we have E · xui = 0 and hence Eui · xui = −E· xui ui . Together with the formula (2.17) relating the Laplacian of x to the mean curvature vector field H we obtain Eu1 · xu1 + Eu2 · xu2 = −E· (xu1 u1 + xu2 u2 ) = −2μE· H. Using this formula we get from (2.42) that √   det G t = μ 1 − 2tE· H + O(t 2 ). Since the area measure on M defined by the metric g t = (x t )∗ ds2 has local √ expression dAt = det G t du1 du2 (see (1.91)), this gives   dAt = 1 − 2tE· H + O(t 2 ) dA, (2.44) which provides the first variational formula (2.38).   Remark 2.2.5. For an arbitrary variation x t , we split the variational vector field E (2.40) into the tangential and the normal component to x = x0 , E = E T + E ⊥ . By (1.94) the divergence of E T in isothermal coordinates (u1 , u2 ) equals div(E T ) =

 1 T Eu1 · xu1 + EuT2 · xu2 . μ

(2.45)

The above calculation then leads to the formula

  dAt = 1 + div(E T ) − 2E· H t + O(t 2 ) dA. t is fixed on bM, the vector field E vanishes on bM and we have that Since x T M div(E )dA = 0 by the divergence theorem (see Theorem 1.6.7), so we end up

with the same variational formula (2.38).  

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We now consider the second variation of area. Assume that M is a compact surface with boundary and x : M → Rn is a smooth minimal immersion. As shown in the proof of Theorem 2.2.2, it suffices to consider normal variations x t : M → Rn of x0 = x which are fixed on bM, so we may assume that the vector fields E and E 2 defined by (2.40) are normal to x. Take isothermal coordinates (u1 , u2 ) on a subset U ⊂ M. Let us simplify the formula (2.43) for c2 . A computation gives 2   2 E · x = u ∑ ui i ∑ E 2 · xui u − E 2 · (xu1 u1 + xu2 u2 ). 2

i=1

i=1

i

(2.46)

Since E 2 · xui = 0 and xu1 u1 + xu2 u2 = μΔ x = 0 as x is minimal, we obtain 2

∑

i=1



2  |Eui |2 + Eu2i · xui = ∑ |Eui |2 = μ|∇E|2 .

(2.47)

i=1

Set N = E/|E| on M ∗ = {p ∈ M : E(p) = 0} and denote by H E := H N and K E := K N the corresponding mean and Gaussian curvature, respectively. Taking into account the formulas (2.11), (2.12), and (2.13) we obtain on U ∗ = U ∩ M ∗ that −(Eu1 · xu2 + Eu2 · xu1 )2 + 4(Eu1 · xu1 )(Eu2 · xu2 ) = 4(E· xu1 u1 )(E· xu2 u2 ) − 4(E· xu1 u2 )2 = 4|E|2 det F E = 4|E|2 μ 2 K E . From this and (2.47) we see that the coefficient c2 (2.43) is given on U ∗ by   (2.48) c2 = μ 2 4|E|2 K E + |∇E|2 , and we have c2 = μ 2 |∇E|2 on U \ U ∗ since the first term in c2 is continuous on U and it vanishes on U \U ∗ . Taking the square root of (2.42), this gives   √  t2  det G t = μ 1 + 4|E|2 K E + |∇E|2 + o(t 2 ) , 2 √ √ where det G t = μ = det G on U \U ∗ . It follows that the expression    t2  2 E 2 2 dAt = 1 + 4|E| K + |∇E| + o(t ) dA (2.49) 2 makes sense on M if we put 4|E|2 K E ≡ 0 on M \ M ∗ and dAt = dA on M \ M ∗ . This gives the following second variational formula for a minimal surface:     d 2  t 4|E|2 K E + |∇E|2 dA. (2.50) Area(x (M)) =  2 dt t=0 M Remark 2.2.6. Our calculation was simplified by assuming that the second order variational vector field E 2 is normal to the immersion x : M → Rn . In general we observe that the expression in (2.46) is nothing but ∑2i=1 Eu2i · xui = μdiv((E 2 )T )

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97

where (E 2 )T is the tangential part of E 2 (see (2.45)). This gives the extra term μ 2 div((E 2 )T ) in the coefficient c2 (2.48), and the term div((E 2 )T ) in the quadratic part of dAt (2.49). Since E 2 vanishes on bM, the divergence theorem shows that this term disappears in the second variational formula (2.50), just like what happens with div(E T ) in the first variational formula (see Remark 2.2.5).   If we write f = |E| and N = E/|E| on M ∗ = {p ∈ M : f (p) = 0}, then |∇E|2 = |∇ f |2 + f 2 |∇N|2 and the formula (2.50) assumes the form    N  d 2  t 2 2 2 4K dA. |∇ f | Area(x (M)) = + f + |∇N| dt 2 t=0 M Definition 2.2.7 (Stable minimal surfaces). An immersed minimal surface x : M → Rn is stable if for any smooth function f : M → R with compact support contained in the interior of M and unit normal field N to x defined on supp f we have    |∇ f |2 + f 2 4K N + |∇N|2 dA ≥ 0. (2.51) M

The minimal surface x is strictly stable if strict inequality holds in (2.51) whenever f is not identically zero. In other words, a minimal surface is (strictly) stable if it is a (strict) local minimum of the area functional. Remark 2.2.8. In R3 , the inequality (2.51) assumes the simpler form   M

 |∇ f |2 + 2K f 2 dA ≥ 0,

(2.52)

where K is the Gaussian curvature of the metric x∗ ds2 on M. To see this, note that there is locally a unique (up to sign) unit normal vector field N to x. Since x is a minimal surface, its mean curvature equals zero, so its principal curvatures at any point p ∈ M are ±λ for some λ ≥ 0, and the Gaussian curvature equals K p = −λ 2 . Lemma 2.1.4 shows that ±λ are the eigenvalues of the differential dN p , and hence  |∇N p |2 = 2λ 2 = −2K p . Inserting this into (2.51) gives (2.52).  Remark 2.2.9. If S ⊂ Rn is a smooth embedded hypersurface and d denotes the signed distance function to S, then the mean curvature of S at any point x ∈ S equals 1 1 2 Δ d(x), and the mean curvature vector is 2 Δ d(x)∇d(x) (see e.g. G. Bellettini [60, Sect. 1.2]). This applies in particular to surfaces in R3 . However, as we mainly work  with parameterized surfaces in Rn , these equations will be of limited interest.  We refer to H. B. Lawson [216, Theorem 4, p. 7] for the corresponding variational formulas in the more general situation of submanifolds in any Riemannian manifold. Variational formulas also exist for certain types of varifolds and currents in Riemannian manifolds, in particular, for rectifiable currents which amount to integration of differential forms over compact oriented submanifolds with sufficiently mild singularities; see H. B. Lawson and J. Simons [217, Theorem 1] or H. Federer [124, 5.1.8]. Currents will be considered in Chapter 9 in the study of minimal hulls of compact sets in Rn .

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2.3 The Enneper–Weierstrass Formula Let M be an orientable surface. We have seen in Section 1.8 that a C 2 immersion x : M → Rn determines on M a unique Riemann surface structure in the given orientation class which renders x conformal. Hence, there is no loss of generality in assuming that M is a Riemann surface and the immersion is conformal. There is a deeper reason for considering conformal parameterizations of minimal surfaces. Given a piecewise C 1 immersion x : D → Rn from the closed unit disc with coordinates (u1 , u2 ), one defines its Dirichlet (energy) integral by D(x) =

 D

|∇x|2 du1 du2 =

 

 |xu1 |2 + |xu2 |2 du1 du2 .

D

It is elementary to see that 2Area(x) ≤ D(x), with equality if and only if x is conformal (see H. B. Lawson [216, p. 61]). Consider a map x whose restriction to T = bD is a monotone parameterization x : T → Γ ⊂ Rn of a given smooth oriented Jordan curve. A map in this class minimizing the Dirichlet integral D(x) also minimizes the area and provides a conformally parameterized minimal surface with boundary Γ (see [216, pp. 60–63]). In other words, conformal parameterization gives a least energy spreading of the surface over a geometric configuration of least area in Rn . This is in analogy to minimization of the energy integral of curves in a Riemannian manifold which produces geodesics parameterized by arc length. We begin by summarizing equivalent conditions on a conformal immersion to parameterize a minimal surface. The resulting Enneper-Weierstrass formula for conformal minimal immersions into Rn and their complex analogues, holomorphic null curves in Cn (see Theorem 2.3.4), brings complex analysis into the picture in a natural way, and is the basis for most developments in this book. Theorem 2.3.1. Let M be an open Riemann surface and x = (x1 , . . . , xn ) : M → Rn (n ≥ 3) be a C 2 conformal immersion. The following conditions are equivalent: (a) x is a minimal surface (a stationary point of the area functional, see (2.38)). (b) x has vanishing mean curvature vector field: H = 0 (see (2.14)). (c) x is harmonic: Δ x = 0. Equivalently, dd c x = 0 (see (1.130)). (d) The Cn -valued (1, 0)-form ∂ x = (∂ x1 , . . . , ∂ xn ) is holomorphic and satisfies (∂ x1 )2 + · · · + (∂ xn )2 = 0.

(2.53)

(e) Let θ be a nowhere vanishing holomorphic 1-form on M (see Theorem 1.10.5). The map f = 2∂ x/θ : M → Cn is holomorphic and has range in the null quadric   A = (z1 , . . . , zn ) ∈ Cn : z21 + z22 + · · · + z2n = 0 . (2.54) The Riemannian metric on M induced by a conformal immersion x : M → Rn equals   (2.55) g = x∗ (ds2 ) = |dx12 | + · · · + |dxn |2 = 2 |∂ x1 |2 + · · · + |∂ xn |2 .

2.3 The Enneper–Weierstrass Formula

99

Proof. Conditions (a) and (b) are equivalent by Theorem 2.2.3, and (b) and (c) are equivalent in view of the formula Δ x = 2H (2.16), where Δ is the intrinsic Laplacian with respect to the metric g = x∗ ds2 and H is the mean curvature vector field of x. By Remark 2.1.3, Δ x = 0 is equivalent to dd c x = 0. By the equivalence (a)⇔(c) in Lemma 1.8.4, an immersion x : M → Rn is conformal if and only if, in any local holomorphic coordinate ζ = u + iv on M, the vectors xu ± i xv ∈ Cn belong to the null quadric (2.54). Clearly, this holds if and only if the Cn -valued (1, 0)-form ∂ x = 21 (xu − i xv )dζ satisfies the nullity condition (2.53). From dd c x = −2i∂ (∂ x) (cf. (1.129)) we see that dd c x = 0 if and only if ∂ x is a holomorphic 1-form. This proves (c)⇔(d), and (d)⇔(e) is obvious. The map f = 2∂ x/θ is well defined as a quotient of (1, 0)-forms, it assumes values in the null quadric A in view of (2.53), and its range avoids 0 since x is an immersion. Finally we show that (2.55) holds. Let ζ = u + iv be a holomorphic coordinate on M. Since x is conformal, we have xu · xv = 0 and |xu |2 = |xv |2 by Lemma 1.8.4. From 2∂ x = (xu − i xv )dζ we obtain 2|∂ x|2 =

 1 2 |xu | + |xv |2 |dζ |2 = |xu |2 |dζ |2 . 2

On the other hand, |dx|2 = |xu du + xv dv|2 = |xu |2 du2 + |xv |2 dv2 + 2xu · xv dudv = |xu |2 |dζ |2 , so (2.55) holds. (This last formula does not require that x be harmonic.)   Recall (see Section 1.10) that a harmonic map x : M → Rn admits a harmonic conjugate if and only if its conjugate differential d c x = i(∂ x − ∂ x) (see (1.126)) is an exact 1-form on M. If this holds then d c x = dy for any harmonic conjugate y : M → Rn , and z = x + iy : M → Cn is a holomorphic map, also called a complex curve in Cn . The obstruction to exactness of d c x is given by the flux of x. Definition 2.3.2. The flux of a harmonic map x : M → Rn is the group homomorphism Fluxx : H1 (M, Z) → Rn given by 

Fluxx ([C]) =

d c x, C

[C] ∈ H1 (M, Z).

(2.56)

Since d c x is a closed 1-form, the integral only depends on the homology class of a path, and hence we shall write Fluxx (C) for Fluxx ([C]) in the sequel. Thus, a harmonic map x admits a harmonic conjugate on M if and only if 

Fluxx (C) =

dcx = 0 C

for every closed curve C ⊂ M.

(2.57)

If γ : [0, L] → M is a piecewise smooth closed curve parameterized by arc length and ν(s) = 2ℑ (∂ x(γ  (s))) for s ∈ [0, L], then the formula (2.56) takes the form

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2 Basics on Minimal Surfaces



Fluxx (γ) = 2ℑ

γ

∂x =

 L 0

ν(s)ds.

(2.58)

If x : M → Rn is a conformal minimal immersion and y is a harmonic conjugate of x on an open subset U ⊂ M, then the holomorphic immersion z = x + iy : U → Cn is a holomorphic null curve according to the following definition. Definition 2.3.3. Let M be an open Riemann surface. A holomorphic immersion z = (z1 , . . . , zn ) : M → Cn (n ≥ 3) is a holomorphic null curve if it satisfies the nullity condition (∂ z1 )2 + (∂ z2 )2 + · · · + (∂ zn )2 = 0. The following result, which is an immediate consequence of Theorem 2.3.1 and of the above discussion, provides a very useful method for constructing conformal minimal immersions M → Rn and holomorphic null curves M → Cn . Theorem 2.3.4 (The Enneper–Weierstrass representation formula). Let n ≥ 3, and let Φ = (φ1 , . . . , φn ) be a nowhere vanishing Cn -valued holomorphic 1-form on an open Riemann surface M such that n

∑ φ j2 = 0

(2.59)

j=1

and



ℜ C

Φ = 0 for all [C] ∈ H1 (M, Z).

(2.60)

Then, for every pair of points p0 ∈ M and x0 ∈ Rn the well defined map x : M → Rn ,

 p

(p ∈ M)

(2.61)

1 g = x∗ (ds2 ) = |dx|2 = |Φ|2 . 2

(2.62)

x(p) = x0 + ℜ

p0

Φ

is a conformal minimal immersion satisfying 2∂ x = Φ

and

Every conformal minimal immersion M → Rn is of this kind. If in addition  C

Φ = 0 for all [C] ∈ H1 (M, Z),

(2.63)

then for any p0 ∈ M and z0 ∈ Cn the well defined map z : M → Cn ,

 p

z(p) = z0 +

p0

Φ

(p ∈ M)

(2.64)

is a holomorphic null curve satisfying ∂z = Φ

and

z∗ (ds2 ) = |dz|2 = |∂ z|2 = |Φ|2 .

Every holomorphic null curve M → Cn (n ≥ 3) is of this kind.

(2.65)

2.3 The Enneper–Weierstrass Formula

101

The analogous statements hold if M is a compact bordered Riemann surface. Given a conformal minimal immersion x : M → Rn , the Cn -valued holomorphic 1-form Φ = 2∂ x is called the Weierstrass data of x. Likewise, if z : M → Cn is a holomorphic null curve then Φ = ∂ z = dz is the Weierstrass data of z. By Theorem 1.10.5, every open Riemann surface M admits a nowhere vanishing holomorphic 1-form θ . For any such θ , we can write a nonvanishing holomorphic 1-form Φ on M satisfying the nullity condition (2.59) as Φ = f θ , where f = Φ/θ = ( f1 , . . . , fn ) : M → A∗ = A \ {0} is a holomorphic map into the punctured null quadric (2.54). If the real 1-form ℜ( f θ ) has vanishing periods over closed curves in M, then the map x : M → Rn p given by x(p) = ℜ f θ (p ∈ M) is a conformal minimal immersion satisfying 2∂ x = f θ (see (2.61)). If in addition the 1-form f θ has vanishing periods, then the  map z : M → Cn given by z(p) = p f θ is a holomorphic null curve (see (2.64)). In dimension n = 3 the Weierstrass formula can be written in the form  p

i 1

 1 1 x(p) = ℜ −g , + g , 1 φ3 , 2 g 2 g where φ3 = 2∂ x3 and g=

∂ x3 φ3 = : M −→ CP1 φ1 − i φ2 ∂ x1 − i ∂ x2

is a holomorphic map called the complex Gauss map of x. The details are given in Section 2.5; see in particular (2.79) and (2.83). Conjugate and associated minimal surfaces. If z = x + iy : M → Cn is a holomorphic null curve then its real part x : M → Rn and its imaginary part y : M → Rn are called conjugate minimal surfaces. Furthermore, minimal surfaces in the 1-parameter family x t = ℜ(eit z) : M → Rn (t ∈ R) are said to be associated minimal surfaces of the null curve z. In particular, every conformal minimal disc D → Rn is the real part of a holomorphic null disc D → Cn . Example 2.3.5. Consider the holomorphic null curve given by z(ζ ) = (cos ζ , sin ζ , −iζ ) ∈ C3 ,

ζ = u + iv ∈ C,

(2.66)

and its associated minimal surfaces in R3 : ⎛ ⎞ ⎛ ⎞ cos u · cosh v sin u · sinh v

x t (ζ ) = ℜ eit z(ζ ) = cost ⎝ sin u · cosh v ⎠ + sint ⎝− cos u · sinh v⎠ . v u At t = 0 we have a catenoid (see Subsection 2.8.1), and at t = ±π/2 we have a helicoid (see Subsection 2.8.2). Hence, these are conjugate minimal surfaces in R3 . The holomorphic null curve (2.66) is called helicatenoid.  

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The following is a consequence of Theorem 2.3.1 and the classical Schwarz reflection principle for harmonic functions. It will be used in Chapter 4. Corollary 2.3.6 (Schwarz reflection principle for minimal surfaces). Assume that x = (x1 , . . . , xn ) : M → Rn , n ≥ 3, is a conformal minimal immersion from an open Riemann surface M. If α ⊂ M is an embedded real analytic arc such that x(α) lies in an affine line L ⊂ Rn , then there is a neighborhood U of α in M such that x(U) is invariant under the reflection about the line L. Proof. Up to a rigid motion we may assume that L is the xn -axis. Writing x = (x , xn ) with x = (x1 , . . . .xn−1 ), we thus have x |α = 0 and the reflection about L is given by T (x , xn ) = (−x , xn ). In a local holomorphic coordinate on M around the real analytic arc α we may assume that α is an open interval in R ⊂ C. Let z = u + iv be the complex coordinate on C and τ(z) = z¯. Pick a connected open neighborhood U ⊂ C of α in the domain of x, with U ∩ R = α and τ(U) = U. Note that ∂x ∂ x j = ∂ zj dz, τ ∗ (dz) = d z¯ (where τ ∗ denotes the pull-back by τ), and hence τ ∗ (∂ x j ) =



 ∂xj ◦ τ d z¯, ∂z

j = 1, . . . , n.

(2.67)

Since the real harmonic function x j for j = 1, . . . , n − 1 vanishes on α, the Schwarz reflection principle implies that x j ◦ τ = −x j on U and hence   ∂xj ∂xj −τ ∗ (∂ x j ) = τ ∗ ∂ (x j ◦ τ) = τ ∗ ◦ τ · dz = d z¯ = ∂ x j . ∂ z¯ ∂ z¯ (We used that τ ◦ τ = IdC and ∂ τ/∂ z = 0, ∂ τ/∂ z¯ = 1.) From (2.53) it follows that

τ ∗ (∂ xn ) = ±∂ xn . By (2.67) this is equivalent to ∂∂xzn (u − iv) = ± ∂∂xzn (u + iv). If the sign onthe right-hand side is negative, then by comparing the real components we get ∂∂xun α = 0 and hence x is constant on α, contradicting the assumption that x is an immersion. Hence, the sign is positive and a comparison of the imaginary components gives ∂∂xvn (u − iv) = − ∂∂xvn (u + iv). Clearly this implies xn (u + iv) = xn (u − iv) and hence xn ◦ τ = xn . This shows that T ◦ x = x ◦ τ on U, and in particular that x(U) is invariant under the reflection about the line L.   Remark 2.3.7 (Minimal surfaces with singularities). The condition that the holomorphic 1-form Φ = (φ1 , . . . , φn ) in Theorem 2.3.4 be nowhere vanishing guarantees that the minimal surface (2.61), and the null curve (2.64), are immersed. If Φ is not identically zero but has zeros, these formulas define a branched minimal surface and a branched null curve, respectively. Their branch points are precisely the zeros of Φ, hence isolated. See Remark 3.12.6 and [174] for further information. Isolated branch points and the mildest singularities of minimal surfaces, and one may consider much worse singularities. In fact, the theory of minimal submanifolds is often treated in the wider setting of rectifiable varifolds and currents. The basic reason for this is a fundamental lack of compactness for ordinary submanifolds of a Riemannian manifold when applying variational methods, since these methods tend

2.4 Nonorientable Minimal Surfaces

103

to introduce singularities. Therefore one is led to consider generalized submanifolds, especially in the study of boundary value problems for minimal submanifolds, such as the Plateau problem. A similar situation occurs in complex analysis where one must deal with analytic chains and currents. We shall consider currents in Chapter 9 when discussing minimal hulls of compact sets in Rn ; see Sections 9.4–9.7. In the rest of the book we work with classical immersed or embedded minimal surfaces.

2.4 Nonorientable Minimal Surfaces In this section we consider nonorientable minimal surfaces in Rn . We show how the connection to complex analysis, explained in Section 2.3, can still be partially retained by considering the orientable double cover of the given nonorientable surface. This is the main vantage point for our analysis of nonorientable minimal surfaces in Euclidean spaces which is developed in the separate publication [29]. By Corollary 2.2.4, a nonorientable minimal surface S in Rn is the image of a conformal minimal immersion x : M → Rn from a conformal nonorientable surface M, where the conformal structure and the Laplacian on M are determined by the metric g = x∗ ds2 on M. Let ι : M → M denote the 2-sheeted oriented covering of M. Then, M is an orientable surface, and hence it carries the unique structure of a Riemann surface which makes ι a conformal map (see Corollary 1.8.7). Let I : M → M be the two-sheeted deck transformation such that M = M/I is the orbit space induced by the group of transformations {IdM , I} acting on M. Note that I is a fixed-point-free antiholomorphic involution. It is then immediate that any conformal minimal immersion x : M → Rn arises from an I-invariant conformal minimal immersion x : M → Rn , in the sense that x ◦ I = x.

(2.68)

Given an I-invariant conformal minimal immersion x : M → Rn , the Cn -valued holomorphic 1-form Φ = 2∂ x on M is I-invariant in the sense that I∗ (Φ) = Φ.

(2.69)

Conversely, if Φ = (φ1 , . . . , φn ) is a nowhere vanishing Cn -valued holomorphic 1form on M satisfying the nullity condition (2.59), the period-vanishing conditions (2.60), and the invariance condition (2.69), then the Rn -valued 1-form ℜΦ is exact and satisfies I∗ (ℜΦ) = ℜΦ. Hence, for any fixed p0 ∈ M the integral  p

x(p) = x(p0 ) +

ℜΦ, p0

p∈M

(2.70)

is a well defined I-invariant conformal minimal immersion M → Rn satisfying 2∂ x = Φ. (See Theorem 2.3.4 and in particular (2.61).)

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Let us fix a nowhere vanishing I-invariant holomorphic 1-form θ on M; such exists by [44, Corollary 6.5]. Given Φ as above, the holomorphic map f = Φ/θ = ( f1 , . . . , fn ) : M −→ A∗ ⊂ Cn is I-invariant in the sense that f ◦ I = f¯. Conversely, an I-invariant holomorphic map f : M → A∗ determines a nowhere vanishing holomorphic 1-form Φ = f θ satisfying conditions (2.59) and (2.69). The period-vanishing condition (2.60) for Φ can now be rewritten as 

ℜ C

Φ =0

for all [C] ∈ H1 (M, Z) with I∗ ([C]) = [C],

(2.71)

where I∗ : H1 (M, Z) → H1 (M, Z) is the group homomorphism induced by I; just take into account that, for any homology class [C] ∈ H1 (M, Z), one has 2[C] = ([C] + I∗ ([C])) + ([C] − I∗ ([C])) 

and ℜ C Φ = 0 for all [C] ∈ H1 (M, Z) with I([C]) = −[C]. This shows that the Weierstrass representation reduces the study of conformal minimal immersions M → Rn from an open nonorientable conformal surface to the study of I-invariant holomorphic maps f : M → A∗ from the 2-sheeted oriented cover (M, I) of M = M/I such that the 1-form Φ = f θ has vanishing real periods (2.71). (Here, θ is a nonvanishing I-invariant holomorphic 1-form on M.) Indeed, the map x : M → Rn given by (2.70) is then an I-invariant conformal minimal immersion which descends to a conformal minimal immersion x : M → Rn .

2.5 The Gauss Map of a Minimal Surface In this and the following sections we discuss some basic properties of the Gauss map of a minimal surface. The Gauss map is of major importance in the theory of minimal surfaces, especially in dimension n = 3. This topic is studied in more depth in Chapter 5 where we prove a fundamental new result in the theory; see Theorem 5.4.1. For historical background and further references on this subject, see R. Osserman [282, Chapter 12]. Let M be an oriented surface and x : M → Rn (n ≥ 3) be an immersion. For each point p ∈ M, the oriented 2-plane Σ p = dx p (Tp M) ⊂ Tx(p) Rn ∼ = Rn is considered as n an element of the Grassmann manifold G2 (R ) of oriented 2-planes in Rn , and the map M p → Σ p ∈ G2 (Rn ) is the classical Gauss map of the immersion x. In dimension n = 3, we can identify the Grassmannian G2 (R3 ) with the unit sphere S2 ⊂ R3 by associating to every oriented 2-plane Σ ∈ G2 (Rn ) the unique cooriented unit normal vector N ∈ S2 ⊂ R3 to Σ . This identification gives rise to the classical Gauss map N : M → S2 of an immersion x : M → R3 .

2.5 The Gauss Map of a Minimal Surface

105

More generally, for any n ≥ 3 we can identify G2 (Rn ) with the projectivization (2.54), the smooth quadric hypersurface of the punctured null quadric An−1 ∗  Q n−2 = [z1 : · · · : zn ] ∈ CPn−1 :

n



.

∑ z2j = 0

(2.72)

j=1

Indeed, given an oriented orthogonal basis v, w ∈ Rn of a 2-plane Σ ∈ G2 (Rn ) such that |v| = |w|, the complex vector z = v − iw ∈ Cn belongs to An−1 ∗ , and vice versa. Given another oriented orthogonal basis v, ˜ w˜ of Σ , it is easily verified that z˜ = v˜ − iw˜ = λ z for some λ ∈ C∗ ; conversely, any two vectors z, z˜ in the same determine the same oriented 2-plane Σ ∈ G2 (Rn ). For more complex ray of An−1 ∗ details we refer to R. Osserman [282, p. 117]. Assume now that M is a Riemann surface and x : M → Rn is a conformal immersion. Then, the partial derivatives xu and xv in any local holomorphic coordinate ζ = u+iv on an open subset U ⊂ M are orthogonal to each other and have the same length at each point (see the equivalence of (b) and (c) in Lemma 1.8.4), so given by ∂∂ζx = 12 (xu − ixv ). If in addition x is a they give rise to the map U → An−1 ∗ minimal immersion then this map is holomorphic by Theorem 2.3.1. A holomorphic change of coordinates on M changes this map by a nowhere vanishing complex multiplicative factor, and hence the induced map to the hyperquadric (2.72) remains the same. This motivates the following definition. Definition 2.5.1. Let M be an open Riemann surface. The Gauss map of a conformal minimal immersion x = (x1 , x2 , . . . , xn ) : M → Rn (n ≥ 3) is the holomorphic map G = [∂ x1 : ∂ x2 : · · · : ∂ xn ] : M −→ Q n−2 ⊂ CPn−1 ,

(2.73)

where Q n−2 is the hyperquadric given by (2.72). Likewise, the Gauss map of a holomorphic null curve z = (z1 , . . . , zn ) : M → Cn is the holomorphic map G = [dz1 : dz2 : · · · : dzn ] : M −→ Q n−2 ⊂ CPn−1 . It follows that all minimal surfaces associated to a given holomorphic null curve have the same Gauss map. Let us explain the geometry a bit more carefully. Consider Cn as the affine part of CPn = Cn ∪ H, where H = CPn−1 is the hyperplane at infinity. Let (z1 , . . . , zn ) be complex coordinates on Cn . The map π : Cn∗ → H = CPn−1 ,

π(z1 , . . . , zn ) = [z1 : · · · : zn ]

(2.74)

is a holomorphic C∗ -bundle. By adding to the total space Cn∗ the hyperplane at infinity H = CPn−1 as the zero section, we obtain a holomorphic line bundle π : Cn∗ ∪ CPn−1 → CPn−1 , the hyperplane section bundle OCPn−1 (+1). Then,  Q n−2 = An−1 ∩ CPn−1 = [z1 : · · · : zn ] ∈ CPn−1 :

 2 z = 0 , ∑ j n

j=1

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2 Basics on Minimal Surfaces

where An−1 = An−1 ∪ Q n−2 is the topological closure of the null quadric in CPn . → Q n−2 is a holomorphic C∗ -bundle. Note that π : An−1 ∗ Writing 2∂ x = f θ , where f = ( f1 , . . . , fn ) : M → An−1 is a holomorphic map and ∗ θ is a nowhere vanishing 1-form on M, the Gauss map of x equals G = π ◦ f = [ f1 : f2 : · · · : fn ] : M → Q n−2 ⊂ CPn−1 .

(2.75)

We introduce the following notions. Definition 2.5.2. Let M be a connected manifold. A map f : M → Cn is: (i) flat if f (M) is contained in an affine complex line in Cn , and nonflat otherwise; (ii) full if the C-linear span of f (M) equals Cn ; (iii) nondegenerate if f (M) is not contained in any affine complex hyperplane. Assume now that M is either a connected open Riemann surface or a compact bordered Riemann surface, and θ is a nowhere vanishing holomorphic 1-form on M (see Theorem 1.10.5). A conformal minimal immersion x : M → Rn is (a) flat if x(M) is contained in an affine 2-plane in Rn , and nonflat otherwise; is full; (b) full if the map f = 2∂ x/θ : M → An−1 ∗ (c) decomposable if, with respect to suitable Euclidean coordinates on Rn , n 2 2 ∑m i=1 (∂ xi ) = 0 for some 1 ≤ m < n (hence ∑i=m+1 (∂ xi ) = 0 as well); (d) nondegenerate if x(M) is not contained in an affine hyperplane of Rn . Part (a) of the following lemma is [282, Lemma 12.2, p. 119], (b) is an immediate consequence of definitions, and (c) is [282, Lemma 12.3, p. 122]. Lemma 2.5.3. Assume that x : M → Rn is a conformal minimal immersion. Set f = 2∂ x/θ : M → An−1 ∗ , and let G be the generalized Gauss map (2.73). Then: (a) The immersion x is flat if and only if the map f : M → An−1 is flat, and this holds ∗ if and only if the Gauss map G : M → CPn−1 is constant. (b) The immersion x is full if and only if f is full, if and only if G is full, i.e., G (M) is not contained in any hyperplane H = {a1 z1 + . . . + an zn = 0} ⊂ CPn−1 . (c) If G (M) is contained in a hyperplane H as in (b) with real coefficients a1 , . . . , an , then x(M) is contained in an affine hyperplane of Rn (i.e., x is degenerate). For a conformal minimal immersion M → R3 , nonflat, full, nondegenerate, and nondecomposable are equivalent notions (see Osserman [282, Lemma 12.4, p. 124]). In dimensions n > 3 we have the implications full =⇒ nondegenerate =⇒ nonflat,

(2.76)

but the converses are not true (see [282, p. 124]). In [282, p. 42] Osserman gives an example of a conformal minimal immersion x = (x1 , x2 , x3 , x4 ) : C → R4 for which 2i∂ x1 + ∂ x2 = 0, so x is degenerate but is not decomposable.

2.5 The Gauss Map of a Minimal Surface

107

The Gauss map in dimension three. The quadric Q1 ⊂ CP2 , defined by (2.72), is an embedded rational curve parameterized by the biholomorphic map ' &   1 1 i 1 τ 1 CP t −→ −t : + t : 1 = 1 − t 2 : i(1 + t 2 ) : 2t ∈ Q1 . (2.77) 2 t 2 t Writing (1 − t 2 , i(1 + t 2 ), 2t) = (a, b, c), we easily find that t=

b−ia c = ∈ CP1 . a−ib ic

(2.78)

Suppose that x = (x1 , x2 , x3 ) : M → R3 is a conformal minimal immersion, and write 2∂ x = 2(∂ x1 , ∂ x2 , ∂ x3 ) = (φ1 , φ2 , φ3 ). In view of (2.78) it is natural to introduce the holomorphic map g=

∂ x3 φ3 = : M −→ CP1 . φ1 − i φ2 ∂ x1 − i ∂ x2

(2.79)

The Gauss map G : M → Q1 ⊂ CP2 (2.73) is then expressed by G = τ ◦g

(2.80)

where τ : CP1 → Q1 is given by (2.77). The map g : M → CP1 given by (2.79) is called the complex Gauss map of the conformal minimal immersion x : M → R3 . This terminology is justified by the following observation. Remark 2.5.4. A calculation (see R. Osserman [282, (8.8), p. 66]) shows that the classical Gauss map N : M → S2 of a conformal minimal immersion x : M → R3 (see (2.18), (2.19)) is related to its complex Gauss map g : M → CP1 (2.79) by   2ℜg 2ℑg |g|2 − 1 , , N = (N1 , N2 , N3 ) = . (2.81) |g|2 + 1 |g|2 + 1 |g|2 + 1 Postcomposing the Gauss map N with the stereographic projection S2 → CP1 from the point (0, 0, 1) ∈ S2 we obtain N1 + i N2 = g : M → CP1 , 1 − N3

(2.82)

thereby justifying the name complex Gauss map for g.   A calculation (cf. Osserman [282, Lemma 8.1, p. 63]) gives the following expression for the Weierstrass data in terms of the pair (g, φ3 ) = (g, 2∂ x3 ): 

i 1

 1 1 −g , + g , 1 φ3 . (2.83) 2∂ x = Φ = (φ1 , φ2 , φ3 ) = 2 g 2 g

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Conversely, given a pair (g, φ3 ) consisting of a holomorphic map g : M → CP1 and a holomorphic 1-form φ3 on M, the meromorphic 1-form Φ = (φ1 , φ2 , φ3 ) on M defined by (2.83) satisfies the nullity condition (2.59). If in addition • the 1-form Φ has no zeros and poles, and • the real part ℜΦ of Φ has vanishing periods over all closed curves in M, then we get a conformal minimal immersion x : M → R3 defined by  p  p

i 1

 1 1 x(p) = −g , + g , 1 φ3 , p ∈ M. ℜΦ = ℜ 2 g 2 g

(2.84)

This is the most classical Weierstrass formula for minimal surfaces in R3 . It is easily seen that a 1-form Φ = (φ1 , φ2 , φ3 ) given by (2.83) has no zeros and poles on M if and only if the following two conditions hold: • if at some point p ∈ M the meromorphic function g has either a zero or a pole of order k ∈ N, then φ3 has a zero at p of the same order k, and • φ3 does not have any other zeros (since those would also be zeros of Φ). Recall that the induced metric g = x∗ ds2 on M is given in terms of the Weierstrass data Φ = (φ1 , φ2 , φ3 ) = 2∂ x by g = 12 |Φ|2 (see (2.62)). A calculation using the formula (2.83) shows that g is expressed in terms of the pair (g, φ3 ) by g=

(1 + |g|2 )2 |φ3 |2 . 4|g|2

(2.85)

This formula will be important in the proof of the Jorge–Xavier and Nadirashvili theorems (see Theorems 7.1.3 and 7.1.5).

2.6 Gaussian Curvature of a Minimal Surface In this section we consider the Gaussian curvature of a minimal surface and its integral, the total curvature. We will see that the total curvature equals the negative area of the Gauss map counted with multiplicities; see (2.91) and (2.94). If the Gauss map is algebraic then the total curvature is finite and can be expressed in terms of the topological degree of the Gauss map (see Corollary 2.6.6). Of particular interest are complete minimal surfaces of finite total curvature (see Definition 2.6.1); these have algebraic Gauss map and are geometrically the simplest ones. Most examples of minimal surfaces in Section 2.8 are of this kind. The study of minimal surfaces of finite total curvature is an important part of the classical global theory of minimal surfaces. In Chapter 4 we prove several fundamental results concerning minimal surfaces of finite total curvature, including the recently discovered Runge and Mergelyan approximation theorems from [37].

2.6 Gaussian Curvature of a Minimal Surface

109

Let x : M → Rn be an immersed minimal surface, and let z be an isothermal coordinate on an open set U ⊂ M. By Theorem 2.3.1 (e) the map f = ( f1 , . . . , fn ) = 2∂ x/dz : U → An−1 ∗ is holomorphic, and by (2.55) the induced metric is given by 1 g = x∗ ds2 = 2|∂ x|2 = | f |2 |dz|2 . 2

(2.86)

In view of (2.32) the Gaussian curvature of g equals K=− where

| f ∧ f  |2 1 2 Δ log | f | = −4 ≤ 0, 0 | f |2 | f |6

(2.87)

| f ∧ f  |2 = | f |2 | f  |2 − | f · f  |2 = ∑ | fi f j − f j fi |2 . i< j

Since the Gauss map of x is given by G = [ f1 : · · · : fn ] : M → CPn−1 (see (2.75)), the equations (2.86) and (2.87) imply | f ∧ d f |2 1 | f ∧ f  |2 1 2 2 2 | f | · |dz| = 2 = G ∗ (σCP n−1 ), | f |6 2 | f |4 2

− Kg = 4

(2.88)

n−1 2 where σCP (see (1.82)). n−1 is the Fubini–Study metric on CP

Definition 2.6.1. Let x : M → Rn be a conformal minimal immersion and g = x∗ ds2 be the induced Riemannian metric on M. The total curvature of x is 

KdA ∈ [−∞, 0],

TC(x) = M

(2.89)

where K ≤ 0 and dA are the Gaussian curvature function and the area element of (M, g), respectively. The immersion x has finite total curvature if 

TC(x) = M

KdA > −∞.

(2.90) 



If Ω is the area form (1.92) defined by the metric g, then M KdA = M KΩ . Let ω denote the fundamental form of the Fubini–Study metric on CPn−1 (see Example 1.7.4). The equation (2.88) clearly implies KΩ = − 21 G ∗ ω. Since the Gauss map G : M → CPn−1 is holomorphic, it follows that 

TC(x) =



KdA = M

M

KΩ = −

1 2

 M

1 G ∗ ω = − Area(G (M)), 2

(2.91)

where the last equality holds by Wirtinger’s Theorem 1.7.5 and the area of the complex curve G (M) ⊂ CPn−1 is counted with multiplicities.

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Gaussian and total curvature of minimal surfaces in R3 . We now express the Gaussian curvature K of a conformal minimal immersion x = (x1 , x2 , x3 ) : M → R3 in terms of its complex Gauss map g : M → CP1 given by (2.79). Choose a local holomorphic coordinate z on an open set U ⊂ M and write 2∂ x = f dz = ( f1 , f2 , f3 )dz. By (2.85) the induced Riemannian metric is given by g=

(1 + |g|2 )2 | f3 |2 |dz|2 = μ |dz|2 , 4|g|2

(2.92)

and by (2.32) its Gaussian curvature equals K=−

2 ∂ 2 log μ 1 Δ0 log μ = − . 2μ μ ∂ z∂ z¯

When calculating Δ0 log μ, we may restrict attention to the complement of zeros and poles of g and f3 since those cancel. Note that for any meromorphic function h we have that Δ0 log |h|2 = 0 on the complement of the set of its zeros and poles (in our case, this applies to g and f3 ). It follows that Δ0 log μ ∂2 ∂ gg |g |2 |g|2 |g |2 |g |2 = log(1+|g|2 ) = = − = . 8 ∂ z∂ z¯ ∂ z 1 + |g|2 1 + |g|2 (1 + |g|2 )2 (1 + |g|2 )2 From this and (2.32) we infer that the Gaussian curvature is given by K=−

4|g |2 16|g|2 |g |2 4|g|2 = − . | f3 |2 (1 + |g|2 )2 (1 + |g|2 )2 | f3 |2 (1 + |g|2 )4

(2.93)

Taking into account (2.92) gives Kg = −

4|dg|2 2 ∗ 2 = −g∗ (σCP 1 ) = −N (dsS2 ), (1 + |g|2 )2

2 2 2 where σCP 1 is the Fubini–Study metric (1.84), dsS2 is the spherical metric on S , 2 and N : M → S is the classical Gauss map (2.81) (see Examples 1.6.4, 1.6.5, and 2 2 the formula (1.86) relating σCP 1 and dsS2 ). This implies the following corollary.

Corollary 2.6.2. The total Gaussian curvature of a conformal minimal immersion x : M → R3 with the complex Gauss map g : M → CP1 (2.79) equals 

TC(x) = M

KdA = −Area(g(M)),

(2.94)

where the area of g(M) ⊂ CP1 is counted with multiplicities. Remark 2.6.3. The formula (2.91) says that TC(x) equals − 12 times the area of the image of the Gauss map G : M → CPn−1 counted with multiplicities. When n = 3, the formula (2.94) says the same thing in terms of the complex Gauss map g : M → CP1 (or the classical Gauss map N : M → S2 ), but without the factor 12 .

2.6 Gaussian Curvature of a Minimal Surface

111

This apparent discrepancy is explained by observing that G = τ ◦ g (2.80) and the quadratic holomorphic map τ : CP1 → Q1 (2.77) increases the area by the factor of two. This is in line with the formula (1.108) which says that the degree two rational  curve Q1 ⊂ CP2 has Area(Q1 ) = 8π, while Area(CP1 ) = 4π.  Complete minimal surfaces of finite total curvature. The beginning point of the study of minimal surfaces of finite total curvature is the following theorem of A. Huber [198] from 1957, which we state without proof. At this point we are not assuming that g is a metric on a minimal surface, so its Gaussian curvature function K may have both signs. Set K − = min{K, 0} : M → (−∞, 0]. Theorem 2.6.4 (Huber [198]). If (M, g) is a complete oriented Riemannian surface  without boundary such that M K − dA > −∞, then M is conformally equivalent to the complement of finitely many points in a compact Riemann surface. The analogous result holds if M includes finitely many boundary curves; in such case, M is of the form M = M \ {p1 , . . . , pk } where M is a compact bordered Riemann surface and p1 , . . . , pk are interior points of M. Definition 2.6.5. Let x : M → Rn be a conformal minimal immersion from a Riemann surface M. The generalized Gauss map G : M → Qn−2 ⊂ CPn−1 of x (see (2.73)) is algebraic if the following two conditions hold. (a) M is conformally equivalent to a compact Riemann surface without boundary, M, punctured at a finite number of points. (b) G extends to a holomorphic map G : M → Qn−2 . In dimension n = 3, condition (b) is clearly equivalent to: (b’) The Gauss map g : M → CP1 extends to a holomorphic map g : M → CP1 . Let deg(G ) ∈ Z+ denote the degree of G (see p. 25). The image G (M) is a compact algebraic curve in CPn−1 whose homology class in H2 (CPn−1 , Z) ∼ =Z equals deg(G )· L, where L is the class of the projective line. By Corollary 1.7.8, the Fubini–Study area of G (M) counted with multiplicities equals 4π deg(G ). Together with the formula (2.91) we obtain the following corollary. Corollary 2.6.6. If x : M → Rn is a conformal minimal immersion with algebraic Gauss map G : M → CPn−1 , then x has finite total curvature equal to TC(x) = −2π deg(G ).

(2.95)

If n = 3 then we also have that TC(x) = −4π deg(g).

(2.96)

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Corollary 2.6.6 does not hold for more general minimal surfaces. In fact, every holomorphic map M → CP1 is the complex Gauss map of a conformal minimal immersion M → R3 (see Theorem 5.4.1), and hence any number in [−∞, 0] is the total Gaussian curvature of some minimal surface in R3 . The following theorem of S. S. Chern and R. Osserman [87] (1967) gives the converse to Corollary 2.6.6 for complete minimal surfaces of finite total curvature. Theorem 2.6.7. If M is an open Riemann surface and x : M → Rn is a complete conformal minimal immersion of finite total curvature, then M is a finitely punctured compact Riemann surface and the Gauss map G : M → CPn−1 of x is algebraic. A more precise version of Theorem 2.6.7 is given by Theorem 4.1.1. Together with Corollary 2.6.6, we infer the following. Corollary 2.6.8. If S = x(M) ⊂ Rn is an orientable complete immersed minimal surface without boundary and of finite total curvature, then the following hold. (a) If n = 3 then the total curvature of S is −4πd for some d ∈ Z+ . (b) If n ≥ 4 then the total curvature of S is −2πd for some d ∈ Z+ . The number d in the above corollary is the degree of the complex Gauss map g : M → CP1 (2.79) when n = 3, and is the degree of the generalized Gauss map G : M → Qn−2 (2.73) when n > 3. The case d = 0 corresponds to the plane. Osserman [282] proved that a complete minimal surface in R3 with total curvature −4π is either the catenoid (Example 2.8.1) or Enneper’s surface (Example 2.8.4). In case (b) when n > 3, any integer d ∈ Z+ is possible. For instance, take the minimal surface x : C → R4 with the Weierstrass data Φ = (1, i, p(z), ip(z)) dz for any polynomial p(z) of degree d; clearly it is a graph over the (x1 , x2 )-plane, hence properly embedded with TC(x) = −2πd. In case (a) when n = 3, every integer d ∈ Z+ is possible as well. Indeed, the conformal minimal immersion x : C → R3 with the Weierstrass data    i  1 1 − z2d , 1 + z2d , zd dz, d ∈ Z+ Φ= 2 2 is complete and has total curvature equals to −4πd. If d = 0 then x is a plane, whereas if d = 1 then x is Enneper’s surface; see Example 2.8.4. The above examples are simply connected. Constructing examples of such surfaces with nontrivial topology is a more involved task. We have seen in Section 2.4 that every immersed nonorientable minimal surface S in Rn is the image of a 2-sheeted conformal minimal immersion x : M → S ⊂ Rn from a Riemann surface M. Hence, the total Gaussian curvature of S is one half of the total Gaussian curvature of x. In particular, if n = 3 and the surface S is complete, without boundary, and of finite total curvature, then by Corollary 2.6.8 its total Gaussian curvature equals

2.7 The Maximum Principle and the Isoperimetric Inequality

1 TC(S) = TC(x) = −2πd, 2

113

d ∈ N.

(2.97)

The case d = 1 cannot arise since the oriented double cover of such a surface S would be a complete oriented minimal surface of total Gaussian curvature −4π, hence a catenoid or the Enneper surface (see R. Osserman [282]). However, none of these two surfaces covers any nonorientable minimal surface. Indeed, Enneper’s surface is simply connected, whereas the catenoid has φ3 = dz/z on C∗ (see (2.106)) and there in no fixed-point-free antiholomorphic involution I of C∗ satisfying I∗ (φ3 ) = φ 3 . (However, the case d = 1 may arise for non-immersed surfaces. Indeed, Henneberg’s surface described in Subsection 2.8.9 is a complete nonorientable branched minimal surface in R3 of total curvature −2π.) The case d = 2 cannot arise either as shown by W. H. Meeks [248, Corollary 2]. Thus, the first possibility in (2.97) is d = 3 and TC(S) = −6π. Such a surface indeed exists — the famous minimal M¨obius strip, discovered by W. H. Meeks in 1981 [248] (see Example 2.8.10), is the unique (up to scaling and rigid motions) complete immersed minimal surface in R3 with total Gaussian curvature −6π. The first known example of a nonorientable complete minimal surface of genus 2 in R3 was the immersed minimal Klein bottle with one puncture, discovered by F. J. L´opez [226] in 1993; see Example 2.8.12. It has total curvature −8π. In 2017 the authors [29] found the first example of a properly embedded algebraic minimal M¨obius strip in R4 ; see Example 2.8.11. Its total curvature equals −4π. By what has been said above, there is no such surface in R3 . We shall discuss minimal surfaces of finite total curvature in much more detail in Chapter 4. There, we shall also present the recent results from [37] on Runge and Mergelyan approximation by minimal surfaces of finite total curvature.

2.7 The Maximum Principle and the Isoperimetric Inequality In this section we collect some basic information regarding two classical topics of interest in the theory of minimal surfaces: the maximum principle and the isoperimetric inequality. Much more precise results regarding the maximum principle for minimal surfaces can be found in the last two chapters. The Maximum Principle for Minimal Surfaces. If M is a compact bordered Riemann surface and x : M → Rn is a continuous map which is harmonic on the interior M˚ = M \ bM, then for any affine linear function λ : Rn → R the composition ˚ and hence it λ ◦ x : M → R is a continuous function which is harmonic on M, assumes its minimum and maximum on the boundary bM. Therefore, x(M) ⊂ Co(x(bM))

(2.98)

where Co(·) denotes the convex hull of a set in Rn . By Corollary 2.2.4 the same holds if x : M → Rn (n ≥ 3) is a continuous map such that x : M˚ → Rn is a minimal

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surface. In particular, there are no compact minimal surfaces without boundary in Rn . Condition (2.98) is known as the convex hull property of minimal surfaces. A more precise argument is that if x : M → Rn is a conformal minimal surface and λ is a subharmonic function on a neighbourhood of x(M) in Rn , then λ ◦ M is a subharmonic function on M as seen from the trace formula (8.6). It follows that if M is a compact bordered Riemann surface and x : M → Rn is a continuous map whose restriction to M˚ is a conformal minimal surface, then  x(M) ⊂ x(bM) Sh(Rn ) ⊂ Co(x(bM)) Sh(Rn ) denotes the hull of a compact set K ⊂ Rn with respect to the set where K n Sh(R ) of all subharmonic functions on Rn . Note that linear functions constitute a rather small subset of the space of subharmonic functions, and the subharmonic hull is in general smaller than the convex hull. An essentially optimal maximum principle for minimal surfaces in Rn is obtained in Section 9.2 where we introduce the minimally convex hull, or minimal hull for short, of a compact set K ⊂ Rn for any n ≥ 3. This is the hull with respect to minimal plurisubharmonic functions on Rn , which are characterized by the property that the sum of the smallest two eigenvalues of their Hessian at any point of the space is nonnegative. This is the largest class of functions λ on Rn whose composition λ ◦ x with any conformal minimal immersion x : M → Rn is a subharmonic function on the Riemann surface M (see Proposition 8.1.2). One of the main results is that a M of a compact set K ⊂ Rn if and only point p ∈ Rn belongs to the minimal hull K if p is the centre of a bounded sequence of conformal minimal discs in Rn whose boundaries converge to K in measure (see Theorem 9.2.1). We also characterize the minimal hull in terms of Green currents (see Theorem 9.7.4). We now describe a maximum principle for minimal graphs in R3 . Let f : D → R be a C 2 function on a domain D ⊂ R2 and denote by x f : D → R3 its graph x f (u, v) = (u, v, f (u, v)),

(u, v) ∈ D.

(Note that this parameterization of the graph is not conformal, except at the critical points of f .) It is easily seen from (2.12)  that the mean curvature of x f in the unit normal direction N = (− fu , − fv , 1)/ 1 + |∇ f |2 is given by   (1 + fv2 ) fuu − 2 fu fv fuv + (1 + fu2 ) fvv ∇f N = . (2.99) H = div  1 + |∇ f |2 2 (1 + |∇ f |2 )3/2 Hence, x f is a minimal surface if and only if f satisfies the minimal graph equation (1 + fv2 ) fuu − 2 fu fv fuv + (1 + fu2 ) fvv = 0.

(2.100)

This second order elliptic quasilinear partial differential equation satisfies the following maximum principle (see e.g. D. Gilbarg and N. S. Trudinger [157]).

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115

Theorem 2.7.1 (Maximum principle for minimal graphs). Let D ⊂ R2 be a connected open domain and let f1 , f2 : D → R be functions of class C 2 (D) satisfying (2.100). If f1 ≥ f2 everywhere on D and f1 = f2 at some point of D, then f1 = f2 . Theorem 2.7.1 shows that if two immersed minimal surfaces in R3 have a common point and locally near this point one of them lies on one side of the other one, then they coincide in a neighbourhood of the point. The following result of T. Rad´o [295] is a consequence of this maximum principle. Corollary 2.7.2. Denote by π : R3 → R2 the projection π(x, y, z) = (x, y). Let D be a compact convex domain in R2 , and let x j : M j → R3 ( j = 1, 2) be a pair of continuous maps from compact connected bordered surfaces M j satisfying the following conditions. • bM j is a single Jordan curve for j = 1, 2, and x1 (bM1 ) = x2 (bM2 ) ⊂ bD × R. • The projection π ◦ x j |bM j : bM j → bD is bijective for j = 1, 2. • The restriction of x j to M j \ bM j is a minimal immersion for j = 1, 2. Then, x1 (M1 ) = x2 (M2 ) and this surface is a graph over D, i.e., the restricted projection π|x1 (M1 ) : x1 (M1 ) → D is a bijection. Proof. For t ∈ R and j ∈ {1, 2} we consider the vertical translation x tj = x j + (0, 0,t) : M j → R3 with x0j = x j . By the convex hull property (2.98) we have that x tj (M j ) ⊂ D × R for j = 1, 2 and all t ∈ R.

(2.101)

By compactness of x1 (M1 ) and x2 (M2 ) there is a number t1 > 0 such that the third component of every point of xt11 (M1 ) is strictly larger than the third component of every point of x2 (M2 ); in particular, xt11 (M1 ) ∩ x2 (M2 ) = ∅. Set t0 := inf {t ∈ (−∞,t1 ] : x1s (M1 ) ∩ x2 (M2 ) = ∅ for all s ∈ [t,t1 ]} . t

A compactness argument shows that x10 (M1 ) ∩ x2 (M2 ) = ∅. Furthermore, (2.101) t and the maximum principle give that x10 (bM1 ) = x2 (bM2 ), and hence t0 = 0 and x1 (M1 ) lies above x2 (M2 ). The symmetric argument shows that x2 (M2 ) lies above x1 (M1 ) as well, so x1 (M1 ) = x2 (M2 ). To see that x1 (M1 ) is a graph over D, it suffices  to apply the above argument but comparing x1 to itself.  The Isoperimetric Inequality for Minimal Surfaces. The isoperimetric problem is an active topic in geometric analysis with implications in other fields such as convex geometry, partial differential equations, and others. It deals with determining domains in a Riemannian manifold (M, g) of largest possible volume among those whose boundaries have a specified volume. The particular case of domains in Rn or, more generally, in minimal submanifolds of Rn , is historically the most interesting

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one. Solutions to this problem rely on isoperimetric inequalities which in dimension two provide upper bounds for the area of a domain in terms of the length of its boundary. It was already known in ancient Greece that the solutions to the isoperimetric problem in the Euclidean plane are circles bounding round discs. This follows from the inequality 4π Area(Σ ) ≤ length(bΣ )2

(2.102)

which holds for any relatively compact domain Σ with rectifiable boundary in R2 . It has been a long-standing conjecture that the same isoperimetric inequality holds for all minimal surfaces in Rn , n ≥ 3. Details about the fascinating history of this problem can be found in many sources, as for instance [178, 281, 56, 75, 197, 300, 82, 108]. In 2019, S. Brendle [72] gave an affirmative solution of this famous conjecture for minimal submanifolds of arbitrary codimension in Rn . This includes the following result for minimal surfaces in Euclidean spaces. Theorem 2.7.3 (Brendle [72]). If x : M → Rn is a compact bordered minimal surface with rectifiable boundary and we endow M with the Riemannian metric x∗ ds2 , then 4π Area(M) ≤ length(bM)2 . Moreover, the equality holds if and only if x(M) is a round disc in an affine plane.

2.8 A Gallery of Minimal Surfaces We conclude this chapter with a brief review of some classical examples of minimal surfaces and describe their basic geometric properties.

2.8.1 The Catenoid The catenoid is obtained by rotating the catenal curve in R2 (the graph of the hyperbolic cosine function) around a suitable axis in R3 . It was described by Leonhard Euler in 1744 [122] and characterized by Pierre Ossian Bonnet in 1860 [68] as the only rotational minimal surface in R3 , besides the plane. For example, by rotating the catenal curve R v → (cosh v, 0, v) ∈ R3 around the x3 -axis we obtain the vertical catenoid in R3 with the axis x1 = x2 = 0 and the implicit equation x12 + x22 = cosh2 x3 . A conformal parameterization of this catenoid (unique up to homotheties) is given by the map x = (x1 , x2 , x3 ) : R2 → R3 ,

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117

Fig. 2.1 The catenoid

x(u, v) = (cos u · cosh v, sin u · cosh v, v) .

(2.103)

This is the real part of the holomorphic null curve z : C → C3 given by z(ζ ) = (cos ζ , sin ζ , −iζ ) ∈ C3 ,

ζ = u + iv ∈ C,

(2.104)

called helicatenoid (see Example 2.3.5). Indeed, a calculation shows that the 1parameter family of minimal surfaces in R3 associated to the helicatenoid is ⎛ ⎞ ⎛ ⎞ cos u · cosh v sin u · sinh v

x t (ζ ) = ℜ eit z(ζ ) = cost ⎝ sin u · cosh v ⎠ + sint ⎝− cos u · sinh v⎠ (2.105) v u with t ∈ R. At t = kπ (k ∈ Z) we have the catenoid (2.103), while at t = ±π/2 + kπ (k ∈ Z) we have a right or left helicoid which will be considered in Subsection 2.8.2. Hence, the catenoid and the helicoid are conjugate minimal surfaces. The Enneper– Weierstrass representation (2.83) of the helicatenoid (2.104) is given by  ζ

(− sin ξ , cos ξ , −i) dξ      ζ  1 1 1 i iξ iξ −e +e = (1, 0, 0) + , , 1 (−i)dξ 2 eiξ 2 eiξ 0

z(ζ ) = (1, 0, 0) +

0

with ζ ∈ C. Comparing with (2.79) we see that the Gauss map of the helicatenoid, and hence of all its associated minimal surfaces (2.105), is g(ζ ) = eiζ . The parameterization of the catenoid given by (2.103) is 2π-periodic in the u variable, hence infinitely sheeted. By introducing the variable w = eiζ = e−v+iu ∈ C∗ , we pass to the quotient C/2π Z ∼ = C∗ and obtain a single sheeted ∗ 3 parameterization x = x(w) : C → R having the Weierstrass representation  w

i 1

 dη 1 1 −η , +η ,1 . (2.106) x(w) = (1, 0, 0) − ℜ 2 η 2 η η 1

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(We introduced the variable η = eiξ into the integral for z(ζ ). This gives the same parametric expression x(u, v) (2.103) in terms of the local conformal coordinates (u, v) = (Arg(w), − log |w|).) From the formulas (2.79), (2.83) we see that the complex Gauss map of the catenoid parameterized by (2.106) is g(w) = w,

w ∈ C∗ ,

so it extends to the identity map C∗ = CP1 → CP1 of degree 1. Hence, by (2.96) the catenoid has total Gaussian curvature equal to −4π. In fact, the catenoid is the only surface in the family (2.104) which factors through C∗ , has algebraic Weierstrass data, and is of finite total curvature; all other surfaces in the family (2.104) with t∈ / πZ are transcendental and of infinite total curvature. The catenoid is a paradigmatic example in the theory of minimal surfaces. Here is a compendium of major results about it. • The catenoid is the only complete minimal surface in R3 foliated by circles; B. Riemann [298, 299]. • Together with Enneper’s surface, the catenoid is the only complete minimal surface in R3 with total Gaussian curvature −4π; R. Osserman [282]. Recall that 4π is the minimal possible absolute value of total Gaussian curvature of a complete nonflat minimal surface in R3 . • The catenoid is the only complete minimal surface in R3 of finite total curvature with two embedded ends; R. Schoen [313]. • Besides the plane, the catenoid is the only complete embedded minimal surface in R3 of genus zero and of finite total curvature; F. J. L´opez and A. Ros [235]. • The catenoid is the only properly embedded minimal surface in R3 of finite topology with two ends; P. Collin [95]. • According to T. H. Colding and W. P. Minicozzi [93], every complete embedded minimal surface in R3 of finite topology is properly embedded. This shows that the results in [313, 235, 95] mentioned above hold true assuming only that the surface is complete (and not necessarily proper). • The only compact minimal annuli in R3 bounded by parallel circles in distinct horizontal planes, with centres in the same vertical, are pieces of a catenoid; M. Shiffman [316].

2.8.2 The Helicoid The helicoid was described by Leonhard Euler [122] (1774) and Jean Baptiste Meusnier [259] (1776). Geometrically, it is generated by rotating a line in a plane of R3 and simultaneously displacing it in the perpendicular direction (the axis of rotation). Therefore, it is invariant under a one parameter family of screw motions around the axis of rotation, and consequently it is foliated by helices (hence its

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119

name). The following figure shows a helicoid whose axis of rotation is the x3 -axis in R3 .

Fig. 2.2 The helicoid

Let z : C → C3 be the helicatenoid (2.104). From (2.105) we obtain the following conformal parameterization of the helicoid x = ℜ(iz) : R2 → R3 : x(u, v) = (sin u · sinh v, − cos u · sinh v, u). Its Weierstrass representation is      ζ  1 1 1 i iξ iξ x(ζ ) = ℜ − e + e , , 1 dξ , 2 eiξ 2 eiξ 0

(2.107)

ζ ∈ C.

Since its complex Gauss map g(ζ ) = eiζ (cf. (2.83)) is transcendental, the total curvature of the helicoid equals −∞. As u increases, the vector (sin u, − cos u) turns in the positive (counterclockwise) direction, so this helicoid is as shown in Fig. 2.2. Note that the catenoid ℜ(z) (2.103) is the conjugate minimal surface of the helicoid (2.107). By taking instead the helicoid ℜ(−iz) and replacing u by −u we get (u, v) → (sin u · sinh v, cos u · sinh v, u). As u increases, the line now rotates in the negative (clockwise) direction. E. Catalan [81] proved in 1842 that the helicoid and the plane are the only ruled minimal surfaces in R3 , i.e., unions of straight lines. Much more recently, W. H. Meeks and H. Rosenberg proved in 2005 [255], using the curvature estimates of T. H. Colding and W. P. Minicozzi [92], that the helicoid and the plane are the only

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properly embedded, simply connected minimal surfaces in R3 . Again, by [93] it suffices to assume that the surface is complete but not necessarily proper.

2.8.3 Scherk’s Surfaces The two Scherk’s minimal surfaces were introduced by Heinrich Scherk [311] in 1835. They were the first new minimal surfaces discovered since Meusnier in 1776. The first Scherk’s surface is invariant under the action of the group of symmetries generated by two orthogonal translations, hence it is doubly periodic. The second Scherk’s surface is invariant under the action of a cyclic group of translations, hence it is singly periodic. The two surfaces are conjugate to each other.

Fig. 2.3 First Scherk’s surface

The implicit equation defining the first Scherk’s surface in R3 with coordinates x = (x1 , x2 , x3 ) is ex3 cos x2 = cos x1 . For determining its Weierstrass data, let ρ : D → C \ {±1, ±i}) denote the holomorphic universal covering of C \ {±1, ±i}. Consider a loop γ ⊂ C \ {±1, ±i}) based at the origin and winding once around the points ±1 and zero times around ±i. Let H1 be the normal subgroup of π1 (C \ {±1, ±i}, 0) generated by the homotopy class of γ. Fix a point z0 ∈ ρ −1 (0) and let G1 be the group of M¨obius transformations L : D → D satisfying the following two conditions: • ρ ◦ L = ρ, and • ρ ◦ c ∈ H1 for all arcs c : [0, 1] → D with c(0) = z0 and c(1) = L(z0 ) ∈ ρ −1 (0). Set

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 Φ=

121

2i 2 4z ,− ,− 2 2 1−z 1+z 1 − z4

 dz

for z ∈ C \ {±1, ±i}. Note that if [α] ∈ π1 (C \ {±1, ±i}, 0) then 

ℜ

α

Φ = 0 ⇐⇒ [α] ∈ H1 .

Hence, if ρ1 : D/G1 → C\{±1, ±i} denotes the unique covering map through which ρ factors, the holomorphic 1-form ρ1∗ (Φ) has vanishing real periods on the orbit space D/H1 . The Weierstrass representation of first Scherk’s surface is given by x = (x1 , x2 , x3 ) : D/G1 → R3 ,

 ζ

x(ζ ) = ℜ 0

ρ1∗ (Φ) .

H. Lazard-Holly and W. H. Meeks [218] characterized the first Scherk minimal surface as the only complete embedded doubly-periodic minimal surface in R3 whose quotient by the associated group of translations has genus zero and finitely many ends. If in addition the surface is proper in R3 then no restriction on the number of ends is required. The second Scherk’s surface is defined by the implicit equation sin x3 = sinh x1 sinh x2 .

Fig. 2.4 Second Scherk’s surface

Its Weierstrass representation is given by x = (x1 , x2 , x3 ) : D/G2 → R3 ,

x(ζ ) = −ℑ

 ζ 0

ρ2∗ (Φ) ,

where G2 is the group of M¨obius transformations of D determined by loops γ1 , γ2 in C \ {±1, ±i} based at the origin and satisfying the following conditions: • the loop γ1 winds once around 1 and i, and zero times around −1 and −i,

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• the loop γ2 winds once around 1 and −i, and zero times around −1 and i, and ρ2 : D/G2 → C \ {±1, ±i} is the only covering map through which ρ factors. The second Scherk’s minimal surface is the only properly embedded singlyperiodic minimal surface in R3 with Scherk-type ends whose quotient by the associated group of translations has genus zero and finitely many ends; see W. H. Meeks and H. Rosenberg [254] for a precise description.

2.8.4 Enneper’s Surface Enneper’s minimal surface was discovered by Alfred Enneper [121] in 1868. It is one of the two most basic minimal surfaces in R3 from the point of view of the Enneper–Weierstrass representation, the other one being the catenoid.

Fig. 2.5 Enneper’s surface

Its parametric equations are u

 v  3(1 + v2 ) − u2 , v2 − 3(1 + u2 ) , u2 − v2 , x : R2 → R3 , x(u, v) = 3 3 and the Weierstrass representation is given by x : C → R3 ,

x(ζ ) = ℜ

 ζ 0

 1 − z2 , i(1 + z2 ), 2z dz.

Hence, the complex Gauss map is g(z) = z, and (2.96) shows that the total Gaussian curvature equals TC(x) = −4π. Here are some of its basic properties. • Enneper’s surface contains two orthogonal straight lines meeting at one point; in the above representation they are the images of the real and the imaginary axes. • Enneper’s surface is conjugate to itself.

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123

• Besides the catenoid, Enneper’s surface is the only complete minimal surface in R3 of total Gaussian curvature −4π; R. Osserman [282]. • Besides the half-plane, half of the Enneper surface is the only orientable stable minimal surface properly embedded in R3 with quadratic area growth and whose boundary is a straight line; J. P´erez [288].

2.8.5 Riemann’s Minimal Examples Bernhard Riemann showed in a posthumously published paper in 1868 [298] that the plane, the helicoid, the catenoid, and a certain family {Rt }t>0 of minimal surfaces of infinite topology in R3 are the only minimal surfaces in R3 foliated by circles and straight lines in horizontal planes. The surfaces {Rt }t>0 , called Riemann’s minimal examples, intersect horizontal planes in lines at precisely integer heights, with the radius of the circles going to infinity near the lines. Each surface Rt is invariant under the reflection about the plane x2 = 0, and under the translation by the vector vt = (t, 0, 2). The complement in Rt of a solid open cylinder in the direction of vt consists of infinitely many pairwise disjoint topological punctured discs, each one close and asymptotic to a horizontal plane at integer height.

Fig. 2.6 A Riemann’s minimal example

For each t > 0 let Mt denote the complex algebraic torus   2 Mt = (z, w) ∈ C : w2 = z(z − t)(tz + 1) . Set Mt = Mt \ {(0, 0), (∞, ∞)}, and let ρt : D → Mt denote the holomorphic universal covering. Fix a point p0 ∈ Mt and take a loop γ1 ⊂ Mt based at p0 such that z(γ1 ) is a loop in C \ {0,t, −1/t} winding once around 0 and t, and zero times around

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−1/t. Likewise, take a loop γ0 ⊂ Mt based at p0 such that z(γ0 ) ⊂ C \ {0,t, −1/t} and z(γ0 ) winds twice around 0 and zero times around t and −1/t. Let Ht be the normal subgroup of π1 (Mt , p0 ) generated by the homotopy classes [γ0 ], [γ1 ]. Fix a point z0 ∈ ρt−1 (p0 ) and let Gt be the group of M¨obius transformations L : D → D satisfying the following two conditions: • ρt ◦ L = ρ, and • ρt ◦ c ∈ Ht for all arcs c : [0, 1] → D with c(0) = z0 and c(1) = L(z0 ) ∈ ρt−1 (p0 ). Set Φt =

      1 1 i dz i 1 −z , +z ,1 2 z 2 z w

on Mt , and note that for every [α] ∈ π1 (Mt , p0 ) we have that 

ℜ

α

Φt = 0 ⇐⇒ α ∈ Ht .

Hence, the holomorphic 1-form ρt∗ (Φ) has vanishing real periods on the orbit space D/Ht , where ρt : D/Gt → Mt is the only covering map through which ρt factors. Up to translations and scaling, the Weierstrass representation of Rt is given by xt : D/Gt → R3 ,

 ζ

xt (ζ ) = ℜ

0

ρt∗ (Φt ) .

M. Shiffman [316] proved that every compact minimal annulus in R3 whose boundary consists of circles in parallel planes is a piece of a catenoid or of a Riemann minimal surface. Furthermore, Riemann’s minimal surfaces are the only properly embedded minimal surfaces of genus zero and infinitely many ends in R3 according to W. H. Meeks, J. P´erez and A. Ros [252]. Together with the uniqueness results for the catenoid and the helicoid mentioned above, it follows that the only properly embedded minimal planar domains in R3 are the planes, the catenoids, the helicoids, and Riemann’s minimal surfaces.

2.8.6 Schwarz’s Surfaces The Schwarz minimal surfaces are periodic minimal surfaces described by Hermann A. Schwarz [315] and Edvard R. Neovius [270] in 1883 (see Fig. 2.7). They were generated by solving certain Plateau problems for polygonal curves and reflecting the solutions across the boundary lines to get triply-periodic minimal surfaces in R3 . Likewise, one can use these reflection arguments for minimal surfaces in a unit polyhedron cell meeting its planar faces at right angles. Any Schwarz surface can be viewed as a compact genus three surface in the flat manifold R3 /G, where G ∼ = Z3 is the group of translations leaving it invariant. Details about their geometry can be found in the papers by A. H. Schoen [312] and W. H. Meeks [249].

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125

Fig. 2.7 From left to right: Schwarz P (primitive) surface and Schwarz D (diamond) surface

2.8.7 The Chen–Gackstatter Surfaces The Chen–Gackstatter surfaces were the first known complete immersed minimal surfaces in R3 of positive genus and finite total curvature. They were discovered in 1982 by Chi Cheng Chen and Fritz Gackstatter [83].

Fig. 2.8 Chen–Gackstatter genus one surface

The first Chen–Gackstatter surface has genus one, one topological end asymptotic to Enneper’s surface, and the same symmetry group as Enneper’s surface. It was characterized by F. J. L´opez [225] as the only complete orientable minimal surface of positive genus and total curvature −8π. Setting M = M \ {(∞, 0)}, where M is the algebraic genus one complex curve   2 M = (z, w) ∈ C : w2 (z2 − 1) = z ,

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its Weierstrass representation is given by   p

iB A2

B A2 3 −w , + w , AB dz x : M → R , x(p) = ℜ 2A w 2A w

(2.108)

for suitable constants A ∈ R \ {0} and B ∈ C, |B| = 1. The second Chen–Gackstatter surface has genus two, one topological end asymptotic to Enneper’s surface, and the same symmetry group as Enneper’s surface. It was characterized by F. J. L´opez, F. Mart´ın, and D. Rodr´ıguez [234] as the only complete orientable minimal surface of genus two, total curvature −12π, and having at least eight symmetries. Its Weierstrass representation is given by (2.108), where now M is the algebraic curve of genus two defined by   2 M = (z, w) ∈ C : w2 (z2 − 1) = z(z2 − a2 ) for a suitable constant a ∈ R \ {0, ±1} and M = M \ {(∞, ∞)}. N. Do Espirito-Santo [111], E. C. Thayer [330], and K. Sato [310] constructed Enneper type complete minimal surfaces of any genus k ≥ 3.

2.8.8 Costa’s Surface Celso J. Costa discovered in 1984 [100] a complete minimal surface in R3 of genus one, with three embedded ends, and total curvature −12π. Soon thereafter, D. Hoffman and W. H. Meeks [191] proved that Costa’s surface is embedded. Until the discovery of this fact, it was believed that the only properly embedded minimal surfaces of finite topology in R3 were planes, catenoids, and helicoids. Costa’s surface has a middle planar end and two catenoidal ends, it contains two horizontal straight lines meeting orthogonally, it is invariant under the reflection about two orthogonal vertical planes, and has the D4 dihedral group of symmetries. Setting M = M \ {(1, 0), (−1, 0), (∞, ∞)}, where M is the algebraic complex curve   2 of genus one (a torus) given by (z, w) ∈ C : w2 z = (z2 − 1) , the Weierstrass representation x : M → R3 of Costa’s surface is given for a suitable A ∈ R \ {0} by  p

i 1

 dz 1 1 − Azw , + Azw , 1 2 . x(p) = ℜ 2 Azw 2 Azw z −1 Costa’s surface belongs to the Costa–Hoffman–Meeks [190] family of embedded minimal tori xa : M a \ {(1, 0), (−1, 0), (∞, ∞)} → R3 , a ∈ (0, 1], where   2 M a = (z, w) ∈ C : w2 z = (z + 1)(z − a) , xa (p) = ℜ

 (mz + 1)dz Azw i mz + 1 Azw − , + ,1 . Azw mz + 1 2 Azw mz + 1 (z + 1)(z − a)

 p 1 mz + 1

2

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127

Fig. 2.9 Costa’s surface

The constants A ∈ R \ {0} and m ∈ R depend on a and are uniquely determined by the condition that the map xa is well defined. Costa [101] proved that, up to rigid motions and scaling, the family {xa : Ma → R3 : a ∈ (0, 1]} contains all complete embedded minimal tori in R3 with total curvature −12π.

2.8.9 Henneberg’s Surface Henneberg’s surface (see Fig. 2.10) is a complete nonorientable minimal surface with two branch points (a branched minimal M¨obius strip), named after Ernst Lebrecht Henneberg [189] who first described it in his doctoral dissertation in 1875. It was the only known nonorientable minimal surface until 1981 when W. H. Meeks discovered an immersed minimal M¨obius strip in R3 (see the next example). The Weierstrass representation of its orientable double cover x : C∗ → R3 is given by x(ζ ) = ℜ

 ζ 1

 2(1 − z4 ) dz. z3

(1/z − z), i(1/z + z), 2

The associated antiholomorphic deck transformation I : C∗ → C∗ is given by I(z) = −1/¯z (see Section 2.4), so x(C∗ ) is a nonorientable minimal M¨obius strip. The map x branches at the points ζ ∈ {1, −1, i, −i}, and these branch points cannot be removed by a deformation; see Remark 3.12.6. Since the Gauss map of its orientable double cover is an algebraic map of degree 1 (the identity map z → z), the total Gaussian curvature of Henneberg’s surface is −2π (see (2.96) and (2.97)).

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Fig. 2.10 Henneberg’s surface

2.8.10 Meeks’s M¨obius Strip The Meeks M¨obius strip, discovered by William H. Meeks in 1981 [248], was the first known example of a nonorientable properly immersed minimal surface in R3 .

Fig. 2.11 Meeks’s M¨obius strip

The Weierstrass representation of the orientable double cover x : C∗ → R3 is the following (see Meeks [248, Theorem 2]):    2  ζ (z + 1)z2 z−1 z2 (z + 1) z−1 i(z − 1) dz, − ,i + 2 x(ζ ) = ℜ ,2 2 (z + 1) z z − 1 z − 1 z (z + 1) 2z2 1 where I : C∗ → C∗ , I(z) = −1/¯z, is the associated antiholomorphic deck 2

of x has degree 3, so x has total transformation. The Gauss map g(z) = (z+1)z z−1 curvature −12π by (2.96) and the M¨obius strip has total curvature −6π (see (2.97)).

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129

As shown by Meeks [248, Sect. 4], this is the only complete nonorientable immersed minimal surface in R3 with the absolute total Gaussian curvature smaller than 8π.

2.8.11 The Alarc´on–Forstneriˇc–L´opez M¨obius Strip The first known example of a properly embedded nonorientable minimal surface in R4 is a minimal M¨obius strip found by the authors in 2017 (see [29, Example 6.1]). No such examples can exist in R3 according to W. H. Meeks [248, Corollary 2].

Fig. 2.12 A projection into R3 of the Alarc´on–Forstneriˇc–L´opez M¨obius Strip

Let I : C∗ → C∗ be the fixed-point-free antiholomorphic involution given by I(ζ ) = −1/ζ¯ . The harmonic map x : C∗ → R4 defined by   1 1 i 2 1 1 2 1 x(ζ ) = ℜ i ζ + ,ζ− , ζ − 2 , ζ + 2 ζ ζ 2 ζ 2 ζ is an I-invariant proper conformal minimal immersion such that x(ζ1 ) = x(ζ2 ) if and only if ζ1 = ζ2 or ζ1 = I(ζ2 ). Hence, the image surface S = x(C∗ ) ⊂ R4 is a properly embedded minimal M¨obius strip in R4 . Note that ∂ x = (∂ x1 , ∂ x2 , ∂ x3 , ∂ x4 ) is a Laurent polynomial in ζ containing terms ζ k of degrees −3 ≤ k ≤ +1. Multiplying ∂ x by ζ 3 we see that the generalized Gauss map G = [∂ x1 : ∂ x2 : ∂ x3 : ∂ x4 ] : CP1 → CP3 of x has degree 4, and hence TC(x) = −8π by Corollary 2.6.6. Since x : C∗ → S is a 2-sheeted covering, we infer that the M¨obius strip S has total Gaussian curvature −4π (see (2.97)).

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2.8.12 The L´opez Klein Bottle This minimal surface in R3 , discovered by Francisco J. L´opez in 1993 (see [226]), is homeomorphic to a Klein bottle minus one point, and is the unique complete minimal Klein bottle in R3 with total curvature −8π (see [227]). It is the simplest complete minimal surface with this topology, in the sense that the absolute value of the total curvature of any other complete minimal finitely punctured Klein bottle in R3 is greater than 8π. Furthermore, any complete nonorientable minimal surface in R3 of higher genus has absolute total curvature greater than 8π.

Fig. 2.13 The L´opez Klein bottle

Its symmetry group is generated by the reflections about two straight lines which are contained in the surface and meet orthogonally at one point of R3 . To describe its Weierstrass representation, fix a number r ∈ R \ {0} and consider the algebraic genus one curve (a torus) given by   M = (z, w) ∈ (CP1 )2 : w2 (rz + 1) = z(z − r) , 1 endowed with the antiholomorphic fixed-point-free involution I(z, w) = ( −1 z , w ). Set M = M \{(0, 0), (∞, ∞)}. For a suitable choice of r ∈ R\{0} the map x : M → R3 given by     2  p  1 w(z − 1) w(z − 1) z+1 z+1 i(z − 1) i dz − + ℜ , ,1 2 w(z − 1) z+1 2 w(z − 1) z+1 z2

is the double cover of the L´opez Klein bottle, with the associated antiholomorphic deck transformation I.

Chapter 3

Approximation and Interpolation Theorems for Minimal Surfaces

In this chapter we develop the first major collection of new results concerning oriented minimal surfaces in Rn and holomorphic null curves in Cn for any n ≥ 3. All of them were obtained since 2012 by using complex analytic methods. The source of our maps is always a connected open Riemann surface, M, with a fixed choice of a complex structure. In some results we consider instead a compact bordered Riemann surface, i.e., a compact smoothly bounded domain in an open Riemann surface. The focus is on approximation, interpolation, general position, completeness, and properness theorems for conformal minimal immersions M → Rn and holomorphic null curves M → Cn for any n ≥ 3. Analogous results for nonorientable minimal surfaces are presented in our Memoir publication [29]. After establishing the notation in Section 3.1, we develop in Section 3.2 an important technical tool used in most subsequent constructions; see Lemma 3.2.1. It says that every holomorphic map f : M → A∗ ⊂ Cn from a compact bordered Riemann surface into the punctured null quadric A∗ = A \ {0} (2.54), whose image is not contained in a single ray of A, can be embedded into a holomorphic family of maps M → A∗ which is period dominating, in the sense that the derivative (with respect to the parameters) of the period map on a basis of the homology group H1 (M, Z) has maximal rank. As an immediate application, we show that for any r ∈ N the space NCrnf (M, Cn ) of nonflat holomorphic null curves M → Cn which are smooth of class C r (M) carries the structure of an infinite-dimensional complex Banach manifold; likewise, the space CMIrnf (M, Cn ) of nonflat conformal minimal immersions M → Rn of class C r (M) is an infinite-dimensional real analytic Banach manifold (see Theorem 3.2.3). Our analysis suggests that the flat maps in NCr (M, Cn ) and CMIr (M, Rn ) are singular points of these spaces. The first major application of Lemma 3.2.1 and the Oka principle for holomorphic maps from Stein manifolds to the punctured null quadric A∗ (see Section 1.13) are approximation and jet-interpolation theorems for conformal minimal immersions and holomorphic null curves, in the spirit of classical theorems of Runge, Mergelyan, and Weierstrass for holomorphic maps; see Theorems 3.6.1, 3.6.2, and 3.7.1. We also prove a general position theorem which enables us to approximate a conformal minimal immersion M → Rn by those with simple double points if n = 4, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_3

131

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3 Approximation and Interpolation Theorems for Minimal Surfaces

and by embedded ones if n ≥ 5 (see Theorem 3.4). Another important technical ingredient, developed in Section 3.5, is the construction of paths in A∗ with given integrals (periods). Here we use some basic ideas of Gromov’s convex integration theory. With these tools in hand, we prove Theorems 3.6.1 and 3.6.2 in Section 3.6. In Section 3.7 we obtain the second main theorem of the chapter, a Runge– Mergelyan–Weierstrass type approximation and interpolation theorem for conformal minimal immersions in which some of the component functions are harmonic on all of M and are kept fixed while approximating the remaining components; see Theorem 3.7.1. This line of results is continued in Section 3.8 with analogues of Mittag-Leffler’s theorem (see Theorem 3.8.2) and Carleman’s approximation theorem (see Theorem 3.8.6) for minimal surfaces in Rn . In Section 3.9 we prove that the approximating conformal minimal immersions in Theorems 3.6.1 and 3.7.1 can be chosen to be complete; see Theorem 3.9.1. Properness of such immersions is discussed in Sections 3.10 and 3.11. Completeness and properness are among the most important and much studied global properties in minimal surface theory. The former is a fundamental condition in Riemannian geometry whereas the latter, being an extrinsic topological property, is crucial in the theory of submanifolds. An immersion x : M → Rn is said to be complete if the path x ◦ γ has infinite Euclidean length in Rn for every divergent path γ in M; cf. Definition 1.6.6. Completeness of x is equivalent to completeness of the Riemannian metric x∗ ds2 on M induced from the Euclidean metric ds2 on Rn by the immersion x. More precisely, we are asking that (M, x∗ ds2 ) be a complete metric space in the sense that Cauchy sequences are convergent. If this is the case then, by the Hopf–Rinow theorem, (M, x∗ ds2 ) is also geodesically complete, in the sense that geodesics are defined for all values of time. A map x : M → Rn is said to be proper if the preimage x−1 (K) is compact in M for any compact set K ⊂ Rn ; equivalently, x ◦ γ is a divergent path in Rn for every divergent path γ in M. Obviously a proper map is complete, but the converse does not hold in general. In Section 3.12 we consider the topological shape of the spaces CMInf (M, Rn ) ⊂ CMI(M, Rn ) of nonflat conformal minimal immersions and nonflat null curves NCnf (M, Cn ) ⊂ NC(M, Cn ) from a given connected open Riemann surface M. Differentiation induces a natural map CMI(M, Rn ) → O(M, A∗ ) → C (M, A∗ ) which provides a weak homotopy equivalence from CMInf (M, Rn ) (or NCnf (M, Cn )) to the space C (M, A∗ ) of continuous maps M → A∗ ; see Theorem 3.12.2. If in addition the Riemann surface M is of finite topological type then these maps are genuine homotopy equivalences. In particular, we identify the path components of these spaces: if n ≥ 4 then the spaces CMInf (M, Rn ) and NCnf (M, Cn ) are connected, while for n = 3 each of them has 2l connected components where H1 (M, Z) = Zl (see Corollary 3.12.4). The same holds for the path components of the bigger spaces CMI(M, Rn ) and NC(M, Cn ) (see Corollary 5.7.7). Most results in this chapter concerning holomorphic null curves generalize to immersed holomorphic curves in Cn whose derivative maps have range in an irreducible conical complex subvariety A ⊂ Cn such that A \ {0} is an Oka manifold; see Remark 3.2.2 (B) and the papers [22, 12].

3.1 Spaces of Conformal Minimal Immersions and Null Curves

133

3.1 Spaces of Conformal Minimal Immersions and Null Curves Most of the notation used in this chapter has already been introduced in Sections 1.1 and 1.12. Thus, if M and X are complex manifolds then O(M, X) denotes the space of holomorphic maps M → X, and O(M) = O(M, C). If K is a compact set in M then A r (K, X) denotes the space of maps K → X of class C r which are holomorphic in the interior K˚ of K. If K is a domain with C 1 boundary in a Riemann surface M and r ∈ Z+ then A r (K, X) is a complex Banach manifold modelled on the Banach space A r (K, Cn ) with n = dim X (see [140, Theorem 8.13.1] or [137, Theorem 1.1]), and every map f ∈ A r (K, X) is a limit of maps in O(K, X), i.e., maps which are holomorphic on open neighbourhoods of K. In this section we introduce some additional notation and terminology pertaining to conformal minimal surfaces and holomorphic null curves. If M is an open Riemann surface and n ≥ 3 is an integer then CMI(M, Rn )

(3.1)

denotes the space of conformal minimal (equivalently, conformal harmonic) immersions M → Rn (see Theorem 2.3.1), and NC(M, Cn )

(3.2)

denotes the space of holomorphic null immersions M → Cn (see Definition 2.3.3). These spaces are endowed with the compact-open topology. Assume now that M is a compact bordered Riemann surface with nonempty smooth boundary bM (see Section 1.10). By Corollary 1.10.9 and Theorem 1.10.10, such M is conformally diffeomorphic to a compact domain with real analytic boundary in a compact Riemann surface. Given r ∈ N we denote by CMIr (M, Rn )

(3.3)

the space of conformal minimal immersions M → Rn of class C r (M). More precisely, an immersion x = (x1 , . . . , xn ) : M → Rn of class C r (M) belongs to CMIr (M, Rn ) if and only if the (1, 0)-form ∂ x = (∂ x1 , . . . , ∂ xn ) is holomorphic on the interior M˚ = M \ bM and it satisfies the nullity condition (2.53). Similarly, NCr (M, Cn )

(3.4)

denotes the space of holomorphic null immersions M → Cn of class A r (M). Recall (cf. Definition 2.5.2 and Lemma 2.5.3) that a conformal minimal immersion x : M → Rn is nonflat or full if and only if the map f = ∂ x/θ : M → A∗ is such, where θ is any nowhere vanishing holomorphic 1-form on M and A∗ is the punctured null quadric (2.54). The analogous definitions apply to null curves, and also in the case when M is a compact bordered surface.

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We denote by CMIfull (M, Rn ) ⊂ CMInf (M, Rn )

(3.5)

CMI(M, Rn )

the open subsets of consisting of immersions which are full or nonflat, respectively, on every connected component of M. The analogous notation CMIrfull (M, Rn ) ⊂ CMIrnf (M, Rn ) ⊂ CMIr (M, Rn ),

r∈N

(3.6)

is used when M is a compact bordered Riemann surface. We denote by NCfull (M, Cn ) ⊂ NCnf (M, Cn ),

NCrfull (M, Cn ) ⊂ NCrnf (M, Cn )

(3.7)

the corresponding spaces of full and nonflat holomorphic null curves in Cn from an open or compact bordered Riemann surface. The tangent space Tz A to the null quadric (1.116) is the kernel at z = (z1 , . . . , zn ) of the (1, 0)-form ∑nj=1 z j dz j ; hence, Tz A = Tw A for z, w ∈ A∗ if and only if z and w are C-collinear. This implies the following lemma. Lemma 3.1.1. A holomorphic map f : M → A∗ from a connected Riemann surface M to the punctured null quadric is nonflat if and only if the linear span of the tangent spaces T f (p) A ⊂ T f (p) Cn ∼ = Cn over all points p ∈ M equals Cn . We now introduce the notion of a generalized conformal minimal immersion and generalized null curve on an admissible set in a Riemann surface. Such maps appear naturally in proofs of approximation and interpolation theorems for conformal minimal immersions and holomorphic null curves. Definition 3.1.2 (Generalized conformal minimal immersions). Let S = K ∪ E be an admissible set in a Riemann surface M (see Definition 1.12.9), and let θ be a nowhere vanishing holomorphic 1-form on a neighbourhood of S in M. A generalized conformal minimal immersion S → Rn (n ≥ 3) of class C r (r ∈ N) is a pair (x, f θ ), where x : S → Rn is a C r map whose restriction to S˚ = K˚ is a conformal minimal immersion, and f ∈ A r−1 (S, A∗ ) satisfies the following conditions: (a) f θ = 2∂ x holds on K, and (b) for any smooth path α in M parameterizing a connected component of E = S \ K we have that ℜ(α ∗ ( f θ )) = α ∗ (dx) = d(x ◦ α). A generalized conformal minimal immersion (x, f θ ) is nonflat or full if and only if the map f ∈ A r−1 (S, A∗ ) is everywhere nonflat or full, in the sense that f is nonflat or full, respectively, on every relatively open subset of S (see Definition 2.5.2). Note that if S = K (i.e., E = ∅) then, by the identity principle, f is everywhere nonflat or full if and only if it is nonflat or full, respectively. Since ℜ(2∂ x) = dx, condition (b) in Definition 3.1.2 is compatible with the requirement f θ = 2∂ x in part (a). Note also that for every immersion x : E → Rn of class C r (r ∈ N) from a smooth embedded arc E ⊂ M there exists a path f : E → A∗ of class C r−1 such that (x, f θ ) is a generalized conformal minimal immersion

3.1 Spaces of Conformal Minimal Immersions and Null Curves

135

E → Rn . Indeed, if α : [0, 1] → E is a smooth regular parameterization and we write (α ∗ θ )(t) = c(t)dt, then condition (b) is equivalent to ℜ( f (α(t))c(t)) = d n n dt x(α(t)) =: g(t) ∈ R∗ for t ∈ [0, 1]. Choosing a map h : [0, 1] → R∗ orthogonal −1 to g(t) and satisfying |g(t)| = |h(t)|, the function f (α(t)) = c(t) (g(t) + i h(t)) fulfills the required condition. Given an admissible set S, the spaces of nonflat, full, and all generalized conformal minimal immersions S → Rn of class C r are denoted respectively by GCMIrfull (S, Rn ) ⊂ GCMIrnf (S, Rn ) ⊂ GCMIr (S, Rn ).

(3.8)

If (x, f θ ) ∈ GCMIr (S, Rn ) and S is connected, then for any point p0 ∈ S the map x : S → Rn can be recovered from f by the Weierstrass formula (2.61):  p

x(p) = x(p0 ) + ℜ

p0

fθ,

p ∈ S.

(3.9)

Conversely, a map f ∈ A r−1 (S, A∗ ) such that the (1, 0)-form f θ has vanishing real periods on all closed curves in S (equivalently, on curves forming a basis of H1 (S, Z)) determines a generalized conformal minimal immersion (3.9). Definition 3.1.3 (Generalized null curves). Let S = K ∪E and θ be as in Definition 3.1.2. A generalized null curve S → Cn (n ≥ 3) of class C r (r ∈ N) is a pair (z, f θ ) where z ∈ A r (S, Cn ), f ∈ A r−1 (S, A∗ ), and the following conditions hold: (a) f θ = dz = ∂ z holds on K (hence z : K˚ → Cn is a holomorphic null curve), and (b) for any smooth path α in M parameterizing a connected component of E we have α ∗ ( f θ ) = α ∗ (dz) = d(z ◦ α). A generalized null curve (z, f θ ) is nonflat or full if and only if the map f is everywhere nonflat or full, respectively (see Definitions 2.5.2 and 3.1.2). The spaces of full, nonflat, and all generalized null curves S → Cn of class C r on an admissible set S are denoted, respectively, by GNCrfull (S, Cn ) ⊂ GNCrnf (S, Cn ) ⊂ GNCr (S, Cn ).

(3.10)

If (z, f θ ) ∈ NCr (S, Rn ) and S is connected, then for any p0 ∈ S the map z : S → Cn can be recovered from f by the Weierstrass formula (2.64):  p

z(p) = z(p0 ) +

p0

fθ,

p ∈ S.

(3.11)

Conversely, a map f ∈ A r−1 (S, A∗ ) such that f θ has vanishing periods on a homology basis of H1 (S, Z) determines a generalized null curve by (3.11).

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3.2 Period Dominating Sprays of Maps into the Null Quadric In this section we show that every nonflat holomorphic map from an admissible set S in a Riemann surface M into the punctured null quadric A∗ = An−1 ⊂ ∗ Cn (2.54) is the core map of a period dominating spray of holomorphic maps S → A∗ ; see Lemma 3.2.1. This is one of the main technical tools in the approximation theory for conformal minimal immersions. The main point is that, when approximating the derivative of a conformal minimal immersion, its periods change. However, approximating a period dominating holomorphic spray, whose core map has vanishing real periods on a homology basis of S, sufficiently closely by a holomorphic spray on a larger domain yields a map in the new spray having the same property, so this map integrates to a conformal minimal immersion by the Weierstrass formula (2.61). The same tool is used for approximating holomorphic null curves and more general directed holomorphic curves (see Remark 3.2.2 (B)). Let M be a connected open Riemann surface. Fix a nowhere vanishing holomorphic 1-form θ on M (see Theorem 1.10.5). Let C = {C1 , . . . ,Cl } be a collection of smooth oriented embedded arcs and closed Jordan curves in M. Set C = li=1 Ci . To any such collection and integer n ∈ N we associate the period map P = (P1 , . . . , Pl ) : C (C, Cn ) −→ (Cn )l = Cln , Pi ( f ) =

 Ci

f θ ∈ Cn ,

f ∈ C (C, Cn ), i = 1, . . . , l.

(3.12)

The original sources for the following result are [22, Lemma 5.1] and [30, Lemma 3.2]; here we give an improved version with interpolation on finite sets. Lemma 3.2.1. Let S = K ∪ E be an admissible set in an open Riemann surface M } be a collection of smooth oriented Jordan (see Def. 1.12.9), let C = {C1 , . . . ,C l curves and arcs in S such that C = li=1 Ci is Runge in S, and let f ∈ A r (S, A∗ ) for some r ∈ Z+ . Assume that every curve Ci ∈ C contains a nontrivial arc Ii disjoint from j=i C j such that f (Ii ) is not contained in a ray of A. (This holds if f : Ii → A∗ is nonflat, cf. Def. 2.5.2.) Then there exist an open neighbourhood U of the origin in Cln and a map Φ f : S ×U → A∗ of class A r (S ×U, A∗ ) such that Φ f (· , 0) = f and ∂   P(Φ f (· ,t)) : (Cn )l −→ (Cn )l ∂t t=0

is an isomorphism.

(3.13)

Furthermore, given a finite set P ⊂ S, we may choose Φ f such that each map Φ f (· ,t) : S → A∗ (t ∈ U) agrees with f at every point of P, and it agrees with f ˚ to any given finite order at every point of P ∩ S. r Every f0 ∈ A (S, A∗ ) satisfying the above assumptions has a neighbourhood Ω ⊂ A r (S, A∗ ) and a holomorphic map Ω f → Φ f having these properties. A map Φ f satisfying Lemma 3.2.1 is called a period dominating spray of maps S → A∗ (for the family of curves C ) with the core Φ f (· , 0) = f .

3.2 Period Dominating Sprays of Maps into the Null Quadric

137

Proof. We first show how to find a spray Φ f satisfying the period domination property (3.13); interpolation is a simple addition which will be explained at the end. We assume that r = 0 since the proof for r > 0 is the same. Recall that Ii is a nontrivial arc in the curve Ci ∈ C which is disjoint from j=i C j and such that f (Ii ) is not contained in a ray of A, so f is nonflat on Ii . By Lemma 3.1.1 we can find for every i = 1, . . . , l points pi,k ∈ Ii and holomorphic vector fields Vi,k on Cn (k = 1, . . . , n) tangent to the null quadric A and such that   span Vi,k ( f (pi,k )) : k = 1, . . . , n = Cn . (3.14) Let φti,k denote the flow of Vi,k . Write t = (t1 , . . . ,tl ) ∈ (Cn )l where ti = (ti,1 , . . . ,ti,n ) ∈ Cn . Pick an open neighbourhood U0 of the origin in (Cn )l such that the map S ×U0 (p,t) −→ φt1,1 ◦ · · · ◦ φtl,n ( f (p)) ∈ A∗ (3.15) 1,1 l,n is well defined for all t ∈ U0 and p ∈ S. We take the composition of flows φti,k for i,k all indices i = 1, . . . , l and k = 1, . . . , n; the order of terms is unimportant. For every such pair (i, k) we choose a smooth function hi,k : C → C supported on a short arc around the point pi,k ∈ Ii ⊂ Ci . Then, the map Φ(p,t) = φh1,1 (p)t 1,1

1,1

◦ · · · ◦ φhl,n (p)t ( f (p)) ∈ A∗ l,n

l,n

(3.16)

is well defined for all p ∈ C and all t ∈ Cln in a neighbourhood U1 ⊂ Cln of the origin (depending on the sizes of the functions hi,k ), and it satisfies Φ(· , 0) = f . Clearly, Φ(p, · ) is holomorphic on U1 for each p ∈ C and  ∂ Φ(p,t)  = h j,k (p)V j,k ( f (p)). ∂t j,k t=0 Let P = (P1 , . . . , Pl ) denote the period map (3.12) associated to the family of curves C in the lemma. Note that   ∂ Pi (Φ(· ,t))  = h j,k · (V j,k ◦ f ) · θ ∈ Cn (3.17)  ∂t j,k C i t=0 for all i, j = 1, . . . , l and k = 1, . . . , n. A suitable choice of the function hi,k supported on a short arc around pi,k in Ii ⊂ Ci ensures that this vector is as close as desired to Vi,k ( f (pi,k )) ∈ Cn if i = j, and it equals zero if i = j. Note that the matrix of the differential (3.13) has block structure, with a block of size n × n for each pair of indices i, j ∈ {1, . . . , l} (its total size is ln × ln). The off-diagonal blocks (i.e., those for i = j) vanish by what has been said above, while the diagonal blocks corresponding to i = j are invertible in view of (3.14), (3.17), and the choice of the functions hi,k . Hence, the differential (3.13) is invertible. Since the set C is Runge in S, Theorem 1.12.7 (the Bishop–Mergelyan theorem) shows that we can approximate each function hi,k as closely as desired uniformly on C by a holomorphic function h˜ i,k ∈ O(S) on a neighbourhood of S. Inserting

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these new functions into (3.16) furnishes a spray Φ f : S × U → A∗ satisfying the conclusion of the lemma, where U ⊂ U1 is a smaller open neighbourhood of the origin in Cln . (More precisely, U is determined by the condition that each vector t˜ = (h˜ i,k (p)ti,k ) ∈ Cln whose components are the products h˜ i,k (p)ti,k for some p ∈ S and t = (ti,k ) ∈ U belongs to the domain U0 of the initial spray (3.15).) Indeed, condition (3.13) holds provided the approximation of the functions hi,k by h˜ i,k is close enough uniformly on C, and the other properties are obvious. It remains to explain how to interpolate on a finite set P ⊂ S. Fix s ∈ N. Let g ∈ O(S) be a holomorphic function vanishing to order s at each point of P and having no other zeros. Choose points pi,k ∈ Ii \ P and vector fields Vi,k as above. As before, we can find continuous functions hi,k : C → C supported on short arcs in Ii around the points pi,k such that, setting hi,k = ghi,k for all i = 1, . . . , l and k = 1, . . . , n, conditions (3.17) hold. We then approximate each hi,k by h˜ i,k ∈ O(S) and set h˜ i,k = gh˜ i,k ∈ O(S). Inserting these functions into (3.16) yields a spray Φ f with the desired properties; in particular, Φ f (· ,t) : S → A∗ (t ∈ U) agrees with f at ˚ (See [12, Lemma 2.2] for the details.) each point p ∈ P, to order s if p ∈ P ∩ S. Clearly, the period dominating spray Φ f constructed above depends holomorphically on f ∈ A r (S, A∗ ) in a neighbourhood of any given map f0 ∈ A r (S, A∗ ); we  can use the same functions h˜ i,k ∈ O(S) for all f near f0 .  Remark 3.2.2. (A) Assume that { fq ∈ A r (S, A∗ ) : q ∈ Q} is a continuous family of maps satisfying the nonflatness condition in Lemma 3.2.1, with the parameter q in a compact Hausdorff space Q. By using additional flows in the definition of the spray Φ f (3.16), we can obtain a family of period dominating sprays Φ fq depending continuously on q ∈ Q. In this case, (3.13) is replaced by the condition that the partial differential of the period map on the variable t is surjective at t = 0. (B) The analogue of Lemma 3.2.1 holds with A replaced by any irreducible conical complex (hence algebraic) subvariety A ⊂ Cn such that A∗ = A \ {0} is smooth and A is full. In this case, the nonflatness condition on f on the arcs Ii must be replaced by the condition that the C-linear span of the union of tangent spaces T f (t) A ⊂ T f (t) Cn ∼ = Cn for t ∈ Ii equals Cn . (See [22, Lemma 5.1].) (C) We are unable to find a period dominating spray whose core f is a flat map with image contained in a complex ray C∗ ν for some ν ∈ A∗ , the reason being that the tangent space Tz A∗ ⊂ Tz Cn ∼ = Cn is independent of the point z ∈ C∗ ν. (D) The vector fields Vi,k in the proof of Lemma 3.2.1 need not be complete. However, for applications it is sometimes convenient to use complete holomorphic vector fields yielding globally defined period dominating sprays Φ : S × CN → A∗ . By Example 1.13.8 the tangent bundle T A∗ is pointwise spanned by linear (hence complete) holomorphic vector fields. Since linear combinations of linear vector fields are still linear, we see that every tangent vector v ∈ Tz A∗ is the value at z of a complete (linear) vector field V on Cn tangent to A. (E) As seen in the proof, one can also ensure that Φ f (· ,t) (t ∈ U) agrees with f to any given order up to r (the smoothness class of f ) at each point of the finite ˚ this however does not give any advantage for subsequent applications. set P \ S;

3.2 Period Dominating Sprays of Maps into the Null Quadric

139

The same situation holds in all subsequent results of this type, as for instance in the approximation theorems with interpolation by conformal minimal immersions in Sections 3.4-3.10.   The following result (see [30, Theorem 3.1]) is an application of Lemma 3.2.1. Theorem 3.2.3. Let M be a compact bordered Riemann surface, and let n ≥ 3 and r ≥ 1 be integers. Then the following hold. (a) The space CMIrnf (M, Rn ) is a real analytic Banach manifold. (b) The space NCrnf (M, Cn ) is a complex Banach manifold. Proof. By [137, Theorem 1.1] the space A r−1 (M, A∗ ) is a complex Banach manifold modelled on the Banach space A r−1 (M, Cn−1 ), where dim A∗ = n − 1. Let C = {C1 , . . . ,Cl } be smooth closed Jordan curves in M˚ forming a homology basis of M as in Lemma 1.12.10. Let P = (P1 , . . . , Pl ) : A r−1 (M, Cn ) → Cln denote the associated period map (3.12) and set   A0r−1 (M, A∗ ) = f ∈ A r−1 (M, A∗ ) : ℜ(P( f )) = 0 . r−1 Denote by A0,nf (M, A∗ ) the open subset of A0r−1 (M, A∗ ) consisting of all nonflat maps (see Definition 2.5.2). Lemma 3.2.1 implies that the differential dP f0 has r−1 (M, A∗ ). (Note that the restriction of maximal rank ln at any point f0 ∈ A0,nf a nonflat holomorphic map M → A∗ to a nontrivial arc I ⊂ M is nonflat on I in view of the identity principle, and hence the hypotheses of the lemma hold.) By the implicit function theorem for Banach spaces, it follows that any map r−1 f0 ∈ A0,nf (M, A∗ ) admits an open neighbourhood Ω ⊂ A r−1 (M, A∗ ) such that r−1 Ω ∩ A0r−1 (M, A∗ ) = Ω ∩ A0,nf (M, A∗ ) is a real analytic Banach submanifold of Ω parameterized by an open set in the kernel of the real part ℜ(dP f0 ) of the differential of P at f0 ; the latter is a real codimension ln subspace of the complex Banach space A r−1 (M, Cn−1 ) (the tangent space at f0 of the complex Banach r−1 (M, A∗ ) is a real analytic Banach manifold A r−1 (M, A∗ )). This shows that A0,nf manifold.  For any base point p0 ∈ M, integration p → v + pp0 ℜ( f θ ) (p ∈ M, v ∈ Rn ) r−1 provides an isomorphism between the Banach manifold A0,nf (M, A∗ ) × Rn and r n CMInf (M, R ) (see (3.9)), so the latter is also a real analytic Banach manifold. This proves part (a). A similar argument gives part (b).  

Remark 3.2.4. (A) We do not know whether the spaces CMIr (M, Rn ) and NCr (M, Cn ) are also Banach manifolds. In fact, it seems that flat conformal minimal immersions and flat holomorphic null curves are singular points of these spaces. (B) The analogous result holds for the space GCMIrnf (S) of nonflat generalized conformal minimal immersions on an admissible set S ⊂ M; we can apply essentially the same proof and take into account Lemma 1.12.10. (C) Note that CMIrfull (M, Rn ) and NCrfull (M, Cn ) are open subsets of Banach manifolds CMIrnf (M, Rn ) and NCrnf (M, Cn ), respectively.

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3.3 A Semiglobal Approximation and Interpolation Theorem The main result of this section is Proposition 3.3.2, which provides the noncritical case in the proof of Theorem 3.6.1. We begin with the following lemma concerning approximation and interpolation of maps into the null quadric A∗ ⊂ Cn (2.54). Lemma 3.3.1. Assume that M is a connected open Riemann surface, S = K ∪ E is a Runge admissible set in M, and C = {C1 , . . . ,Cl } is a collection of smooth oriented Jordan curves and arcs in S such that every Ci ∈ C contains a nontrivial arc Ii disjoint from j=i C j and C = li=1 Ci is Runge in M. Let P denote the period map (3.12) associated to C . Then, given a finite set of points A = {a1 , . . . , am } ⊂ S, every map f ∈ A r (S, A∗ ) (r ∈ Z+ ) can be approximated in C r (S) by full holomorphic maps F ∈ O(M, A∗ ) such that P(F) = P( f ), F agrees with f on A, and F agrees ˚ with f to any given finite order at every point of A ∩ S. In particular, if (x, f θ ) is a generalized conformal minimal immersion S → Rn and the collection C includes a homology basis for S (such a family C exists by Lemma 1.12.10), then any holomorphic map F ∈ O(M, A∗ ) furnished by the lemma also has vanishing real periods on a homology basis of S. Proof. The proof proceeds in two steps. In the first step, we approximate and interpolate f by a full map g ∈ A r (S, A∗ ) with P(g) = P( f ). (If f is full, we proceed directly to step 2.) In the second step, we apply Lemma 3.2.1 to find a period dominating spray of maps in A r (S, A∗ ) with the core g. We then approximate and interpolate this spray by a holomorphic one on M, using the Mergelyan theorem for maps from open Riemann surfaces into A∗ (see Theorems 1.13.1 and 1.13.3). A suitable value of the parameter in the new spray gives a map F ∈ O(M, A∗ ) approximating and interpolating f and having the same periods as f . Step 1: Approximation by a full map. For simplicity of exposition we first consider the case when S = K is a connected compact domain. Let Σ ( f ) = span f (K) denote the C-linear subspace of Cn spanned by f (K). Assume that Σ ( f ) is a proper subspace of Cn . Choose points p1 , . . . , pk ∈ K \ A such that Σ ( f ) = span{ f (p1 ), . . . , f (pk )}. We claim that there is a holomorphic / Σ ( f ) for some point vector field V on Cn , tangential to A, such that V ( f (p0 )) ∈ p0 ∈ K \ ({p1 , . . . , pk } ∪ A). This is obvious if dim Σ ( f ) < n − 1 since the tangent space Tz A∗ at any point z ∈ A∗ is a hyperplane. If dim Σ ( f ) = n − 1, there are pairs of nearby points in K whose images by f lie in different nearby rays of A. The tangent planes to A at these two points (considered as linear hyperplanes in Cn ) are not the same, so at least one of them contains a vector not in Σ ( f ). This vector is the value at the respective point of a vector field V tangent to A, proving the claim. Let t → φt (z) denote the flow of V . Recall that φt (z) = z +tV (z) + O(t 2 ) for small values of |t|. Let s ∈ N. Choose a function h ∈ O(K) which vanishes at the points p1 , . . . , pk , vanishes to order s at the points in A, and satisfies h(p0 ) = 1. For any ξ ∈ A r (K) near the zero function we define the map Θ (ξ ) ∈ A r (K, A∗ ) by Θ (ξ )(p) = φξ (p)h(p) ( f (p)),

p ∈ K.

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141

Clearly, Θ (ξ ) depends holomorphically on ξ . Consider the holomorphic map A r (K) ξ −→ P(Θ (ξ )) ∈ (Cn )l . Since A r (K) is an infinite-dimensional complex Banach space, there is a nonconstant function ξ ∈ A r (K) arbitrarily close to the zero function such that P(Θ (ξ )) = P(Θ (0)) = P( f ). Set g = Θ (ξ ). Since the function ξ h vanishes at the points p1 , . . . , pk , we have that g(p j ) = φ 0 ( f (p j )) = f (p j ) for j = 1, . . . , k, so Σ ( f ) ⊂ Σ (g). As ξ h is nonconstant on K, we can move the point p0 slightly if necessary so that ξ (p0 ) = 0 and h(p0 ) is close to 1. It follows that g(p0 ) is close to f (p0 ) + ξ (p0 )h(p0 )V ( f (p0 )), which does not belong to Σ ( f ) by the choices of V and p0 . Since h vanishes to order s at the points in A, g agrees with f on A, and it agrees with f to order s at every point ˚ The proof is finished by a finite induction, increasing the dimension of the of A ∩ S. span Σ ( f ) at every step. For a general admissible set S = K ∪ E we apply the same argument on every connected component of K, noting also that fullness on a curve in E can be achieved by a small smooth deformation which is fixed on E ∩ (A ∪ K). Step 2: Approximation by maps in O(M, A∗ ). Replacing f by the map g from step 1, we may assume that f is full and hence nonflat. By Lemmas 1.12.10 and 3.2.1 there exists a period dominating spray Φ f : S × CN → A∗ with Φ(· , 0) = f . (We can obtain a spray defined for all parameter values t ∈ CN in view of Remark 3.2.2 (D).) Since S is Runge in M and A∗ is an Oka manifold (see Example 1.13.8), Theorems 1.13.1 and 1.13.3 show that f can be approximated as closely as desired in C r (S) by a holomorphic map f˜ : M → A∗ which agrees with f at A and to a given order ˚ By the Mergelyan theorem for functions on admissible s ∈ N at every point in A ∩ S. sets (see Theorem 1.12.11) we can also approximate the functions hi,k ∈ A r (S) (see (3.16)) by functions h˜ i,k ∈ O(M) which agree with hi,k on A and to order s at every ˚ The associated spray Φ ˜ : M × CN → A∗ with the core f˜, defined by point of A ∩ S. f (3.16) with f replaced by f˜ and hi,k replaced by h˜ i,k , is then close to Φ f on S × BN , where BN is the ball in CN . Assuming that the approximations are close enough, the period domination property of the spray Φ f and the implicit function theorem give a point t0 ∈ BN close to the origin such that the map F = Φ f˜(· ,t0 ) ∈ O(M, A∗ ) satisfies P(F) = P( f ). By the construction, F satisfies the required interpolation condition and can be chosen arbitrarily close to f in C r (S, A∗ ), which in particular guarantees that F is full.   We are now ready to prove the following result which will provide the noncritical case in the proof of Theorem 3.6.1. Proposition 3.3.2. Assume that M is an open Riemann surface, θ is a nowhere vanishing holomorphic 1-form on M, S is a connected admissible set in M such ∼ = that the inclusion S → M induces an isomorphism H1 (S, Z) −→ H1 (M, Z), A = {a1 , . . . , ak } is a finite subset of S, and r, s ∈ N. Then, the following hold.

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(a) Every generalized conformal minimal immersion (x, f θ ) ∈ GCMIr (S, Rn ) can be approximated in C r (S) by full conformal minimal immersions X : M → Rn with FluxX = Fluxx . (b) Every generalized null curve (z, f θ ) ∈ GNCr (S, Cn ) can be approximated in C r (S) by full holomorphic null curves Z : M → Cn . In both cases, the maps X and Z can be chosen to agree with x and z, respectively, ˚ at every point of the given finite set A, and to order s at every point of A ∩ S. Note that the condition on S implies that it is Runge in M. Indeed, S has no holes in M (i.e., it is Runge in M) if and only if the inclusion S → M induces an injective homomorphism H1 (S, Z) → H1 (M, Z) of the first homology groups. Proposition 3.3.2 also holds if the admissible set S is disconnected, but this will be shown in the proof of Theorem 3.6.1 since we shall also need Lemma 3.5.4. The latter result provides a suitable extension of a generalized conformal minimal immersion across arcs connecting different connected components of S. Proof. Write S = K ∪ E as in Definition 1.12.9 and let Ki for i = 1, . . . , m be the connected components of K. Let C be a Runge homology basis of S furnished by Lemma 1.12.10; then C is also a homology basis of M in view of the hypothesis on S. We enlarge the given finite set A in the proposition by adding to it the endpoints of all connected components Ei of E = S \ K (note that each Ei is an arc, unless S is a single closed curve) and also the endpoints ai, j and qi ∈ Ki of all arcs Ai, j ⊂ Ki chosen in the proof of Lemma 1.12.10. We now construct a family C of arcs and closed curves in S as follows. (i) If a curve C ∈ C does not contain any points of A except q1 , we put it in C. Otherwise, we split C into the union of finitely many arcs lying back to back, with the points of A ∩C as the common endpoints of adjacent arcs, and we put all these arcs in C. (ii) If Ek is a connected component of E which is not contained in any of the curves from the previous item, we connect each endpoint of Ek contained in a connected component Ki of K to the vertex qi ∈ Ki as described in case 2 in the proof of Lemma 1.12.10, first going along bKi to a suitable point ai, j and then going along Ai, j to qi . We then split the resulting curve into arcs at the points of A as in the previous case, and we put all these arcs in C. (iii) Let A denote the set of points a ∈ A belonging to at least one of the curves in the family C constructed thus far. Any remaining point a ∈ A \ A lies in a connected component Ki of K. Choose an embedded arc Λa ⊂ Ki connecting a to the vertex qi ∈ Ki such that Λa does not meet any arcs from C, or arcs Λa for a = a ∈ A \ A , other than at qi . We put the arcs Λa for a ∈ A \ A in the family C whose construction is now complete. It is evident from the construction that the union C of all arcs and closed curves in the family C is a connected Runge set in S, and every curve in C

3.4 General Position Theorems

143

contains a nontrivial arc which is not contained in any other curve from the family. Furthermore, C contains a homology basis of S, and hence of M. Let P be the period map (3.12) associated to the family C. Consider case (a). Given (x, f θ ) ∈ GCMIr (S, Rn ), Lemma 3.3.1 shows that the map f : S → A∗ of class A r−1 (S) can be approximated as closely as desired in C r−1 (S) by a full holomorphic map F : M → A∗ which agrees with f to the given order s at all points in the set A ∩ S˚ and satisfies P(F) = P( f ). (We can also ensure ˚ but that is not needed for this proof.) Fix a that F − f vanishes at the points in A \ S, point p0 ∈ A. It follows that the map X : M → Rn given by  p

X(p) = x(p0 ) + ℜ

p0

Fθ ,

p∈M

(3.18)

is a well defined full conformal minimal immersion with FluxX = Fluxx . If p ∈ S then the integral can be computed on a path in S, showing that X|S approximates x in C r (S). If in addition p ∈ A then the integral may be computed on a chain of arcs in C connecting p0 to p, and hence the condition P(F) = P( f ) implies X(p) = x(p). This shows that X = x on A, and jet interpolation at the points of A ∩ S˚ follows from the condition that F agrees with f to order s at those points. The proof in case (b) is the same, except that we do not take real parts in (3.18).  

3.4 General Position Theorems In this section we prove that a generic conformal minimal immersion of a compact bordered Riemann surface M into Rn is an embedding if n ≥ 5, an immersion with simple double points if n = 4, and its boundary is embedded for any n ≥ 3. We also show that a generic holomorphic null curve M → Cn for any n ≥ 3 is embedded. Here, the word generic refers to an element of an open (everywhere) dense subset of the Banach space CMIrnf (M, Rn ) or NCrnf (M, Cn ) for any r ∈ N. By using also the approximation theorems in Section 3.6, we obtain analogous results on any open Riemann surface M, where generic now refers to an element of a countable intersection of dense open subsets of the Fr´echet spaces CMI(M, Rn ) or NC(M, Cn ). Every such set is dense by Baire’s theorem (see Remark 1.4.2). Theorem 3.4.1. Let M be a compact bordered Riemann surface, n ≥ 3, and r ∈ N. (a) Every holomorphic null immersion z : M → Cn of class A r (M) can be approximated in the C r (M, Cn ) topology by holomorphic null embeddings z˜ : M → Cn . Furthermore, z˜ can be chosen such that z˜(M) avoids a given topologically closed smooth submanifold Σ ⊂ Cn with dimR Σ ≤ 2n − 3, and z˜(bM) avoids a given such submanifold Σ  ⊂ Cn with dimR Σ  ≤ 2n − 2. (b) If n ≥ 5 then every conformal minimal immersion x : M → Rn of class C r (M) (i.e., x ∈ CMIr (M, Rn )) can be approximated in the C r (M) topology by conformal minimal embeddings x˜ : M → Rn with Fluxx˜ = Fluxx .

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(c) Every x ∈ CMIr (M, R4 ) can be approximated by conformal minimal immersions x˜ : M → R4 with simple double points such that x˜ : bM → R4 is an embedding and Fluxx˜ = Fluxx . (d) Every x ∈ CMIr (M, R3 ) can be approximated by conformal minimal immersions x˜ : M → R3 such that x˜ : bM → R3 is an embedding and Fluxx˜ = Fluxx . In cases (b)–(d) the map x˜ can be chosen such that x(M) ˜ avoids a given topologically closed smooth submanifold Σ ⊂ Rn with dimR Σ ≤ n − 3, and x(bM) ˜ avoids a given topologically closed smooth submanifold Σ  ⊂ Rn with dimR Σ  ≤ n − 2. In particular, for any n ≥ 3 we can choose x˜ such that x(M) ˜ avoids a given finite set of points in Rn , and x(bM) ˜ avoids a given smooth curve in Rn . In all four cases, the approximating map can also be chosen to agree with the given map on a given finite set in bM, and to a given finite order on a given finite set ˚ provided the properties of the initial map do not contradict the conclusion. in M, Recall that an immersion x : M → Rn is said to have simple double points if for any pair of distinct points p = q in M with x(p) = x(q), the tangent planes dx p (Tp M) and dxq (Tq M) intersect only at 0 ∈ Tx(p) Rn , and there are no triple intersections. Case (a) of Theorem 3.4.1 was proved in [22, Sect. 6], and the remaining cases were obtained in [30, Sect. 4] by similar methods; see also [12, Sect. 5A]. The present statement and proof are more precise and include new interpolation and avoidance conditions. The proofs rely on transversality methods applied to dominating sprays of holomorphic null curves or conformal minimal immersions. Proof. We may assume that M is a compact smoothly bounded domain in an open Riemann surface R such that the inclusion M → R induces an isomorphism ∼ = H1 (M, Z) −→ H1 (R, Z); see Corollary 1.10.9. We begin with case (a). Let z : M → Cn be a holomorphic null curve of class C r , z ∈ NCr (M, Cn ). By Proposition 3.3.2 we may assume that z is holomorphic and full on the ambient Riemann surfaces R. Consider the difference map δ z : M × M → Cn ,

δ z(p, q) = z(q) − z(p),

p, q ∈ M.

(3.19)

Let DM = {(p, p) : p ∈ M} denote the diagonal of M × M. Clearly, the map z is injective if and only if (δ z)−1 (0) = DM . Since an immersion is locally injective, there is an open neighbourhood U ⊂ M × M of the diagonal DM such that δ z does not assume the value 0 ∈ Cn on U \ DM . To prove the first statement in part (a) of the theorem, it suffices to find arbitrarily close to z in C r (M, Cn ) a holomorphic null immersion z˜ : M → Cn whose difference map δ z˜ is transverse to the origin 0 ∈ Cn on M ×M \U. Indeed, since dimC M ×M = 2 < n, this implies that δ z˜ does not assume the value zero on M × M \ U, so z˜(p) = z˜(q) if (p, q) ∈ M × M \ U. On the other hand, if (p, q) ∈ U \ DM then z˜(p) = z˜(q) provided that z˜ is close enough to z in C 1 (M, Cn ). Thus, z˜ is an injective immersion. To satisfy the last two claims, we must ensure in addition that z˜ and z˜|bM

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145

are transverse to given submanifolds Σ , Σ  ⊂ Cn . If dimR Σ ≤ 2n − 3, this implies that z˜(M) ∩ Σ = ∅, and if dimR Σ  ≤ 2n − 2, we infer that z˜(bM) ∩ Σ  = ∅. It will be explained at the end of the proof how to satisfy the interpolation conditions. A map z˜ with these properties will be found by a transversality argument. Let us ˚ In the presence of submanifolds describe the main idea. Fix an interior point p ∈ M.  n  / Σ ∪ Σ  , and we fix a Σ , Σ ⊂ C to be avoided, we choose p such that z(p ) ∈   small closed neighbourhood U ⊂ M of p such that z(U  ) ∩ (Σ ∪ Σ  ) = ∅. (If z(M) ⊂ Σ ∪ Σ  then the interpolation conditions do not apply, so we are allowed to make a small translation of z which removes z(p ) away from Σ ∪ Σ  .) By shrinking the set U ⊂ M × M around DM , we may assume that (p , q) ∈ M × M \U for every q ∈ M \U  . We shall find a neighbourhood Ω ⊂ CN of the origin 0 ∈ CN for some N ∈ N and a holomorphic map H : M × Ω → Cn satisfying the following conditions: 1. 2. 3. 4.

H(· ,t) is a holomorphic null immersion for every t ∈ Ω , H(· , 0) = z, H(p ,t) = z(p ) for every t ∈ Ω , and the difference map δ H : M × M × Ω → Cn , δ H(p, q,t) = H(q,t) − H(p,t),

p, q ∈ M, t = (t1 , . . . ,tN ) ∈ Ω ,

(3.20)

is such that the partial differential ∂t |t=0 δ H(p, q,t) : CN −→ Cn

(3.21)

is surjective for every (p, q) ∈ M × M \U. Assume for a moment that such H exists. Since M × M \ U is compact, there is a neighbourhood Ω  ⊂ Ω of the origin 0 ∈ CN such that the partial differential ∂t δ H is surjective on (M × M \U) × Ω  . The standard transversality argument (see the text preceding Theorem 1.4.3) then shows that for a generic point t ∈ Ω  the difference map δ H(· ,t) restricted to M × M \U is transverse to any submanifold of Cn , in particular, to the origin 0 ∈ Cn . For dimension reasons, it omits the value 0. For such t close enough to 0, the map z˜ = H(· ,t) : M → Cn is a holomorphic null embedding. By fixing the variable q = p and noting that H(p, p ,t) = H(p,t) − z(p ) in view of condition 3, (3.21) implies that ∂t H(p,t) : CN → Cn is surjective for every p ∈ M \ U  and t ∈ Ω  , where U  ⊂ M is a neighbourhood of p chosen above. As before, we infer that for most t ∈ Ω  the map z˜ = H(· ,t) : M → Cn is transverse to Σ on M \ U  , so z˜(M \ U  ) ∩ Σ = ∅ for dimension reasons if dim Σ ≤ 2n − 3. Choosing t sufficiently close to 0 ensures that z˜(U  ) does not intersect Σ either, so z˜(M) ∩ Σ = ∅. A similar argument gives that z˜(bM) ∩ Σ  = ∅ if dim Σ  ≤ 2n − 2. We now show how to find a deformation family H with the required properties. Lemma 3.4.2. (Notation as above.) For every point (p0 , q0 ) ∈ M × M \ DM there exists a deformation family H(· ,t) (t ∈ Cn ) satisfying conditions 1.–3. above such that the partial differential ∂t |t=0 δ H(p0 , q0 ,t) : Cn → Cn is an isomorphism.

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Proof. Since z is holomorphic on the open Riemann surface R ⊃ M, it suffices to prove the lemma for points (p0 , q0 ) ∈ M˚ × M˚ \ DM . Indeed, applying this on a somewhat larger domain M  containing M in the interior will imply the lemma. Let Λ ⊂ M˚ be a smooth embedded oriented arc with the initial point p0 and the terminal point q0 . Choose another embedded arc Λ  ⊂ M connecting p0 to a boundary point of M and otherwise disjoint from bM such that Λ ∩ Λ  = {p0 }. Removing the arc Λ ∪ Λ  from M does not change the topology of M. Hence, given a point q ∈ M \ (Λ ∪ Λ  ), Lemma 1.12.10 gives closed Jordan curves C1 , . . . ,Cl ⊂ M \ (Λ ∪ Λ  ) based at q and forming a basis of the homology group H1 (M, Z) such that C = lj=1 C j is Runge in M and every curve Ci contains a nontrivial arc not contained in any of the curves C j for j = i. The set C ∪ Λ is then also Runge in M since none of the curves Ci surrounds the arc Λ (indeed, such a curve would intersect Λ  ). Let P be the period map (3.12) associated to the family C = {C1 , . . . ,Cl } and a nowhere vanishing holomorphic 1-form θ on R. Write dz = f θ with f : M → A∗ a holomorphic map. (Recall that z, and hence f , are holomorphic on a neighbourhood of M in R.) The idea of the proof is the following. We shall construct a holomorphic spray Ψf : M × Cn → A∗ with Ψf (· , 0) = f = dz/θ such that Ψf is period dominating on Λ , and the periods of Ψf (· ,t) over the curves Ci ∈ C are very close to those of f (i.e., close to zero) for t in a neighbourhood of 0. The difference map of the integral of this spray on Λ therefore satisfies the condition (3.21) at the endpoints p0 , q0 of Λ . However, the integral is not well defined on M since the periods over the curves in C need not vanish. In the second step we change these periods to zero, making sure that the change of the period map on Λ is small enough so that condition (3.21) remains valid. This will be done by superimposing another C -period dominating spray over the first spray Ψf and applying the implicit function theorem. We now proceed with the details of this plan. Since the map z : M → Cn is full, the restriction z : Λ → A∗ is also full because a nontrivial arc in M is a determining set for holomorphic functions on M. Hence, there exist holomorphic vector fields V1 , . . . ,Vn on Cn tangential to A and points p1 , . . . , pn ∈ Λ \ {p0 , q0 } such that   span V1 ( f (p1 )), . . . ,Vn ( f (pn )) = Cn . To simplify the exposition, we choose vector fields Vi to be linear and hence complete (see Example 1.13.8), although this will not be essential for the argument. Let φti (t ∈ C) denote the flow of Vi . Let hi : C ∪ Λ → R+ for i = 1, . . . , n be smooth functions vanishing near the endpoints p0 , q0 of Λ and also on C. Set t = (t1 , . . . ,tn ) ∈ Cn . As in the proof of Lemma 3.2.1, we consider the map ψ(p,t) = φh11 (p)t1 ◦ · · · ◦ φhnn (p)tn ( f (p)) ∈ A∗ ,

p ∈ C ∪ Λ , t ∈ Cn .

Clearly, ψ is holomorphic in t ∈ Cn , ψ(· , 0) = f , and ψ(p,t) = f (p) if p ∈ C or if p is near p0 or q0 since the functions hi vanish there. For i = 1, . . . , n we have that ∂ ψ(p,t)   = hi (p)Vi ( f (p)), ∂ti t=0

p ∈ C ∪Λ.

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A suitable choice of the functions hi : Λ → C supported near the points pi ∈ Λ gives  Λ

hi Vi ( f (· ))θ ≈ Vi ( f (pi )) ∈ Cn ,

i = 1, . . . , n.

(3.22)

Hence, we may assume that the vectors on the left-hand side of (3.22) form a basis of Cn . Fix functions hi with this property and extend them by 0 to C. Let ε > 0; its precise value will be chosen later. Since the compact set C ∪ Λ is Runge in M, Mergelyan’s theorem furnishes functions gi ∈ O(M) such that sup |gi − hi | < ε,

C ∪Λ

i = 1, . . . , n.

(3.23)

In analogy with (3.16) we consider A∗ -valued holomorphic maps Ψ (p, ζ ,t) = φg11 (p)t1 ◦ · · · ◦ φgnn (p)tn (ζ ),

Ψf (p,t) = Ψ (p, f (p),t),

(3.24)

where p ∈ M, ζ ∈ A∗ , and t ∈ Cn . Note that Ψf (· , 0) = f . If ε > 0 is chosen small enough, we see from (3.22) that the vectors    ∂  Ψ (· ,t) θ = gi Vi ( f (· )) θ ∈ Cn (3.25) f ∂ti t=0 Λ Λ are so close to the vectors Vi ( f (pi )) that they form a basis of Cn . The Cn -valued 1-form Ψf (· ,t)θ on M need not be exact. We shall now correct its periods to zero and thereby make it exact. From (3.24) and the Taylor expansion of the flow of a vector field we see that n

Ψf (p,t) = f (p) + ∑ ti gi (p)Vi ( f (p)) + O(|t|2 ). i=1

In view of (3.23) we obtain the following uniform estimates for some c > 0 and for all t ∈ Cn in a neighbourhood of the origin: |Ψf (p,t) − f (p)| ≤ c|t|, |Ψf (p,t) − f (p)| ≤ cε|t|,

p ∈ Λ, p ∈ C.

It follows that the periods over the loops Ci ∈ C satisfy the estimate           Ψf (· ,t) θ  =  Ψf (· ,t) θ −  ≤ η0 ε|t| f θ     Ci

Ci

(3.26)

(3.27)

Ci

for some constant η0 > 0 and for all t ∈ Cn in a neighbourhood of the origin. Lemma 3.2.1 (see in particular (3.16)) gives A∗ -valued holomorphic maps Φ(p, ζ , t˜) and Φ f (p, t˜) = Φ(p, f (p), t˜) for p ∈ M and ζ ∈ A∗ , depending holomorphically on t˜ ∈ CN for some N ∈ N, such that Φ(p, ζ , 0) = ζ and the differential of the period map t˜ → P(Φ f (· , t˜)) ∈ Cln (see (3.12)) at t˜ = 0 has maximal rank equal to ln. We now consider the holomorphic map

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M × Cn × CN (p,t, t˜) −→ Φ(p,Ψf (p,t), t˜) ∈ A∗ . Note that for t˜ = 0 we have Φ(p,Ψf (p,t), 0) = Ψf (p,t), while for t = 0 this equals Φ(· , f , t˜) = Φ f (· , t˜), and Φ f (· , 0) = f . Since the period map t˜ → P(Φ f (· , t˜)) ∈ Cln is submersive at t˜ = 0, the implicit function theorem shows that the equation P(Φ(· ,Ψf (· ,t), t˜)) = P( f )

(3.28)

can be solved on t˜ as a function of t on a neighbourhood of the origin. More precisely, there is a holomorphic map t˜ = ρ(t) ∈ CN near t = 0 ∈ Cn , with ρ(0) = 0 ∈ CN , such that the family of A∗ -valued holomorphic maps Θ f (p,t) := Φ(p,Ψf (p,t), ρ(t)), satisfies

 Ci

Θ f (· ,t)θ =

 Ci

f θ = 0,

p∈M

(3.29)

i = 1, . . . , l

(3.30)

for every t ∈ Cn near 0. Hence, the integral H(p,t) = z(p ) +

 p p

Θ f (· , · ,t)θ ,

p∈M

(3.31)

is well defined. (Here, p is the chosen base point.) Clearly, H(· , 0) = z, and H(· ,t) : M → Cn is a holomorphic null immersion for every t ∈ Cn close to 0. The solution t˜ = ρ(t) of the period equation (3.28) becomes unique if we restrict the variable t˜ to a complex linear subspace of CN on which the differential of the period map t˜ → P(Φ f (· , t˜)) ∈ Cln is an isomorphism. In view of (3.27), this solution satisfies the estimate |ρ(t)| ≤ η1 ε|t|

(3.32)

for some constant η1 > 0 and for t ∈ Cn in a neighbourhood of the origin. The map Φ(p, ζ , t˜) is of the form (3.16), i.e., it is obtained by composing flows of certain holomorphic vector fields W j tangential to A for the times t˜j gj (p). From the Taylor expansion of the flow and the estimates (3.26), (3.32) we see that     Φ(p,Ψf (p,t), ρ(t)) −Ψf (p,t) = ∑ ρ j (t) g j (p)W j (Ψf (p,t)) + O(|t|2 ) ≤ η2 ε|t| holds for some constant η2 > 0 and for all p ∈ C ∪ Λ and all t ∈ Cn near the origin. (Note however that this estimate need not hold for p ∈ M \ (C ∪ Λ ) because, as ε → 0, there is no control on the size of the functions gi in the definition of Ψf (p,t) (3.24). There was a misprint at the corresponding place both in [22, p. 752] and in [30, p. 13], but the estimate was used only for p ∈ C ∪ Λ .) Applying this estimate on the arc Λ and taking into account (3.29) gives       Θ f (· ,t)θ − Ψf (· ,t)θ  ≤ η3 ε|t|   Λ

Λ

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149

for some η3 > 0. If ε > 0 is chosen small enough, it follows that ∂  the vectors  ∂ti t=0

 Λ

Θ f (· ,t)θ ∈ Cn (i = 1, . . . , n) span Cn

(3.33)

(because they are close enough to the vectors (3.25)). In view of (3.31) we have  Λ

Θ f (· ,t)θ = H(q0 ,t) − H(p0 ,t) = δ H(p0 , q0 ,t).

Hence, the above says that the partial differential ∂   δ H(p0 , q0 ,t) : Cn −→ Cn ∂t t=0 is an isomorphism. This proves Lemma 3.4.2.   We see from the proof of Lemma 3.4.2 that the deformation family H = Hz can be chosen to depend holomorphically on z locally near a given initial full holomorphic null immersion z0 : M → Cn . Equivalently, the map Θ f (p,t) given by (3.29) and satisfying the period conditions (3.30) depends holomorphically on f in a neighbourhood of the initial map f0 = dz0 /θ . Letting f vary through maps with vanishing periods on the curves Ci ∈ C , H f (p,t) given by (3.31) is a family of holomorphic null curves depending holomorphically on f . This allows us to compose any finite number of deformation families given by Lemma 3.4.2, increasing the number of parameters each time. For example, if H = Hz (· ,t) and G = Gz (· , τ) are families of null curves with Hz (· , 0) = Gz (· , 0) = z, we define the composed family by (H # G)z (p,t, τ) = GHz (·,t) (p, τ),

p ∈ M.

Clearly, we have that (H # G)z (· , 0, τ) = Gz (· , τ),

(H # G)z (· ,t, 0) = Hz (· ,t).

The operation # extends by induction to finitely many factors; it is associative, but not commutative. This operation is similar to the composition of sprays introduced by Gromov [171]; see also [140, Def. 6.3.5, p. 268]. We can now complete the proof of Theorem 3.4.1 (a). The above construction gives a finite open covering U = {Ui }m i=1 of the compact set M × M \ U and deformation families H i = H i (· ,t i ) : M → Cn , with H i (· , 0) = z and H i (p ,t) = z(p ) for t i = (t1i , . . . ,tki i ) ∈ Ωi ⊂ Cki , so that the difference map δ H i (p, q,t i ) is submersive at t i = 0 for all (p, q) ∈ Ui and i = 1, . . . , m. Taking t = (t 1 , . . . ,t m ) ∈ CN with N = ∑m i=1 ki and setting H(p,t) = (H 1 # H 2 # · · · # H m )(p,t 1 , . . . ,t m ),

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we obtain a deformation family such that H(· , 0) = z and δ H is submersive on M × M \U for all t ∈ CN sufficiently close to the origin. By the argument prior to Lemma 3.4.2, this completes the proof of Theorem 3.4.1 in case (a), except for the interpolation conditions stated at the end. Let us explain the necessary modifications in this case. Suppose that we wish to interpolate the given holomorphic null curve z at the points A = {a1 , . . . , ak } ⊂ M by an embedded null curve z˜, assuming that z is injective on this set. We may assume that the base point p is one of these points, say p = a1 . Choose an open set U ⊂ M × M which is a neighbourhood of the diagonal DM and also of every pair (ai , a j ) (i, j = 1, . . . , k) so that the difference map δ z (3.19) does not assume the value 0 on U \ DM . Fix a point (p0 , q0 ) ∈ M ×M \U. Pick an arc Λ ⊂ M connecting p0 to q0 such that Λ contains at most one of the points a1 , . . . , ak , and if it does, the corresponding point ai is an endpoint of Λ (either p0 or q0 ; note that both endpoints of Λ cannot be contained in A = {ai }ki=1 since (p0 , q0 ) ∈ / U while U contains all pairs (ai , a j ).) Fix a point q ∈ M˚ \ (A ∪ Λ ). As before, we choose a period basis C = {C1 , . . . ,Cl } of H1 (M, Z) based at q such that C ∪ Λ is Runge in M and C ∩ (A ∪Λ ) = ∅, where C = li=1 Ci . Finally, we add to the family C smooth embedded arcs E2 , . . . , Ek ⊂ M connecting the point p = a1 to the respective points a2 , . . . , ak ∈ A. These arcs are chosen pairwise disjoint, disjoint from C, and such that all except perhaps one of them, say E j , are disjoint from Λ . (The exceptional arc E j could share its endpoint a j with an endpoint of Λ .) Let E = kj=2 E j . Note that C ∪ E ∪ Λ is Runge in M since E does not add any loop to the Runge set C ∪ Λ . The proof of Lemma 3.4.2 now proceeds as before and yields a holomorphic spray of maps Θ f (· ,t) : M → A∗ (3.29) for t ∈ Cn near the origin which satisfies (3.33), the period conditions (3.30), and  Ej

Θ f (· ,t)θ = 

 Ej

fθ,

j = 2, . . . , k.

The integral Hz (p,t) = z(p ) + pp Θ f (· ,t)θ , p ∈ M (see (3.31)) is then a holomorphic null curve for each t which agrees with z at the points in A, and  ∂  n n ∂t t=0 δ H(p0 , q0 ,t) : C → C is an isomorphism. We finish the proof by composing finitely many such sprays as explained above. ˚ we build into If we wish to interpolate to a higher order at the points of A ∩ M, the construction of sprays a nonconstant holomorphic function g ∈ O(M) vanishing to a desired order at these points, as was done in the proof of Lemma 3.2.1. The proofs of cases (b)–(d) in Theorem 3.4.1 are quite similar; we briefly explain the modifications. (See [30, Sect. 4] for more details.) By Proposition 3.3.2 we may assume that the given conformal minimal immersion x : M → Rn is defined and full on a neighbourhood of M in R. By a similar method as before we find a spray of holomorphic maps Θ f (· ,t) : M → A∗ (3.29), depending real analytically on parameters t = (t1 , . . . ,tn ) ∈ Rn in a neighbourhood of the origin, satisfying the period equations (3.30) and the following analogue of condition (3.33):

3.5 Paths with Given Periods in Affine Algebraic Varieties

151

  ∂  the vectors ℜ Θ f (· ,t)θ ∈ Rn (i = 1, . . . , n) span Rn . ∂ti t=0 Λ It follows that for every t near 0 ∈ Rn the integral Hx (p,t) = x(p ) + ℜ

 p p

Θ f (· ,t)θ ,

p∈M

(3.34)

is a conformal minimal immersion M → Rn satisfying FluxHx (·,t) = Fluxx . Let U be a neighbourhood of the diagonal DM in M × M, chosen such that the difference map δ x : M × M → Rn does not assume the value 0 ∈ Rn on U \ DM . By composing finitely many such deformation families as explained in the proof of case (a) we get a family H(· ,t), where t ∈ RN for some big N ∈ N, whose difference map δ H(· , · ,t) : M × M → Rn (see (3.19)) is submersive in the t variable on M × M \U for all t ∈ RN near the origin. The same transversality argument as in the case (a) shows that for a generic t ∈ RN near the origin, the conformal minimal immersion x˜ = H(· ,t) : M → Rn is such that the difference map δ x˜ : M × M → Rn is transverse to 0 ∈ Rn on M × M \ U. If n ≥ 5 > 4 = dimR M × M, it follows that δ x˜ does not assume the value 0 on M × M \ U for dimension reasons. Assuming as we may that x˜ is close enough to x, it follows that x˜ is an embedding. If n = 4, the conclusion is that δ x˜ assumes the value 0 ∈ R4 only at finitely many points in M × M \ U, hence x˜ has at most finitely many pairs of double points. Furthermore, the condition that δ x˜ is transverse to the origin implies that any double point of x˜ is a transverse self-intersection. We can remove multiple intersection points by a further small deformation using these methods. For any n ≥ 3 (and in particular for n = 3 and n = 4 as in cases (c) and (d)), we can arrange in addition that x˜ : bM → Rn is an embedding since dim(bM × bM) = 2 < n. Interpolation at finitely many points and the avoidance conditions are handled exactly as before.  

3.5 Paths with Given Periods in Affine Algebraic Varieties In this section we construct paths with given integrals in any irreducible algebraic subvariety of Cn ; see Lemma 3.5.4. This will be used in the proof of approximation and interpolation theorems given in the following two sections. We begin with the following result concerning convex hulls of affine varieties. Lemma 3.5.1. The convex hull Co(A) of a closed connected algebraic subvariety A ⊂ Cn is the smallest affine complex subspace of Cn containing A. In particular, the convex hull of the null quadric A ⊂ Cn (2.54) equals Cn . This is [22, Lemma 5.1] where the hypothesis that the subvariety A be connected was accidentally omitted. Note that every irreducible subvariety is connected. The lemma is false is general for disconnected affine varieties, such as the union of two parallel affine complex hyperplanes in Cn . The lemma also fails in general for nonalgebraic (transcendental) complex subvarieties.

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We shall base the proof on the following result from convex geometry (see S. Boyd and L. Vandenberghe [69, Sect. 2.5]). Theorem 3.5.2 (Separating hyperplane theorem). If A and B are nonempty disjoint convex (not necessarily closed) subsets of an affine space Rn , there exists an affine hyperplane H ⊂ Rn separating them, i.e., such that A and B are contained in different closed half-spaces of Rn defined by H. In particular, every proper convex subset of Rn is contained in a halfspace. Proof of Lemma 3.5.1. If Co(A) = Cn , then Theorem 3.5.2 gives a nonconstant real linear function u on Cn and a number c ∈ R such that A ⊂ {u ≤ c}. The projective  → A denote its closure A ⊂ CPn of A is an algebraic subvariety of CPn . Let η : A  is a desingularization (see J. M. Aroca, H. Hironaka and J. L. Vicente [53]), so A −1  compact algebraic manifold and Λ = η (A \ A) is a proper closed subvariety of A. −1  . Then, u◦η : Ω → R is a bounded above plurisubharmonic Let Ω = η (A) = A\Λ  → R given function which therefore extends to the plurisubharmonic function u˜ : A at each point z0 ∈ Λ by u(z ˜ 0 ) = lim supΩ z→z0 u ◦ η(z) (see M. Klimek [207,  is compact, the maximum principle for plurisubharmonic Theorem 2.9.22]). Since A  Hence, u functions implies that u˜ is constant on each connected component of A. is constant on each irreducible component Ai of A, so Ai lies in a real hyperplane {u = ci }, ci ∈ R, and therefore in one of the complex hyperplanes foliating {u = ci }. Since these hyperplanes are pairwise disjoint while A is connected, A lies in a complex hyperplane of Cn . Now proceed inductively to conclude that Co(A) is the  smallest affine complex subspace Σ of Cn containing A.  Remark 3.5.3. Lemma 3.5.1 has an elementary proof if A is a complex cone; in particular, if A is the null quadric A (2.54). Indeed, let Σ ⊂ Cn denote the C-linear span of A, a complex vector subspace of Cn . Every vector v ∈ Σ is of the form v = ∑kj=1 c j v j where c j ∈ C∗ and v j ∈ A for j = 1, . . . , k. Setting k

c=

∑ |c j |2 > 0,

rj =

j=1

|c j |2 > 0, c

wj =

cj v j ∈ A, rj

we have ∑kj=1 r j = 1 and v = ∑kj=1 r j w j , so v ∈ Co(A) and therefore Co(A) = Σ .   Lemma 3.5.4. Assume that A is an irreducible nondegenerate algebraic subvariety of Cn (i.e., A is not contained in any affine hyperplane of Cn ). Given a continuous map f0 : [0, 1] → Areg , a continuous function g : [0, 1] → C∗ , a vector v ∈ Cn , and a connected domain Ω in Cn containing 0 and v, there is a homotopy fτ : [0, 1] → Areg (τ ∈ [0, 1]) that is fixed at 0 and 1 such that f0 is the given map and the map f = f1 is smooth, nonflat near the endpoint f (0) (see Definition 2.5.2), and we have that  1

 t

f (s)g(s) ds = v 0

and 0

f (s)g(s) ds ∈ Ω for all t ∈ [0, 1].

(3.35)

In particular, any pair of points in Areg can be connected by a smooth path f : [0, 1] → Areg satisfying conditions (3.35).

3.5 Paths with Given Periods in Affine Algebraic Varieties

153

Lemma 3.5.4, which is the main result of this section, is similar to Gromov’s convex integration lemma [173, Sect. 2.1] and our proof follows the same line of ideas. The presence of the weight function g requires additional attention. Proof. The proof will be accomplished in four steps. Step 1: Reduction to an approximate problem. We begin by showing that it suffices to solve the problem with the exact condition in the first integral in (3.35) replaced by the approximate condition  1     f (s)g(s) ds − v < ε (3.36)  0

for any given ε > 0, while keeping the second condition in (3.35). This is done by writing [0, 1] = [0, c] ∪ [c, 1] for a suitable number 0 < c < 1 and setting up a period dominating spray of deformations of f0 supported in (0, c), in analogy to the spray (3.16) in the proof of Lemma 3.2.1. Solving the approximate problem by a deformation of f0 supported on [c, 1], the small error in (3.36) is then offset by a suitable choice of the parameter in the spray over [0, c]. Let us explain the details. Since the subvariety A is irreducible, its regular locus Areg is connected. Furthermore, since A is nondegenerate, Lemma 3.5.1 implies that its convex hull equals Cn . By a small deformation of the path f0 : [0, 1] → Areg we may therefore assume that f0 is smooth and nonflat near 0. (In fact, we can even arrange that f0 is everywhere nonflat, but we shall not keep track of this condition there is a number in the proof.) SinceΩ ⊂ Cn is an open set containing the origin,  0 < c < 1 such that 0t f0 (s)g(s) ds ∈ Ω for all t ∈ [0, c]. Set v0 = 0c f0 (s)g(s) ds ∈ Ω . After a small deformation of f0 which is fixed on [0, c ] ∪ [c, 1] for some c ∈ (0, c), there are numbers s1 , . . . , sn ∈ (c , c) and tangent vectors wi ∈ T f (si ) Areg (i = 1, . . . , n) such that spanC {w1 , . . . , wn } = Cn . (We are identifying the tangent space Ta Cn with Cn .) By Cartan’s Theorem A there exists for every i = 1, . . . , n a holomorphic vector field Wi on Cn tangent to A and vanishing on Asing such that Wi ( f (si )) = wi . Let φζi denote the flow of Wi . Choose a smooth function hi : [0, 1] → R+ supported on a small neighbourhood of si in (c , c) for i = 1, . . . , n and consider the family of deformations of f0 given by Φ f0 (ζ , s) = φζ11 h1 (s) ◦ · · · ◦ φζnn hn (s) ( f0 (s)) ∈ Areg

s ∈ [0, 1],

(3.37)

for ζ = (ζ1 , . . . , ζn ) in a ball B ⊂ Cn centred at 0 ∈ Cn . (Compare with (3.16) and note that the deformation is supported on (c , c).) We see as in the proof of Lemma 3.2.1 that the functions hi can be chosen such that the map   1 ∂  Φ f0 (ζ , s)g(s) ds : Cn −→ Cn  ∂ ζ ζ =0 0 

is an isomorphism. It follows that the family of integrals 0c Φ f0 (ζ , s)g(s)ds ∈ Cn for ζ ∈ B is an open neighbourhood of 0c f0 (s)g(s)ds = v0 in Cn .

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3 Approximation and Interpolation Theorems for Minimal Surfaces

This reduces the problem to finding a homotopy fτ : [0, 1] → Areg (τ ∈ [0, 1]) which is fixed on [0, c] such that the path f = f1 satisfies (3.36) for a small enough ε > 0. Indeed, we can then remove the error by replacing f with the path Φ f (ζ , · ) : [0, 1] → Areg (see (3.37)) for a suitably chosen ζ ∈ B. To simplify the notation, we now rescale the interval [c, 1] back to [0, 1] and replace the vector v by v − v0 and the domain Ω by Ω − v0 . In the remainder of the proof we explain how to obtain a deformation f = f1 satisfying (3.36) and the second condition in (3.35) for a given ε > 0. Step 2: Reduction to the case when the set Ω in (3.35) contains the line segment {sv : 0 ≤ s ≤ 1}. Choose a polygonal path Γ = mj=1 Γj ⊂ Ω connecting 0 and v, where each Γj is a line segment in Cn of the form Γj = {p j−1 + sv j : s ∈ [0, 1]},

p j−1 ∈ Ω , v j ∈ Cn .

The initial point of Γ1 is p0 = 0, the final point of Γm is pm−1 + vm = v, and we have p j = p j−1 + v j for all j = 1, . . . , m − 1, i.e., the initial point of Γj+1 coincides with the final point of Γj . Note in particular that v = ∑mj=1 v j . Replacing f0 (s) by f0 (s/m) for s ∈ [0, m] and likewise for g, we rescale the parameter interval to [0, m]. The problem now is to find a deformation f (s) of f0 (s) which is fixed near each point s ∈ {0, 1, . . . , m} such that for every j = 1, . . . , m we have that  j    ε    j−1 f (s)g(s) ds − v j  < m 

t and the integral j−1 f (s)g(s) ds lies in a given open neighborhood Ω j ⊂ Cn of the line segment {sv j : 0 ≤ s ≤ 1} for all t ∈ [ j − 1, j]. Indeed, since v = ∑mj=1 v j , choosing the neighbourhoods Ω j small enough it is clear that f satisfies (3.36) and the second condition in (3.35).

Step 3: Solving the problem in the case when g is constant. We now assume that Ω in (3.35) contains the straight line segment {sv : 0 ≤ s ≤ 1} (see step 2). We first give the argument in the special case when g = 0 is constant. Replacing the vector v by v/g, we reduce the problem to the case g = 1. Dividing the interval [0, 1] into a big number N of subintervals I j and performing the same procedure on each of them, with g replaced on I j by a constant g j such that |g − g j | is small, the error goes to zero as N → +∞ (see step 4 for the details). Since Co(A) = Cn by Lemma 3.5.1, the closed convex sets n

KR = Co(A ∩ RB ),

R>0

increase to Cn as R → +∞. It follows that for every η > 0 there are R > 0 and finitely n many points v1 , . . . , vk ∈ Areg ∩ RBn such that ηB ⊂ Co({v1 , . . . , vn }). (Indeed, by Carath´eodory’s theorem every point p ∈ KR is the convex combination of 2n + 1 n points in A ∩ RB , and for R > 0 big enough there are finitely many p1 , . . . , pm ∈ KR n such that ηB ⊂ Co({p1 , . . . , pm }). Since Areg is dense in A, the points v j may be chosen in Areg .) By choosing η > |v| and then choosing R > 0 big enough, we can

3.5 Paths with Given Periods in Affine Algebraic Varieties

155

ensure that the set f0 ([0, 1]) ∪{v1 , . . . , vk } belongs to the same connected component of Areg ∩ RBn (recall that Areg is connected). We now fix these choices. We have that v = ∑ki=1 ri vi where ri > 0 and ∑ki=1 ri = 1. (We omit the terms with ri = 0.) Let si = ∑ij=1 r j for i = 1, . . . , k, so 0 = s0 < s1 < . . . < sk = 1 with si − si−1 = ri . Choose a number δ with 0 < δ < 12 mini ri and consider the pairwise disjoint intervals Ii = [si−1 + δ , si − δ ] for i = 1, . . . , k. Their complement J = [0, 1] \ ki=1 Ii has total length 2kδ . Let f : [0, 1] → Areg ∩ RBn be chosen such that f = f0 near 0 and 1, and f (t) = vi for t ∈ Ii (i = 1, . . . , k). On the remaining intervals contained in J we choose f so that it is continuous and homotopic to f0 . More precisely, on [si−1 , si ] we spend the time s ∈ [si−1 , si−1 + δ ] to travel from f0 (si−1 ) to vi along a path in Areg ∩ RBn , remain at vi for the time s ∈ Ii , and use the time s ∈ [si − δ , si ] to first return along the same path back to f0 (si−1 ) and then continue to the next point f0 (si ) along the trace of f0 . Clearly, the path f obtained in this way is homotopic to f0 by a homotopy in Areg ∩ RB. We have that  1 0

k

f (s) ds = ∑ (ri − 2δ )vi + i=1

 J

k

f (s) ds = v − 2δ ∑ vi + i=1



f (s) ds. J



enough Note that 2δ ∑ki=1 |vi | < 2δ kR and | J f (s) ds| < 2δ kR. Choosing δ > 0 small  we get the estimate (3.36). The same estimates show that the distance from 0t f (s)ds to Γ does not exceed 4δ kR for any t ∈ [0, 1], so by choosing δ > 0 small enough we ensure the second condition in (3.35). This completes the proof if g is constant. Step 4: The general  a number σ > 0 and split [0, 1] into a big number N  case.j Fix , of subintervals I j = j−1 N N such that for each j there is a constant g j = 0 with as in step 3 on each |g − g j | < σ on I j . We then perform the same procedure  subinterval I j , with g replaced by g j , to ensure that  I j f (s)g j ds − v/N  < ε/N. Since | f | < R and I j has length 1/N, we have that       f (s)g j ds − f (s)g(s)ds ≤ R max |g − g j | < R σ ,  N Ij  I N Ij j so the cumulative error over the whole interval [0, 1] is bounded by Rσ . By choosing σ > 0 small enough, this error is as small as desired. (Note that we can use the same constant R > 0 on all intervals I j independently of their number N by the argument at the beginning of the proof, choosing η > max j |v|/|g j | and determining R > 0 accordingly.)   Remark 3.5.5. Lemma 3.5.4 still holds if As ⊂ Cn is a family of nondegenerate irreducible algebraic subvarieties depending smoothly on s ∈ [0, 1] and we consider a path f : [0, 1] → Cn satisfying f (s) ∈ (As )reg for all s ∈ [0, 1]. Indeed, by partitioning [0, 1] into subintervals I1 , . . . , IN of equal length, it suffices to solve the problem on each I j in the case when both As and gs are constant for s ∈ I j and the  vector v is replaced by N −1 v. It is also possible to allow g to have zeros. 

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3.6 The First Main Theorem In this section we prove the first main result of this chapter, Theorem 3.6.1, which is a Mergelyan type approximation theorem combined with a Weierstrass type interpolation theorem for conformal minimal immersions from open Riemann surfaces into the Euclidean space Rn for any n ≥ 3. We also prove the corresponding result for null curves in Cn ; see Theorem 3.6.2. In the following section we give the second main theorem in which some of the component functions of a given generalized conformal minimal immersion extend harmonically to the whole Riemann surface and are kept fixed when approximating the remaining components. Theorem 3.6.1 (Mergelyan theorem for conformal minimal surfaces). Assume that M is an open Riemann surface, θ is a nowhere vanishing holomorphic 1-form on M, S is an admissible Runge set in M, Λ is a closed discrete subset of M, n ≥ 3 and r ≥ 1 are integers, and x is a generalized conformal minimal immersion S → Rn of class C r (S, Rn ) and a conformal minimal immersion on a neighbourhood of each point p ∈ Λ . Given a number ε > 0, a map k : Λ → N, and a group homomorphism p : H1 (M, Z) → Rn with p|H1 (S,Z) = Fluxx , there is a conformal minimal immersion x˜ : M → Rn satisfying the following conditions. (a) x˜ − xC r (S) < ε. (b) The difference x˜ − x vanishes to order k(p) at every point p ∈ Λ . (c) Fluxx˜ = p on H1 (M, Z). (d) If n ≥ 5 and the map x : Λ → Rn is injective, then x˜ is an injective immersion. (e) If n = 4 and x has simple double points on Λ , then x˜ is an immersion with simple double points. Note that for basic interpolation on Λ matching only the values, we may begin with a map x : S ∪ Λ → Rn such that x|S ∈ GCMIr (S, Rn ). The assumption on x in condition (e) means that no three distinct points of Λ are mapped to the same point in R4 , and if x(p) = x(q) for a pair of distinct points p, q ∈ Λ then the tangent planes dx p (Tp M) and dxq (Tq M) have trivial intersection in Tx(p) R4 . The latter condition is irrelevant for basic interpolation on Λ . Proof. By the hypothesis we have that x ∈ GCMIr (S, Rn ). Choose a number ε0 > 0 with 2ε0 < ε. By Proposition 3.3.2 there exist a neighbourhood U ⊂ M of S and a full conformal minimal immersion x0 : U → Rn such that x − x0 C r (S) < ε0 , x0 − x vanishes to order k(p) at each point p ∈ Λ ∩ S, and Fluxx0 = Fluxx on H1 (S, Z). We may choose U to have the same first homology group as S and to be small enough such that Λ ∩U ⊂ S. By Theorem 3.4.1 we may also assume that x0 satisfies condition (d) or (e) for the appropriate values of n. Proposition 1.12.5 furnishes a strongly subharmonic Morse exhaustion function ρ : M → R such that S ⊂ {ρ < 0} ⊂ {ρ ≤ 0} ⊂ U. The critical points of ρ form a closed discrete subset of M. A generic choice of ρ ensures that c0 = 0 is a regular

3.6 The First Main Theorem

157

value of ρ and every level set {ρ = c} for c > 0 contains at most one critical point of ρ or at most one point of Λ , but not both. Let M0 = {ρ ≤ 0}. We have that Fluxx0 = Fluxx on H1 (M0 , Z) since this holds on the bigger domain U ⊃ M0 . Choose an increasing sequence 0 = c0 < c1 < c2 < · · · with limi→∞ ci = +∞ such that the following conditions hold for every i ∈ N. (i) ci is a regular value of ρ. (ii) {ρ = ci } ∩ Λ = ∅. (iii) The set Ai = {ci−1 < ρ < ci } contains at most one critical point of ρ or at most one point of Λ , but not both. We shall inductively construct a sequence of conformal minimal immersions x i from a neighbourhood of the compact domain Mi = {ρ ≤ ci } into Rn and a decreasing sequence εi > 0 such that the following conditions hold for all i ∈ N. (ai ) (bi ) (ci ) (di ) (ei ) (fi )

x i − x i−1 C r (Mi−1 ) < εi−1 . The difference x i − x vanishes to order k(p) at every point p ∈ Λ ∩ Mi . Fluxx i = p on H1 (Mi , Z). If n ≥ 5 and the map x : Λ → Rn is injective, then x i is injective. If n = 4 and x has simple double points on Λ , then x i has simple double points. We have that 0 < εi < εi−1 /2, and every map y : Mi → Rn of class C r (Mi ) such that y − x i C r (Mi ) < 2εi is a full injective immersion (in case (di )) or a full immersion with simple double points (in case (ei )).

Assuming that we have such sequences x i and εi , the sequence x i converges in the C r topology on compacts in M to a conformal minimal immersion x˜ = lim x i : M → Rn i→∞

satisfying the theorem. Indeed, convergence is ensured by conditions (ai ) and (fi ), interpolation on Λ is ensured by (bi ), condition (c) follows from (ci ), and the general position properties (d) and (e) follow from (di ), (ei ), and (fi ). So, the theorem will be proved once we find sequences x i and εi as above. The base of the induction is the map x0 : U → Rn and the number ε0 chosen at the beginning of the proof. Assume inductively that i ∈ N and we have already found maps x j and numbers ε j satisfying the required conditions for j = 0, . . . , i − 1. Let us now explain how to find the next map x i . We consider three distinct cases. Case 1: the domain Ai = {ci−1 < ρ < ci } does not contain any critical point of ρ or point from Λ . In this case, Ai = M˚ i \ Mi−1 is a finite union of annuli. A full conformal minimal immersion x i : Mi → Rn satisfying conditions (ai )–(ci ) is provided by Proposition 3.3.2, while conditions (di ) and (ei ) are ensured by Theorem 3.4.1 (the general position theorem). Finally, every sufficiently small number εi > 0 satisfies condition (fi ) for this map x i . Case 2: the domain Ai = {ci−1 < ρ < ci } contains a point p ∈ Λ . Hence, p is the unique point of Ai ∩ Λ , and Ai is a finite union of annuli. If x : Λ → Rn is injective,

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then by Theorem 3.4.1 we can deform x i−1 slightly, keeping all required conditions, such that x i−1 (Mi−1 ) does not contain the point x(p) ∈ Rn . Let U p ⊂ Ai be a closed disc containing p as interior point and such that x|Up is a conformal minimal immersion (in case of basic interpolation set U p = {p}). Choose a point q ∈ bMi−1 and connect it to a point q ∈ bU p by a smooth embedded arc E ⊂ (Ai \ U p ) ∪ {q, q } intersecting bMi−1 and bU p transversely at q and q , respectively (q = p in case of basic interpolation). The union Si = Mi−1 ∪ E ∪U p is then a Runge admissible set in M. We parameterize E by a path γ : [0, 1] → E with γ(0) = q and γ(1) = q . By the inductive hypothesis, the map f i−1 = 2∂ x i−1 /θ : Mi−1 −→ A∗

(3.38)

is holomorphic on a neighbourhood of Mi−1 . By Lemma 3.5.4 we can extend f i−1 smoothly to E ∪U p , mapping it to A∗ , such that f i−1 |Up = (2∂ x/θ )|Up and 

ℜ E

f i−1 θ = ℜ

 1 0

˙ f i−1 (γ(s))θ (γ(s), γ(s)) ds = x(q ) − x i−1 (q).

We then extend x i−1 from Mi−1 to Si by setting x i−1 |Up = x|Up and x i−1 (γ(t)) = x i−1 (q) + ℜ

 t 0

˙ f i−1 (γ(s))θ (γ(s), γ(s)) ds,

t ∈ [0, 1].

(3.39)

It follows that (x i−1 , f i−1 θ ) ∈ GCMIr (Si ). Since the sets Mi−1 ⊂ Si ⊂ Mi clearly have the same homology basis, Proposition 3.3.2 and Theorem 3.4.1 provide the next conformal minimal immersion x i : Mi → Rn satisfying conditions (ai )–(ei ). We finish as before by choosing εi > 0 satisfying condition (fi ). Case 3: the domain Ai = {ci−1 < ρ < ci } contains a critical point p of ρ. Hence, p is the unique critical point of ρ on Ai , it is a Morse point, and Ai does not contain any points of Λ . The topology of the sublevel set {ρ ≤ c} changes at the value c = ρ(p). The Morse index of ρ at p is either 0, in which case p is a local minimum of ρ, or 1, in which case p is a saddle point of ρ. Subcase 3a: The Morse index of ρ at p equals 0. In this case, a new connected component of the sublevel set {ρ ≤ c} appears at p when c passes the value ρ(p). Let Δ ⊂ Ai be a small smoothly bounded disc at p. We extend x i−1 to Si = Mi−1 ∪ Δ by taking x i−1 : Δ → Rn to be an arbitrary conformal minimal immersion. The set Si is admissible and has the same topology as Mi , so we obtain the next map x i : Mi → Rn as before by applying Proposition 3.3.2 and Theorem 3.4.1. Assume now that p has Morse index 1. The change of topology of the sublevel set {ρ ≤ c} at p is described by adjoining to Mi−1 a smoothly embedded arc E ⊂ Ai ∪ bMi−1 attached with both endpoints q0 , q1 transversely to bMi−1 and otherwise contained in Ai . The admissible set Si = Mi−1 ∪ E has the same topology as Mi . We orient E from q0 to q1 . We must consider two possibilities according to whether q0 and q1 belong to the same or different connected components of Mi−1 . In the first subcase which we call 3b, a new nontrivial curve appears in the homology.

3.6 The First Main Theorem

159

If on the other hand the arc E connects two different components of Mi−1 then the first homology group and the genus do not change; this will be subcase 3c. Subcase 3b: the Morse index of ρ at p equals 1 and the endpoints of the arc E belong to the same connected component of Mi−1 . Connecting q1 to q0 by an oriented arc E  ⊂ Mi−1 gives a new closed Jordan curve C = E ∪ E  in the homology of Si = Mi−1 ∪ E. We proceed as in case 2. Let f i−1 be given by (3.38). By Lemma 3.5.4 we can extend f i−1 smoothly to E, mapping it to A∗ , such that 

ℜ C

f i−1 θ = 0



and ℑ C

f i−1 θ = p(C).

By (3.39) we extend x i−1 from Mi−1 to a generalized conformal minimal immersion (x i−1 , f i−1 θ ) on Si with Fluxx i−1 (C) = p(C), and hence Fluxx i−1 = p on H1 (Si , Z). Subcase 3c: the Morse index of ρ at p equals 1 and the endpoints of the arc E belong to different components of Mi−1 . In this case, no new curve appears in the homology basis of Si = Mi−1 ∪ E. By Lemma 3.5.4 we can extend f i−1 to a smooth map E → A∗ such that 

ℜ E

f i−1 θ = x i−1 (q1 ) − x i−1 (q0 ),

(3.40)

and by (3.39) we extend x i−1 to a generalized conformal minimal immersion Si → Rn . (There is no additional condition on the flux.) The proof in subcases 3b and 3c is now completed as in the previous cases by appealing to Proposition 3.3.2 and Theorem 3.4.1.   Theorem 3.6.2 (Mergelyan theorem for holomorphic null curves). Assume that M is an open Riemann surface, S is an admissible Runge set in M, Λ is a closed discrete subset of M, n ≥ 3 and r ≥ 1 are integers, and z is a generalized null curve S → Cn of class A r (S, Cn ) and a holomorphic null curve on a neighbourhood of each point p ∈ Λ . Given ε > 0 and a map k : Λ → N, there is a holomorphic null curve z˜ : M → Cn satisfying the following conditions. (a) ˜z − zC r (S) < ε. (b) z˜ − z vanishes to order k(p) at every point p ∈ Λ . (c) If the map z : Λ → Rn is injective, then z˜ is an injective immersion. Theorem 3.6.2 corresponds to the case when p : H1 (M, Z) → Rn in Theorem 3.6.1 is the zero homomorphism. The proof proceeds by the same type of induction. The only change is that, when integrating the derivative maps f i = dz i /θ in the induction process, we consider full complex integrals and not only their real parts. (Here, θ is a fixed nowhere vanishing holomorphic 1-form on M.) In particular,  condition (3.40) is replaced by E f i−1 θ = z i−1 (q1 ) − z i−1 (q0 ). Remark 3.6.3. An analogue of Theorem 3.6.2 holds for any holomorphic immersion z : M → Cn whose derivative dz/θ = f : M → Cn lies in A∗ = A \ {0}, where A ⊂ Cn is an irreducible conical complex subvariety such that A∗ is an Oka manifold. We refer to [22, 12] for this generalization.  

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3.7 The Second Main Theorem: Fixed Component Functions In this section we prove the second main result of this chapter, Theorem 3.7.1. The difference from Theorem 3.6.1 is that up to n − 2 component functions of a given generalized conformal minimal immersion S → Rn are now globally defined harmonic functions on the given Riemann surface, and they will remain fixed when approximating the remaining components. The main interest of this result is in the construction of conformal minimal immersions with a control of the placement of the image surface in the ambient space; in particular, in the construction of proper conformal minimal surfaces in Rn given in the following section. Let n = k + m ≥ 3 where k ≥ 1 and m ≥ 2. Let M be a connected open Riemann surface and S = K ∪ E be a Runge admissible set in M. Fix a nowhere vanishing holomorphic 1-form θ on M. Let (x, f θ ) ∈ GCMIr (S, Rn ) be a generalized conformal minimal immersion for some r ≥ 1. Write x = (x , x ), where x = (x1 , . . . , xk ) and x = (xk+1 , . . . , xn ), f = ( f  , f  ), where f  = ( f1 , . . . , fk ) and f  = ( fk+1 , . . . , fn ).

(3.41) (3.42)

We assume that the component x extends to a harmonic map x : M → Rk . Let f  = 2∂ x /θ ∈ O(M, Ck ) and consider the holomorphic function   2 (3.43) + · · · + fn2 ∈ O(M). g = f12 + · · · + fk2 = − fk+1 We distinguish the following three cases: 1. The function g is identically zero and k = n − 2, m = 2. 2. The function g is identically zero and k ≤ n − 3, m ≥ 3. 3. The function g is not identically zero. Given a number c ∈ C we let   Σc = (zk+1 , . . . , zn ) ∈ Cm : z2k+1 + · · · + z2n = −c .

(3.44)

This is a smooth quadric hypersurface in Cm for each c = 0, and Σ0 is the null quadric Am−1 ⊂ Cm . The hyperquadrics Σc for c ∈ C∗ are biholomorphic to each other by dilation of coordinates. We have seen in Example 1.13.8 (cf. (1.159)) that Σ0 \ {0} and Σc for c ∈ C∗ are Oka manifolds, and hence Theorem 1.13.3 applies to holomorphic maps from open Riemann surfaces (and more generally from Stein manifolds) into any one of them. Note that f  (p) ∈ Σg(p) ,

p ∈ S.

In case 1, the set Σ0 = {z2n−1 + z2n = 0} = {zn−1 + izn = 0} ∪ {zn−1 − izn = 0}

(3.45)

3.7 The Second Main Theorem: Fixed Component Functions

161

is the union of two complex lines in C2 intersecting at the origin. Since the image of any holomorphic map M → Σ0 is contained in one of these two lines, we can approximate a given map f  : S → Σ0 by a holomorphic map f˜ : M → Σ0 if and only if f  (S) ⊂ Σ0 is contained in one of the two lines; this gives a nonfull holomorphic map f˜ = ( f  , f˜ ) : M → An−1 . If f˜ is chosen such that it has zeros only at points where f  is nonzero, then f˜ has range in An−1 ∗ . A suitable choice of ˜ for a (nonfull) conformal minimal immersion periods of f˜ ensures that f˜ = 2∂ x/θ x˜ = (x , x˜ ) : M → Rn . In case 2, Σ0 = Am−1 ⊂ Cm and (x, f θ ) ∈ GCMIr (S, Rn ) is decomposable (see Definition 2.5.2). By Theorem 3.6.1 we can approximate (x , f  θ ) ∈ GCMIr (S, Rm ) by x˜ ∈ CMI(M, Rm ), and (x , x˜ ) ∈ CMI(M, Rn ) solves the problem. Neither of these two cases is of much interest since a generic (generalized) conformal minimal immersion is full and nondecomposable. In the remainder of the section we consider case 3 when the function g in (3.43) is not identically zero on M. We have the following analogue of Theorem 3.6.1 in which the first k component functions of x are harmonic on M and remain fixed while approximating and interpolating the remaining components. Theorem 3.7.1. Assume that M is an open Riemann surface, θ is a nowhere vanishing holomorphic 1-form on M, S = K ∪ E is an admissible Runge set in M, Λ is a closed discrete subset of M, n = k + m ≥ 3 (with k ≥ 1 and m ≥ 2) and r ≥ 1 are integers, x = (x , x ) (3.41) is a generalized conformal minimal immersion S → Rn of class C r (S, Rn ) and a conformal minimal immersion on a neighbourhood of each point p ∈ Λ , and x = (x1 , . . . , xk ) extends to a harmonic map x : M → Rk such that the function g = f12 + · · · + fk2 ∈ O(M) (3.43) is not identically zero and ˚ Given ε > 0, a map its zero set Q = {g = 0} intersects S only at points in S˚ = K. k : Λ → N, and a group homomorphism p = (p , p ) : H1 (M, Z) → Rk × Rm = Rn with p|H1 (S,Z) = Fluxx and p = Fluxx on H1 (M, Z), there is a conformal minimal immersion x˜ = (x , x˜ ) : M → Rn satisfying the following conditions. (a) x˜ − x C r (S) < ε. (b) The difference x˜ − x vanishes to order k(p) at every point p ∈ Λ . (c) Fluxx˜ = p on H1 (M, Z). (Equivalently, Fluxx˜ = p on H1 (M, Z).) Proof. Let f = 2∂ x/θ = ( f  , f  ), where f  = 2∂ x /θ ∈ O(M, Ck ) and f  = 2∂ x /θ is such that f  |S ∈ A r−1 (S, Ck ) and it is holomorphic on a neighbourhood of each point p ∈ Λ . Since the function g ∈ O(M) defined by (3.43) is not identically zero on M, its zero set g−1 (0) = Q = {q1 , q2 , . . .} is a discrete subset of M. We define the complex analytic subvariety Σ of M × Cm by     Σ = (p, z ) ∈ M × Cm : z ∈ Σg(p) = (p, z ) : z2k+1 + · · · + z2n = −g(p) . (3.46) The projection π : Σ → M, π(p, ζ ) = p, is a holomorphic fibre bundle over M \ Q with smooth Oka fibre Σ1 (3.44), and it has singular fibre Σ0 over points in Q. Condition (3.45) says that the map p → (p, f  (p)) is a section of Σ over S, and also over a small neighbourhood of any point p ∈ Λ . With an obvious abuse of

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language we shall say that f  is a section of Σ . To prove the theorem, we must approximate f  in C r−1 (S) by a global holomorphic section f˜ of Σ (3.46) over M satisfying suitable interpolation conditions at points p ∈ Λ and period conditions on a homology basis of H1 (M, Z) (to ensure condition (c) in the theorem), and also on certain other arcs in M to ensure the interpolation of values on Λ . The real part of the integral of f˜ θ will then provide a harmonic map x˜ : M → Rm such that (x , x˜ ) ∈ CMI(M, Rn ) satisfies the theorem. The problem of approximating f  on S and interpolating it on Λ by a global holomorphic section of Σ (without paying attention to period conditions) can be solved by Theorem 1.13.4 (the Oka principle for sections of ramified holomorphic fibrations whose regular fibres are Oka). In order to apply that theorem, we need a globally defined continuous section of Σ (3.46) over M which is holomorphic on a small neighbourhood of any point in M over which the projection π : Σ → M is ramified, i.e., at points in the discrete set Q = {g = 0}. If a point p ∈ Q also ˚ then by the hypothesis belongs to Λ (the set on which we must interpolate) or to K,  of the theorem f is already a holomorphic section on a neighbourhood of p. Near any other point p ∈ Q \ Λ we can find a local holomorphic section of Σ by setting fk+1 = 1 and fi = 0 for k + 2 ≤ i ≤ n − 1; the equation (3.46) then gives  fn (p ) = i 1 + g(p ) for p ∈ M near p. Since fn (p) = i, we have that f (p) = ( f  (p), f  (p)) ∈ A∗ . To simplify the notation, we replace Λ by Λ ∪ Q. This reduces the problem to that of approximating f  on S, interpolating it on Λ , and fulfilling the period conditions. Note that the pair (M, S ∪ Λ ) is homotopy equivalent to a relative CW-complex of dimension 1, so we obtain a skeleton of M from S ∪ Λ by successive attachments of points and edges. Since the regular fibres Σc (c = 0} of the projection π : Σ → M are connected, any section of Σ over S ∪ Λ can be extended to a continuous section over all of M. Hence, all conditions of Theorem 1.13.4 are satisfied. It remains to explain how to find a holomorphic section f˜ of Σ which also satisfies the required period conditions. The proof proceeds in several steps, following those in the proof of Theorem 3.6.1 given in the previous section. of Step 1: Approximating f  by a full map. Let C = {C1 , . . . ,Cl } be a collection curves in S introduced in the proof of Proposition 3.3.2. The union C = li=1 Ci is a Runge set in S which contains a homology basis of S as well as arcs used to interpolate on the finite set Λ ∩ S. In particular, any curve in C intersects Λ only at its endpoints. Let P = (P1 , . . . , Pl ) : C (C, Cm ) → (Cm )l , with 

Pi ( f ) =

 Ci

f  θ ∈ Cm ,

f ∈ C (C, Cm ), i = 1, . . . , l,

(3.47)

be the associated period map (3.12), but applied only to the component f  of a map f = ( f  , f  ). The point f  (p) ∈ Cm belongs to the smooth full quadric hypersurface Σg(p) ⊂ Cm (3.44) for all p ∈ Ci , except perhaps the endpoints where g may vanish and hence the quadric may degenerate to Σ0 . Consider the linear holomorphic vector

3.7 The Second Main Theorem: Fixed Component Functions

163

fields on Cn defined by Vi, j = zi

∂ ∂ −zj , ∂zj ∂ zi

k + 1 ≤ i, j ≤ n

(3.48)

(see (1.157)). Their flows given by (1.158) preserve the fibres {z } × Cm and leave the function z2k+1 + · · · + z2n invariant, so the hyperquadrics (3.44) are invariant sets. By using these flows and applying the same proof as in Lemma 3.3.1 we can find a map f˜ ∈ A r−1 (S, Cm ) which is full on every curve Ci ∈ C , it approximates f  as closely as desired in C r−1 (S), the difference f˜ − f  vanishes to a given order k(p) at the points p ∈ Λ ∩ S, and it satisfies P( f˜ ) = P( f  ) and ( f  , f˜ )(S) ⊂ A∗ . Step 2: Finding a period dominating spray. Replace f = ( f  , f  ) by the map ( f  , f˜ ) from step 1, so f  is full on every curve Ci ∈ C . By following the proof of Lemma 3.2.1 and using flows of vector fields (3.48) we find a holomorphic spray Φ f = (Φ f  , Φ f  ) : S × (Cm )l −→ A∗ of class A r−1 such that Φ f (· , 0) = f , Φ f  (· ,t) = f  for all t, Φ f  (· ,t) agrees with f  to order k(p) at each point p ∈ Λ ∩ S for all t, and ∂   P(Φ f  (· ,t)) : (Cm )l −→ (Cm )l is an isomorphism. ∂t t=0 Step 3: The noncritical case. We now obtain an analogue of Proposition 3.3.2. Assume that S is a connected admissible set in an open Riemann surface M such ∼ = that the inclusion S → M induces an isomorphism H1 (S, Z) −→ H1 (M, Z); hence S is Runge in M. Let Λ ⊂ S be a finite subset and C be a family of curves in S as in step 2. (At this point, the pair S ⊂ M and the set Λ are not assumed to be the ones in the statement of the theorem, but some ad hoc ones arising in the proof.) Assume that π : Σ → M is as above (see (3.46)) with all its ramification values contained in ˚ Let f = ( f  , f  ) and the spray Φ f be as in step 2 for this particular pair S ⊂ M. K. Next, we apply Theorem 1.13.1 (the Mergelyan theorem for manifold-valued maps) and then Theorem 1.13.4 (the Oka principle for sections of ramified maps with Oka fibres) in order to find a holomorphic section g : M → Σ which approximates f  in C r−1 (S) and agrees with f  to order k(p) at every point p ∈ Λ . as the core of the spray Φ f . We replace f = ( f  , f  ) by fˆ = ( f  , g ) : M → An−1 ∗ We also approximate the functions in C r−1 (S), used in the construction of Φ f , by ⊂ Cn global holomorphic functions on M. The new spray Φ fˆ : M × (Cm )l → An−1 ∗ has the same properties as Φ f ; in particular, the first component Φ f  = f  has not changed, while the second component Φg is C -period dominating provided that g is sufficiently close to f  on S. This gives a parameter value t ∈ (Cm )l near 0 such that the map f˜ = Φg (· ,t) is a section of Σ satisfying all required conditions, including the period conditions P( f˜ ) = P( f  ).

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Step 4: Completion of the proof. We follow the proof of Theorem 3.6.1. There are three cases which may arise in an induction step: (a) the noncritical case, which is handled by step 3 above; (b) interpolation at an additional point p ∈ Λ in the annulus Ai = {ci−1 < ρ < ci } (see case 2 in the proof of Theorem 3.6.1); and (c) dealing with a critical point p ∈ Ai of the strongly subharmonic exhaustion function ρ on M. In cases (b) and (c) we must sometimes add new arcs E to the given collection C on which the periods are controlled. The initial step amounts to extending the map ( f  )i−1 , given in the inductions step as in the proof of Theorem 3.6.1, smoothly across an arc E whose interior is contained in the annulus Ai so that the extended  map is a section of Σ (3.46) and the integral E ( f  )i−1 θ equals a prescribed vector in Cm . This is possible in view of Remark 3.5.5 on p. 155, following the proof of Lemma 3.5.4. We leave the remaining details to the reader.  

3.8 Mittag-Leffler and Carleman Theorems for Minimal Surfaces In 1884, G. Mittag-Leffler [262] proved that for any closed discrete subset A of C and meromorphic function f on a neighbourhood of A there is a meromorphic function f˜ on C which is holomorphic on C \ A and such that f˜ − f is holomorphic at every point of A. In 1948, H. Florack [127] extended this result to functions on any open Riemann surface. The natural counterpart in minimal surface theory of a meromorphic function on an open Riemann surface M is a conformal minimal immersion x : M \ A → Rn , where A is a closed discrete subset of M, such that the Cn -valued 1-form ∂ x is meromorphic on M with poles in A. This extends to admissible sets as follows. Definition 3.8.1. Assume that M is an open Riemann surface, θ is a nowhere vanishing holomorphic 1-form on M, S is an admissible set in M (see Definition 1.12.9), and A ⊂ S˚ is a finite set. A generalized complete conformal minimal immersion S \ A → Rn of class C r (S \ A) and of finite total curvature is a pair (x, f θ ) satisfying the following conditions. (i) The map x : S˚ \ A → Rn is a conformal minimal immersion such that the vectorial 1-form 2∂ x = f θ has an effective pole at each point of A. (ii) If W ⊂ S˚ is a smoothly bounded compact neighbourhood of A such that S = S \ W˚ is an admissible set, then (x, f θ ) is a generalized conformal minimal immersion S → Rn of class C r (S , Rn ); see Definition 3.1.2. Condition (i) implies that the length of the path x ◦ γ is infinite for any path γ : [0, 1) → S \ A with limt→1 γ(t) ∈ A, and hence x : S \ A → Rn is complete in the Riemannian sense. In fact, we shall see in Chapter 4 that x : S \ A → Rn is a proper map (by a result of Jorge and Meeks [202]). Furthermore, condition (i) also implies that the total curvature of x is finite on W \ A for any W as in (ii); see Section 2.6 for

3.8 Mittag-Leffler and Carleman Theorems for Minimal Surfaces

165

more details. These facts justify the assumption that ∂ x has an effective pole at each point of A, and they motivate the above definition. The first main result in this section is the following analogue of the MittagLeffler theorem for conformal minimal surfaces in Rn , including approximation and interpolation, which was obtained in [37]. Theorem 3.8.2 (Mittag-Leffler theorem for conformal minimal surfaces). Assume that M is an open Riemann surface, A ⊂ M is a closed discrete subset, and U ⊂ M is a neighbourhood of A. If x : U \ A → Rn (n ≥ 3) is a conformal minimal immersion such that ∂ x extends meromorphically to A with an effective pole at each point of A, then there is a full conformal minimal immersion x˜ : M \ A → Rn such that the map x˜ − x is harmonic at every point of A. In particular, ∂ x˜ extends meromorphically to M with effective poles at all points of A. Furthermore, if we assume that U is a union of pairwise disjoint connected admissible sets which form a locally finite family and (x, f θ ) is a generalized complete conformal minimal immersion of class C r (r ∈ N) and of finite total curvature on each component of U \ A, then for any number ε > 0, closed discrete ˚ map k : Λ → N, and group homomorphism subset Λ of M with A ⊂ Λ ⊂ U, p : H1 (M \ A, Z) → Rn with p|H1 (U\A,Z) = Fluxx there is a conformal minimal immersion x˜ : M \ A → Rn as above satisfying the following conditions. (a) x˜ − xC r (U) < ε. (b) The difference x˜ − x vanishes to order k(p) at every point p ∈ Λ . (c) Fluxx˜ = p on H1 (M \ A, Z). The first part of Theorem 3.8.2 is a natural counterpart to the Mittag-Leffler theorem for conformal minimal immersions from open Riemann surfaces. The analogous result for holomorphic null curves in Cn holds true with essentially the same proof. In particular, we have the following analogue of the Weierstrass–Florack interpolation theorem (see Theorem 1.12.14). Theorem 3.8.3 (Weierstrass interpolation theorem for conformal minimal surfaces). Let A be a closed discrete subset of an open Riemann surface M. If z : U → Cn (n ≥ 3) is a meromorphic map on a neighborhood U of A such that z|U\A is a holomorphic null curve and z has an effective pole at each point of A, then there is a meromorphic map z˜ : M → Cn whose restriction to M \ A is a full holomorphic null curve such that z˜ − z is holomorphic at every point of A. Theorem 3.8.2 can be seen as a version of Theorem 3.6.1 (the Mergelyan approximation theorem for conformal minimal surfaces) in which the initial immersion is defined on a punctured admissible subset and its Weierstrass data extend meromorphically to the punctures; in other words, the Weierstrass data are allowed to have poles. The proof follows closely the one of Theorem 3.6.1, but with an additional ingredient: a semiglobal approximation and interpolation result for meromorphic maps into the null quadric (see Lemma 3.8.4), which we now explain.

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Given an admissible compact set S in an open Riemann surface M, a finite set ˚ and an integer r ∈ Z+ we denote by A ⊂ S, A∞r (S | A, A∗ ) the space of maps f : S \ A → A∗ ⊂ Cn into the punctured null quadric (2.54) ˚ it has an effective pole at each point of A, such that f is meromorphic on S, r   and f ∈ A (S , A∗ ) for any S as in Definition 3.8.1 (ii). In particular, every map f ∈ A∞r (S | A, A∗ ) is holomorphic on S˚ \ A. We have the following extension of Lemma 3.3.1. Lemma 3.8.4. Assume that M, S, and C are as in Lemma 3.3.1, and let A ⊂ S˚ be a finite set. Given a map f ∈ A∞r (S | A, A∗ ) (r ∈ Z+ ), a number ε > 0, a finite set Λ ⊂ S˚ with A ⊂ Λ , and an integer s ∈ N there is a full holomorphic map F ∈ O(M \ A, A∗ ) satisfying the following conditions. (i) F − f is holomorphic at every point of A, and hence F − f lies in A r (S, Cn ). In particular, F is meromorphic with an effective pole at each point of A. (ii) F − f C r (S) < ε. (iii) F − f vanishes to order s at every point of Λ . (iv) P(F − f ) = 0, where P is the period map (3.12) associated to C . Note that the period map P in (iv) may be applied in view of condition (i). Proof. Since A ⊂ S˚ is finite and f |S\A takes values in A∗ , the Weierstrass theorem (see Theorem 1.12.13) furnishes a holomorphic function w on M vanishing nowhere on M \ A such that w f has no zeros or poles on A and hence w f ∈ A r (S, A∗ ). Following step 1 in the proof of Lemma 3.3.1, we may approximate w f uniformly on S by a full map g ∈ A r (S, A∗ ) such that g − w f vanishes to any given order s at every point of Λ ⊃ A. It follows that g˜ = g/w ∈ A∞r (S | A, A∗ ) and, assuming that s is chosen sufficiently large, g˜ − f is holomorphic at every point of A, it vanishes to order s at every point of Λ , and is close to 0 in the C r (S). Since g is full, so is g. ˜ Finally, as in the proof of Lemma 3.3.1 we can choose g such that, in addition to the above, we have that P(g˜ − f ) = 0. Replacing f by g, ˜ we may thus assume that f and hence w f are full and therefore nonflat. To complete the proof, we follow step 2 in the proof of Lemma 3.3.1. Let Cw f (C, Cn ) denote the space of continuous maps u : C \ A → Cn such that u/w − f ∈ C (C, Cn ). A standard modification of the proof of Lemma 3.2.1 furnishes an open neighbourhood U of the origin in CN for some large integer N ∈ N and a map Φw f : S ×U → A∗ of class A r (S ×U, A∗ ) such that Φw f (·, 0) = w f , Φw f (·,t) agrees with w f to a given finite order s at each point of Λ , and ∂   P w f (Φw f (·,t)) : CN → (Cn )l ∂t t=0

is submersive,

where P w f = (P1w f , . . . , Plw f ) : Cw f (C, Cn ) −→ (Cn )l = Cln is the period map

3.8 Mittag-Leffler and Carleman Theorems for Minimal Surfaces

P w f (u) = P

u w

−f ,

with P given by (3.12). Explicitly: 

u Piw f (u) = − f θ ∈ Cn , Ci w

167

u ∈ Cw f (C, Cn ),

u ∈ Cw f (C, Cn ), i = 1, . . . , l.

(Compare with (3.12); we refer to [37, Proof of Theorem 3.1] for more details.) As in the proof of Lemma 3.3.1, we may approximate w f by a holomorphic map f˜ : M → A∗ such that f˜ − w f vanishes to order s at every point of Λ . Furthermore, choosing s sufficiently large and reasoning as in that proof, we obtain a map F of the form F = Φ f˜(·,t0 ) ∈ O(M, A∗ ), for some t0 ∈ BN close to the origin, such that  = 0.  F/w− f is holomorphic at every point of A (hence, F ∈ C (C, Cn )) and P w f (F) (Actually, the spray Φ f˜ is such that Φ f˜(·,t)/w − f is holomorphic at every point of A for all t ∈ BN .) If the approximation of w f by f˜ is close enough and s is chosen  satisfies the conclusion of the lemma.  sufficiently large, then F = F/w  Lemma 3.8.4 gives the following extension of Proposition 3.3.2; the proof requires only minor modifications and is omitted. For the second part, one combines the proof of Lemma 3.8.4 with the argument in the proof of Theorem 3.7.1. With this proposition in hand, the proof of Theorem 3.8.2 follows very closely that of Theorem 3.6.1 and we omit the details. Proposition 3.8.5. Assume that M is an open Riemann surface, θ is a nowhere vanishing holomorphic 1-form on M, and S = K ∪Γ is a connected admissible set in ∼ = M such that the inclusion S → M induces an isomorphism H1 (S, Z) −→ H1 (M, Z). Let A and Λ be a pair of finite sets in S˚ with A ⊂ Λ , and fix r, s ∈ N. Every generalized conformal minimal immersion (x, f θ ) : S \ A → Rn of class C r (S \ A) and finite total curvature (see Definition 3.8.1) can be approximated in C r (S \ A) by full conformal minimal immersions X : M \ A → Rn satisfying the following conditions. (i) The difference X − x is harmonic at every point of A. In particular, ∂ X extends meromorphically to M with an effective pole at every point of A. (ii) X − x vanishes to order s at every point of Λ ⊃ A. (iii) FluxX = Fluxx . Furthermore, if n = k + m with integers k ≥ 1 and m ≥ 2, x = (x , x ) is as in (3.41), and x = (x1 , . . . , xk ) extends to a harmonic map x : M \ A → Rk such that the function f12 + · · · + fk2 ∈ O(M \ A) (cf. (3.42) and (3.43)) is not identically zero and ˚ then there is a conformal minimal its zero set intersects S only at points in S˚ = K, immersion X = M \ A → Rn as above of the form X = (x , X  ). A different extension of Theorem 3.6.1 goes in the direction of ensuring better than uniform approximation on certain closed unbounded subsets of the given open Riemann surface, instead of just uniform approximation on compacts; i.e., Carleman type approximation.

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T. Carleman [77] proved in 1927 that for any pair of continuous functions f : R → C and ε : R → (0, +∞) there is an entire function F : C → C such that |F(x) − f (x)| < ε(x) for all x ∈ R. Note that the approximation takes place in the fine C 0 (R) topology. A. A. Nersesjan [271] and A. Boivin [67] characterized the Carleman sets in an arbitrary open Riemann surface M, i.e., those closed subsets E ⊂ M for which every function in A (E) can be approximated in the fine C 0 (E) topology by holomorphic functions on M. See [128, Sec. 3] for more information. The second main result in this section is an analogue of Carleman’s theorem for conformal minimal surfaces in Rn (see Theorem 3.8.6) which was obtained by I. Castro-Infantes and B. Chenoweth in 2019, [80]. Before stating their theorem, we describe the closed sets on which the approximation takes place. Let E be a closed subset in an open Riemann surface M. The holomorphic hull of E is the union E = j∈N Ej , where {E j } j∈N is any exhaustion of E by compact sets and Ej denotes the holomorphically convex hull of E j (1.41) (i.e., the union of E j and its holes). It is easily seen that the hull E is independent of the choice of  The same definition applies exhaustion. The set E is called O(M)-convex if E = E. in more general complex manifolds; see [238] and [84]. The closed set E ⊂ M has bounded exhaustion hulls if for every compact set  K ⊂ M, the set K ∪ E \ K ∪ E is relatively compact in M. A closed set S in M is said to be a Carleman admissible set if S = S, S has bounded exhaustion hulls, and S = K ∪ E where K is the union of a locally finite pairwise disjoint collection of compact domains with piecewise smooth boundaries and E = S \ K is the union of a locally finite pairwise disjoint collection of smooth Jordan arcs, so that each component of E intersects the boundary of K only at its endpoints (if at all) and all such intersections are transverse. Note that a compact set S ⊂ M is a Carleman admissible set if and only if it is a Runge admissible set in M. A generalized conformal minimal immersion S → Rn of class C r (r ∈ N) on a Carleman admissible set S ⊂ M is a pair (x, f θ ) such that for any compact smoothly bounded domain M  in M, the restriction of (x, f θ ) to the admissible set S = S ∩ M  is a generalized conformal minimal immersion S → Rn of class C r (S ). If S is compact then this agrees with our usual notion (see Definition 3.1.2). The following result of I. Castro-Infantes and B. Chenoweth [80] generalizes the Mergelyan theorem for conformal minimal immersions (see Theorem 3.6.1). Theorem 3.8.6 (Carleman theorem for conformal minimal surfaces). Let S = K ∪ E be a Carleman admissible set in an open Riemann surface M and (x, f θ ) be a generalized conformal minimal immersion S → Rn of class C r , r ∈ N. Given a continuous function ε : S → (0, +∞), a closed discrete subset Λ of M with ˚ a map k : Λ → N, and a group homomorphism p : H1 (M, Z) → Rn with Λ ⊂ S, p|H1 (S,Z) = Fluxx , there is a conformal minimal immersion x˜ : M → Rn satisfying the following conditions. (i) |x(p) ˜ − x(p)| < ε(p) for all p ∈ S. Furthermore, x˜ can be chosen to approximate x as closely as desired in the fine Whitney C r topology on S. (ii) x˜ − x vanishes to order k(p) at every point p ∈ Λ .

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169

(iii) Fluxx˜ = p on H1 (M, Z). (iv) If n ≥ 5 and the map x : Λ → Rn is injective, then x˜ is an injective immersion. (v) If n = 4 and x has simple double points on Λ , then x˜ is an immersion with simple double points. The second condition in (i) means that the derivatives of x˜ − x of order up to r, when measured in a fixed metric on the space of r-jets of maps M → Rn , can be made smaller than ε(p) at any point p ∈ S for a given continuous positive function ε : S → (0, ∞). Thus, the first condition in (i) refers to the approximation in the fine C 0 topology, which is what the authors proved in [80]. The analogous result holds true for null curves and for a more general family of directed holomorphic immersions in Cn ; see [80, Theorem 1.2]. We give a considerably simpler proof of Theorem 3.8.6 by using the Mergelyan theorem with interpolation for conformal minimal immersions; see Theorem 3.6.1. This also gives an approximation in the fine C r topology. Proof. The conditions on S imply that M is exhausted by smoothly bounded compact Runge domains M1  M2  · · ·  i∈N Mi = M such that for every i ∈ N we have that (a) bMi ∩ S ⊂ E \ K and the intersection bMi ∩ E is transverse, and (b) Hi := S ∪ Mi \ S ∪ Mi ⊂ Mi+1 . (Note that Hi is the union of holes of S ∪ Mi .) Set S0 = S and ∪ Mi = S ∪ Mi ∪ Hi ⊂ S ∪ Mi+1 , Si = S

i ∈ N.

(3.49)

Note that Si is a closed Carleman admissible set in M and we have Si ⊂ Si+1 ,

Si \ Mi+1 = S \ Mi+1 ,

∞ !

Si = M.

i=0

We shall inductively construct a sequence of generalized conformal minimal immersions (xi , fi θ ) of class C r from Si to Rn . We begin with (x0 , f0 θ ) = (x, f θ ) on S = S0 . Assume inductively that (xi−1 , fi−1 θ ) is a generalized conformal minimal immersion Si−1 → Rn of class C r which agrees with (x0 , f0 θ ) on Si−1 \ Mi+1 . Since Si−1 has no holes, Si−1 ∩ Mi+1 has no holes either. Theorem 3.6.1 (Mergelyan’s theorem for conformal minimal immersions) furnishes a generalized conformal minimal immersion (y, hθ ) from Mi+1 to Rn which approximates (xi−1 , fi−1 θ ) as closely as desired in the C r topology on Si−1 ∩ Mi+1 (which means that h ∈ O(Mi+1 , A∗ ) approximates fi−1 in C r−1 (Si−1 ∩ Mi+1 )) and it agrees with (xi−1 , fi−1 θ ) to order r at every point of the finite set Si−1 ∩ Mi+1 = {a1 , . . . , am }; see condition (a). (It suffices to ensure that h agrees with fi−1 to order r − 1 at each point a j and y(a j ) = xi−1 (a j ) for j = 1, . . . , m.) Hence, the pair (xi , fi θ ), which equals (y, hθ ) on Si ∩ Mi+1 and equals (xi−1 , fi−1 θ ) on Si \Mi+1 = S \ Mi+1 , is a generalized conformal minimal immersion Si → Rn of class C r . Clearly, the induction may

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3 Approximation and Interpolation Theorems for Minimal Surfaces

proceed. Assuming that the approximation is close enough at every step, the sequence (xi , fi θ ) converges to a conformal minimal immersion x˜ : M → Rn , with ˜ : M → A∗ , such the sequence fi converging to the holomorphic map f˜ = 2∂ x/θ that x˜ is arbitrarily close to x in the fine C r Whitney topology on S. The additional conditions in the theorem are obtained directly from the corresponding results in Theorem 3.6.1, applied at every step of the induction process.   Theorems 3.8.2 and 3.8.6 can be joined into a single Mittag-Leffler–Carleman type theorem for conformal minimal surfaces. Let S = K ∪ E be a closed Carleman admissible set in an open Riemann surface M, and let A be a closed discrete subset ˚ A generalized conformal minimal immersion S \ A → Rn of locally of M with A ⊂ K. finite total curvature is a pair (x, f θ ) such that for any compact admissible set S ⊂ S with A ∩ S ⊂ S˚ , (x, f θ ) is a generalized conformal minimal immersion S \ A → Rn of finite total curvature in the sense of Definition 3.8.1. Combining the proofs of Theorems 3.8.2 and 3.8.6, one easily obtains the following compilation result. Corollary 3.8.7. Let S, M, ε, Λ , and k be as in Theorem 3.8.6. Also, fix a set A ⊂ Λ ⊂ S˚ and assume that (x, f θ ) is a generalized conformal minimal immersion S \ A → Rn (n ≥ 3) of locally finite total curvature. For any group homomorphism p : H1 (M \ A, Z) → Rn with p|H1 (S\A,Z) = Fluxx and function k : Λ → N there is a conformal minimal immersion x˜ : M \ A → Rn such that (i) x˜ − x is harmonic at every point of A, (ii) |x(p) ˜ − x(p)| < ε(p) for all p ∈ S, (iii) x˜ − x vanishes to order k(p) at every point p ∈ Λ , and (iv) Fluxx˜ = p on H1 (M \ A, Z).

3.9 Global Properties I: Completeness In this section we prove that the approximating conformal minimal immersions in Theorems 3.6.1, 3.7.1, and 3.8.2 can always be chosen to be complete. Theorem 3.9.1. The following assertions hold true. i) The conformal minimal immersions x˜ : M → Rn in Theorem 3.6.1 can be chosen to be complete. ii) The holomorphic null curves z˜ : M → Cn in Theorem 3.6.2 can be chosen to be complete. iii) The conformal minimal immersions x˜ : M → Rn in Theorem 3.7.1 can be chosen to be complete. iv) The conformal minimal immersions x˜ : M \ A → Rn in Theorem 3.8.2 can be chosen to be complete. v) The holomorphic null curves z˜ : M \ A → Cn in Theorem 3.8.3 can be chosen to be complete.

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171

The statement iii) will be used in Section 5.6 for constructing complete minimal surfaces in Rn whose Gauss maps fail to intersect a certain maximal possible family of hyperplanes. The analogue of this result for holomorphic null curves holds true as well. Theorem 3.9.1 is a slight generalization of [12, Theorem 1.2] (items i)–iii)) and of [37, Theorem 7.1] (items iv) and v)); earlier results in this direction can be found in [16, 17, 18]. Before embarking on proofs of these results, we give some historical comments in order to place them in the proper context. As late as in the 1980s the prevailing thought was that hyperbolic Riemann surfaces (i.e., open Riemann surfaces carrying nonconstant negative subharmonic functions, see Definition 1.10.11) play only a marginal role in the global theory of minimal surfaces. This belief was partially refuted by the pioneering works of L. P. Jorge and F. Xavier [203] from 1980, N. Nadirashvili [268] from 1996, and S. Morales [264] from 2003. The constructions in these works combined the Enneper–Weierstrass formula (2.84) for minimal surfaces in R3 with the classical Runge theorem for holomorphic functions from 1885 (see Theorem 1.12.1). For the statements and discussion of these results, see Sections 3.10 and 7.1. The following obvious consequences of Theorem 3.9.1 give further evidence against the aforementioned thought. Corollary 3.9.2 ([230]). Every open Riemann surface is the complex structure of a complete immersed minimal surface in R3 . Corollary 3.9.3 ([37]). If A is a closed discrete subset of an open Riemann surface M, then M \ A is the complex structure of a complete immersed minimal surface in R3 such that every point of A is an end of finite total curvature. The proof of Theorem 3.9.1 uses Theorems 3.6.1, 3.7.1, and 3.8.2. It relies on an intrinsic-extrinsic version of the method developed by Jorge and Xavier [203] for constructing complete minimal surfaces in R3 between two parallel planes (see Section 7.1). We shall prove assertion i); ii)–iii) are completely analogous, while iv) and v) require an extra comment (see the last paragraph of the proof). Proof of Theorem 3.9.1 i). By Theorem 3.6.1 we may assume without loss of generality that x is a full conformal minimal immersion on M (see Definition 2.5.2). Choose a strongly subharmonic Morse exhaustion function ρ : M → R and an increasing sequence of real numbers c0 < c1 < c2 < · · · which diverges to +∞ and satisfies conditions (i), (ii), and (iii) in the proof of Theorem 3.6.1 (see p. 157). Fix a point p0 ∈ M˚ 0 = {ρ < c0 }. Set x0 = x. We shall inductively construct a sequence of conformal minimal immersions x i from a neighbourhood of the compact domain Mi = {ρ ≤ ci } to Rn and a sequence of positive numbers εi satisfying conditions (ai )–(fi ) in the proof of Theorem 3.6.1 and also the following one for all i ∈ N. (gi ) distx i (p0 , bMi ) > i, where distx i denotes the distance function associated to the metric (x i )∗ ds2 induced by the Euclidean metric ds2 via x i . Given such sequences, the limit map x˜ = limi→∞ x i : M → Rn is a conformal minimal immersion satisfying Theorem 3.6.1 (see p. 157). Moreover, condition (gi ) clearly

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implies that the image by x˜ of any divergent path in M has infinite Euclidean length, and hence x˜ is complete, provided that x i is close enough to x i−1 in C r (Mi−1 ) for each i ∈ N. So, to complete the proof it remains to explain the induction. Fix a nowhere vanishing holomorphic 1-form θ on M. It is clear that condition (g0 ) holds true since p0 ∈ M˚ 0 . For the inductive step, assume that for some i ∈ N we have x j and ε j satisfying the required conditions for all j ∈ {0, . . . , i − 1}. By Theorem 3.6.1 we may assume that x i−1 = (x1i−1 , . . . , xni−1 ) is defined everywhere on M and is a full conformal minimal immersion. Choose a pair of numbers di−1 and di with ci−1 < di−1 < di < ci and so close to ci−1 that the compact domain {ci−1 ≤ ρ ≤ di } does not contain any critical point of ρ or point from Λ . In addition, we i−1 ) does not vanish choose these numbers such that the map ( f i−1 ) = ( f1i−1 , . . . , fn−2 i−1  i−1 i−1 = ( f1 , . . . , fn ) = 2∂ x i−1 /θ . anywhere on Ai = {di−1 ≤ ρ ≤ di }, where f (Note that ( f i−1 ) has finitely many zeros in the compact set {ci−1 ≤ ρ ≤ ci } since f i−1 : M → A∗ is a full holomorphic map.) Fix a number μ satisfying 0 < μ < min |( f i−1 ) |. 

(3.50)

Ai

Note that Ai is a finite union of smoothly bounded compact annuli. Take a finite collection Γ of pairwise disjoint Jordan arcs in A˚ i with the property that if α ⊂ M \Γ is a path with the initial point in {ρ < di−1 } and the final point in {ρ > di }, then  α

|θ | >

i . μ

(3.51)

To find such a family of arcs, one can follow the idea in the construction of the labyrinth by Jorge and Xavier in [203] (see the proof of Theorem 7.1.3). Also, since xni−1 is continuous and both Mi−1 and Γ are compact, there is a λ > 0 so big that   i−1 xn (p) − (xni−1 (q) + λ ) > i for all pairs of points p ∈ Mi−1 and q ∈ Γ . (3.52) The union S = Mi−1 ∪ Γ is a (disconnected) Runge admissible set in M (see Definition 1.12.9). Consider the map x˜ i−1 = (x˜1i−1 , . . . , x˜ni−1 ) : S → Rn given by x˜li−1 = xli−1 |S for all l ∈ {1, . . . , n − 1} and

i−1 xn on Mi−1 , i−1 (3.53) x˜n = λ + xni−1 on Γ . Note that (x˜ i−1 , f i−1 θ ) ∈ GCMI(S, Rn ). Hence, Theorem 3.7.1 provides a conformal minimal immersion x i = (x1i , . . . , xni ) from a neighbourhood of Mi = {ρ ≤ ci } to Rn which is as close as desired to x˜ i−1 in C r (S) and satisfies (ai ), (bi ), (ci ), and xli = xli−1 for all l ∈ {1, . . . , n − 2}.

(3.54)

We claim that x i also satisfies condition (gi ) provided the approximation of (x˜ i−1 , f θ ) by x i on S is close enough. It suffices to verify that distx i (Mi−1 , bMi ) > i.

3.9 Global Properties I: Completeness

173

Let γ be a path in Mi connecting Mi−1 to bMi . If γ ∩ Γ = ∅, then we have that 

(2.55)

length(x i ◦ γ) = 2 (3.54)

=

 γ

γ

|∂ x i | ≥ 2

 γ

i |(∂ x1i , . . . , ∂ xn−2 )|

(3.50)

|( f i−1 ) θ | > μ

 γ

(3.51)

|θ | > i.

If on the other hand γ ∩ Γ = ∅ then for any p ∈ γ ∩ Mi−1 and q ∈ γ ∩ Γ we have length(x i ◦ γ) ≥ |x i (p) − x i (q)| ≥ |xni (p) − xni (q)| ≈ (3.53)

(3.52)

≈ |x˜ni−1 (p) − x˜ni−1 (q)| = |xni−1 (p) − xni−1 (q) − λ | > i. This shows (gi ). Finally, by the general position theorem (see Theorem 3.4.1) we may assume that conditions (di ) and (ei ) hold. To complete the proof, it remains to choose εi > 0 so small that condition (fi ) holds true for this x i . This proves part i); the proof of ii) is entirely analogous and is omitted. Concerning part iii), recall that Theorem 3.7.1 is a version of Theorem 3.6.1 in which up to n − 2 component functions of x are harmonic on all of M. Under this assumption, we may follow the proof of part i) but using Theorem 3.7.1 instead of Theorem 3.6.1 each time when the latter is invoked. This minor modification of proof gives part iii) in the theorem. In order to prove assertions iv) and v) we need to distinguish cases. Assume that γ : [0, 1) → M \ A is a divergent path. If γ diverges in M, then we ensure that x˜ ◦ γ has infinite length for x˜ given in iv) by the arguments above. If on the contrary γ converges to a point a ∈ A, then x˜ ◦ γ has infinite length since ∂ x˜ has an effective pole at a. The same discussion applies for assertion v) in the theorem. In these cases, we use the second part of Proposition 3.8.5 instead of Theorem 3.7.1.   We also point out the following extensions of Theorem 3.8.6 and Corollary 3.8.7. Theorem 3.9.4. The following assertions hold true. i) The conformal minimal immersion x˜ : M → Rn in Theorem 3.8.6, with fine C 0 approximation in part (i), can be chosen to be complete. ii) The conformal minimal immersion x˜ : M \ A → Rn in Corollary 3.8.7 can be chosen to be complete. We emphasize that the above theorem does not require completeness of the initial immersion x : E → Rn (or x : E \ A → Rn ). Indeed, since x˜ is required to approximate x only in the fine C 0 (E) topology, it is possible to increase the length of the image of E as much as desired during the process. These results fail in general if we ask for fine C r approximation for any r > 0. Assertion i) in the theorem was proved by I. Castro-Infantes and B. Chenoweth in [80]. Assertion ii) follows by combining their argument with Proposition 3.8.5. The proof of this theorem broadly follows that of Theorem 3.9.1, but the proof of the inductive step requires a different idea given by the following lemma.

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Lemma 3.9.5. Let K and V be smoothly bounded, connected, compact domains in an open Riemann surface M such that K ⊂ V˚ and K is a deformation retract of V . Let Γ be a finite collection of pairwise disjoint smooth Jordan arcs in V \ K˚ such that each arc in Γ has an endpoint in bK, the other endpoint in bV , it meets bK and bV transversely at its endpoints and is otherwise disjoint from bK ∪ bV , and ˚ and let S = K ∪ Γ is an admissible set. Let A ⊂ Λ be a pair of finite sets in K, n x : S \ A → R (n ≥ 3) be a generalized complete conformal minimal immersion of class C r (S \ A) (r ∈ N) and of finite total curvature (see Definition 3.8.1). Given numbers ε > 0 (small), δ > 0 (big), and s ∈ N there is a conformal minimal immersion x˜ : V \ A → Rn satisfying the following conditions. (a) x˜ − x extends harmonically to every point of A. (b) x˜ − xC r (K) < ε and x˜ − xC 0 (Γ ) < ε. (c) distx˜ (bK, bV ) > δ . (d) x˜ − x vanishes to order s at every point of Λ . (e) Fluxx˜ = Fluxx on H1 (S \ A, Z). Proof. For simplicity of exposition we assume that V \ K˚ is connected (an annulus) and Γ is a single arc. Write x = (x1 , . . . , xn ). Up to a deformation of x outside a small neighbourhood of K in S = K ∪ Γ , we may assume that the Euclidean length of the path x1 |Γ : Γ → R is bigger than δ ; this deformation must be highly zigzagging but small in the C 0 (Γ ) topology. Further, by Proposition 3.8.5 we may assume that x is a full conformal minimal immersion on V \ A. Thus, there is a smooth Jordan arc β ∈ V \ S with endpoints in bV and otherwise disjoint from bV such that length(x1 ◦ γ) > δ for any path γ in V \ β connecting bK and bV .

(3.55)

(We may choose β as the boundary of a small smoothly bounded neighbourhood of S = K ∪ Γ in V , removing the part of the boundary contained in bV ; see p. 307 and in particular (7.9).) Note that S ∪ β is an admissible set. Choose a generalized conformal minimal immersion x = (x1 , . . . , xn ) : S ∪ β → Rn of class C r (S ∪ β ) such that x = x on S, x1 = x1 on S ∪ β , and xn (p) > xn C 0 (V ) + δ for all p ∈ β .

(3.56)

Fix a number ε  with 0 < ε  < ε. Proposition 3.8.5 furnishes a conformal minimal immersion x˜ = (x˜1 , . . . , x˜n ) : V \ A → Rn with x˜1 = x1 such that x˜ satisfies conditions (a), (d), and (e), and it satisfies the following stronger version of condition (b): (b’) x˜ − xC r (K) < ε  and x˜ − xC 0 (Γ ∪β ) < ε  . If ε  > 0 is chosen sufficiently small then x˜ also satisfies condition (c). Indeed, let γ ⊂ V \ K be a path connecting bK and bV . If γ ∩ β = ∅ then length(x˜ ◦ γ) ≥ length(x˜1 ◦ γ) = length(x1 ◦ γ) > δ by (3.55). If on the contrary γ ∩ β = ∅ then (3.56) and (b’) imply that length(x˜ ◦ γ) > δ provided ε  > 0 is small enough.

3.10 Global Properties II: Properness

175

In the general case, the difference V \ K˚ is a finite union of topological annuli and the above proof easily adapts by choosing Γ to contain at least one arc in every ˚  connected component of V \ K.  With Lemma 3.9.5 in hand, the proof of Theorem 3.9.4 can be completed by following the proof of Theorem 3.9.1. We refer to [80] for the details.

3.10 Global Properties II: Properness As mentioned in the previous section, as late as in the 1980s the prevailing opinion among the experts was that hyperbolic Riemann surfaces play only a marginal role in the global theory of minimal surfaces in Euclidean spaces. This is seen in particular from the following well known conjectures from that time. Conjecture 3.10.1 (Sullivan). There is no properly immersed minimal surface in R3 with finite topology and hyperbolic conformal type. Conjecture 3.10.2 (Schoen–Yau [314, p. 18]). No hyperbolic open Riemann surface admits a proper harmonic map into R2 . In particular, every immersed minimal surface in R3 with a proper projection to R2 is parabolic. Although Sullivan’s Conjecture 3.10.1 was well known at the time, a precise reference does not seem to be available in the literature. The first counterexample was given in 2003 by S. Morales [264] who constructed a proper conformal minimal immersion D → R3 from the open unit disc by using the L´opez-Ros deformation [235] and the Runge approximation theorem in a highly intricate way. Subsequently, immersed proper minimal surfaces in R3 with arbitrary topology and hyperbolic conformal type were constructed by L. Ferrer, F. Mart´ın and W. H. Meeks [126]; see also A. Alarc´on, L. Ferrer, and F. Mart´ın [19] for the case of finite topologies and the first known examples with uncountably many ends. The first and more ambitious part of the Schoen–Yau Conjecture 3.10.2 was refuted in 1999 by V. Boˇzin [70] who gave an explicit example of a proper harmonic map D → R2 . Another example was given in 2001 by F. Forstneriˇc and J. Globevnik [144] who constructed a proper holomorphic map ( f1 , f2 ) : D → C2 from the disc such that f1 and f2 are nowhere vanishing; it follows that lim max{log | f1 (ζ )|, log | f2 (ζ )|} = +∞,

|ζ |→1

and hence (log | f1 |, log | f2 |) : D → R2 is a proper harmonic map. A third counterexample to the first assertion in Conjecture 3.10.2 was obtained by A. Alarc´on and J. A. G´alvez [33] in 2011 by a simplification of Morales’s construction from [264]. On the other hand, the second part of the Schoen–Yau conjecture concerning minimal surfaces remained open at that time.

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Compared with the results on completeness discussed in the previous section, ensuring properness of the approximating conformal minimal immersions in Theorems 3.6.1 and 3.7.1 is a considerably more arduous task. Moreover, it is a necessary condition that the prescription of values on the interpolation set Λ ⊂ M be proper in Rn . We now show that this requirement is also sufficient. In particular, we prove that every open Riemann surface admits a proper conformal minimal embedding into R5 and a proper conformal minimal immersion into R3 with proper projection to R2 (see Corollary 3.10.4). The following theorem is a compilation of the main results on this subject. Theorem 3.10.3. Let M, S, Λ , and x = (x1 , . . . , xn ) be as in Theorem 3.6.1. • If the map x|Λ : Λ → Rn is proper, then the conformal minimal immersions x˜ : M → Rn in Theorem 3.6.1 can be chosen to be proper. Furthermore, if m ≥ 3 and the map x = (x , x ) into Rn−m × Rm is such that x extends harmonically to M and x |Λ : Λ → Rm is proper, then x˜ = (x˜ , x˜ ) : M → Rn−m × Rm can be chosen such that x˜ = x and x˜ : M → Rm is proper. • The holomorphic null curves z˜ : M → Cn in Theorem 3.6.2 can be chosen to be proper provided that the map z|Λ : Λ → Cn is proper. • The conformal minimal immersions x˜ : M → Rn in Theorem 3.8.6 can be chosen to be proper provided that the initial immersion x : S → Rn is proper. The assumption m ≥ 3 is necessary, for otherwise one could obtain proper nonflat minimal surfaces in R3 contained between two parallel planes, contradicting the halfspace theorem of D. Hoffman and W. H. Meeks III [192]. Theorem 3.10.3 is a culmination of a series of works by the authors and I. CastroInfantes and B. Chenoweth [38, 43, 30, 12, 80]. The optimal result with interpolation on a discrete set was obtained in [12], and the second statement in the theorem is a slight extension of the results in that paper. The last statement is from [80]. The proof of Theorem 3.10.3 is given in the following section, except for the last two statements which are analogous and whose proof is omitted. In the remainder of this section we discuss corollaries of this theorem and connections to several other well known conjectures in the field. We also pose a couple of new conjectures which are justified by these results. The following corollary of Theorem 3.10.3 settles in the negative the second part of the Schoen–Yau conjecture and furnishes an optimal solution to both Conjecture 3.10.1 and Conjecture 3.10.2. Corollary 3.10.4. The following assertions hold for every open Riemann surface M. (i) There is a proper conformal minimal immersion M → R3 with a proper orthogonal projection into an affine plane. In particular, there exists a proper harmonic map M → R2 . (ii) There is a proper conformal minimal immersion M → R4 with simple double points. (iii) There is a proper conformal minimal embedding M → R5 .

3.10 Global Properties II: Properness

177

The analogous results also hold for nonorientable conformal surfaces and were proved by the authors in [29, Theorem 6.2]. Assertion (i) in the corollary (which clearly implies Corollary 3.9.2) does not follow directly from Theorem 3.10.3, but from an easy modification of its proof and is due to Alarc´on and L´opez [38]. A different proof of the existence of a proper harmonic map M → R2 from any open Riemann surface M was given by R. Andrist and E. F. Wold [51, Theorem 5.6] as a consequence of their result that every open Riemann surface admits a proper holomorphic immersion into (C∗ )2 . A particularly simple proof of the existence of a proper harmonic map M → R2 , using only the classical Mergelyan approximation theorem on open Riemann surfaces (see Theorem 1.12.7), was given by I. Castro-Infantes [79]. The approach of Andrist and Wold was extended to higher dimensions by F. Forstneriˇc, who proved in [142, Corollary 3.5] that every Stein manifold M of complex dimension n ≥ 1 admits a proper pluriharmonic map into R2n . Recall that a pluriharmonic map is locally the real part of a holomorphic map, and an open Riemann surface is the same thing as a 1-dimensional Stein manifold. Assertions (ii) and (iii) in Corollary 3.10.4 were obtained by the authors [30] in a more precise form, with approximation on any compact O(M)-convex subset of M. Since every complex curve in Cn is a minimal surface by a result of W. Wirtinger [346] (see Corollary 1.7.12), assertion (ii) also follows from the classical result that every open Riemann surface admits a proper holomorphic immersion with simple double points into C2 (see [140, Theorem 2.4.1 (a)]). Conformal minimal immersions satisfying conditions (ii) or (iii) can also be chosen to have a proper orthogonal projection into an affine plane. In fact, a minor modification of the proof of Theorem 3.10.3 shows that for any real number δ , 0 < δ < π/2, there is a conformal minimal immersion x = (x1 , x2 , . . . , xn ) : M → Rn satisfying either of the assertions (i), (ii), or (iii) in Corollary 3.10.4 and such that the function x2 + tan(δ )|x1 | : M → R is positive and proper (see [38] for the details). This implies the following corollary from [38]. Corollary 3.10.5. For any open Riemann surface M and any wedge W in R3 of angle strictly bigger than π there is a proper conformal minimal immersion x : M → R3 with x(M) ⊂ W . By the already mentioned halfspace theorem of Hoffman and Meeks [192], the only properly immersed minimal surfaces in R3 contained in a halfspace (a wedge of angle π) are affine planes; hence Corollary 3.10.5 is sharp in this sense. In view of Corollary 3.10.4 (iii), it is a natural problem to determine the smallest dimension d ≥ 3 for which every open Riemann surface embeds as a proper conformal minimal surface in Rd . It is well known that only a few open Riemann surfaces embed as proper minimal surfaces in R3 (see [235, 95, 96, 255, 252, 253] for a number of restrictions on the topological and conformal types of properly embedded minimal surfaces in R3 ). Hence, the smallest embedding dimension for open Riemann surfaces as minimal surfaces satisfies d ≥ 4. By Corollary 3.10.4 (iii), we have that d ∈ {4, 5}; we conjecture that d = 4.

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Conjecture 3.10.6 ([26, Conjecture 1.2]). Every open Riemann surface admits a proper conformal minimal embedding into R4 . Conjecture 3.10.6 is related to the Forster–Bell–Narasimhan Conjecture [129, 59] which asks whether every open Riemann surface embeds as a smooth closed complex curve in C2 . Indeed, an affirmative answer to the latter would settle Conjecture 3.10.6 in the positive. This seems one of the most difficult open problems in complex analysis. It has been known since the early 1960s that every open Riemann surface immerses properly holomorphically into C2 and embeds properly holomorphically into C3 (see [140, Theorem 2.4.1] and the references therein). In the last two decades, powerful new methods for constructing proper holomorphic embeddings of open Riemann surfaces into C2 have been developed, using in particular the technique of exposing boundary points (see Section 6.7) and pushing the boundary of the curve in C2 to infinity by applying holomorphic automorphisms; see the papers by F. Forstneriˇc and E. F. Wold [147, 148] and the survey in [140, Sections 9.10–9.11]. For instance, every bordered Riemann surface whose closure embeds holomorphically into C2 , and every (possibly infinitely connected) circle domain in C, are known to embed properly holomorphically into C2 . There are no topological restrictions to the embedding problem, hence neither to Conjecture 3.10.6, for Alarc´on and L´opez proved in [41] that every smooth open orientable surface embeds properly into C2 as a complex curve. However, their proof does not give any control of the complex structures on such curves. These results, and the absence of any conceptual obstructions, provide support to the Forster–Bell–Narasimhan Conjecture. Since the family of minimal surfaces in R4 contains immersed complex curves in C2 as a small subfamily, Conjecture 3.10.6 has in principle a better chance of being true. Nevertheless, every open Riemann surface which is known to properly embed as a conformal minimal surface in R4 is also known to admit a proper holomorphic embedding into C2 . The main difficulty when approaching Conjecture 3.10.6 is that no automorphisms of R4 other than the rigid motions map minimal surfaces to minimal surfaces. This means that one of the major techniques used in the construction of embedded complex curves in C2 , namely, holomorphic automorphisms of C2 and in particular the Anders´en–Lempert theory, is not available in the construction of properly embedded minimal surfaces. Since a conformal minimal immersion from an open Riemann surface into Rn is minimal if and only if it is harmonic (see Theorem 2.3.1), the following conjecture has an even better chance of being true. Conjecture 3.10.7. Every open Riemann surface admits a proper harmonic embedding into R4 . Every Riemann surface (either open or closed) admits a smooth, proper, conformal (not necessarily harmonic) embedding in R3 according to R. A. R¨uedy [306]. It was proved by R. E. Greene and H. Wu [166] that every open Riemannian manifold of dimension k ≥ 2 admits a proper harmonic embedding into R2k+1 ; in particular, every open Riemann surface admits a proper harmonic embedding into R5 . The latter fact is implied by Corollary 3.10.4 (iii) which shows that there is

3.11 Proof of Theorem 3.10.3

179

such an embedding which is also conformal, hence a minimal surface. Conjecture 3.10.7 proposes a different improvement of this particular instance of the results by Greene and Wu, reducing the embedding dimension by one. Of course, an affirmative answer to Conjecture 3.10.6 would confirm Conjecture 3.10.7.

3.11 Proof of Theorem 3.10.3 We shall explain the proof of the first (and the main) statement in the theorem. To obtain the version with some fixed component functions in the second statement, one follows the same line of arguments but appeals to Theorem 3.7.1 instead of Theorem 3.6.1 every time when the latter is invoked. As said before, the proof of the last two statements is analogous and will be omitted. A standard procedure to obtain a proper map from an open manifold M into Rn is to choose a normal exhaustion M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ j∈Z+ M j = M by smoothly bounded compact domains, with Mi−1 ⊂ M˚ i for every i ∈ N, and construct a sequence of maps x i : Mi → Rn having the following three main properties: • x i is close to x i−1 on Mi−1 in an appropriate topology, • x i maps bMi further and further away from the origin as i goes to ∞, and • the set x i (Mi \ M˚ i−1 ) should not be much closer to the origin than x i−1 (bMi−1 ). Carrying out this process in the right way, the limit map limi→∞ x i : M → Rn will be proper. In the context of Theorem 3.10.3, the induction step realizing this program is given by the following lemma. We denote by  · ∞ the infinity norm on Rn : y∞ = max{|y j | : j = 1, . . . , n},

y = (y1 , . . . , yn ) ∈ Rn .

Lemma 3.11.1. Let M, θ , n, and r be as in Theorem 3.6.1, and let K and L be smoothly bounded, compact Runge domains in M such that K ⊂ L˚ and K is a deformation retract of L. Assume that x is a conformal minimal immersion from a neighbourhood of K to Rn mapping bK into Rn \ {0}. Pick a number   0 < δ < min x(p)| ∞ : p ∈ bK . (3.57) Given a finite set Λ ⊂ K˚ and numbers ε > 0 and k ∈ N, there is a conformal minimal immersion x˜ from a neighbourhood of L to Rn satisfying the following conditions. (i) x˜ − xC r (K) < ε. (ii) The difference x˜ − x vanishes to order k at every point of Λ . (iii) Fluxx˜ = Fluxx . ˚ (iv) x(p) ˜ ∞ > δ for all p ∈ L \ K. (v) x(p) ˜ ∞ > 1/ε for all p ∈ bL. Proof. The assumptions imply that L \ K˚ is a finite union of annuli. For simplicity of exposition we assume that L \ K˚ is connected; in the general case we apply the

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3 Approximation and Interpolation Theorems for Minimal Surfaces

same argument to each component. We assume without loss of generality that 1 > δ. ε

(3.58)

Write x = (x1 , . . . , xn ). By (3.57) there are an integer l ≥ 3, subsets I1 , . . . , In of Zl = Z/lZ = {0, 1, . . . , l − 1}, and a family of compact connected subarcs {α j : j ∈ Zl } of bK satisfying the following conditions.

(A1) j∈Zl α j = bK. (A2) α j and α j+1 have a common endpoint p j and are otherwise disjoint, j ∈ Zl . (A3) na=1 Ia = Zl and Ia ∩ Ib = ∅ for all a = b ∈ {1, . . . , n}. (A4) If j ∈ Ia for some a ∈ {1, . . . , n}, then |xa (p)| > δ for all p ∈ α j . (It may happen that Ia = ∅ for some a ∈ {1, . . . , n}.) For each j ∈ Zl we connect the point p j ∈ bK to some point q j ∈ bL by a smooth embedded arc γ j ⊂ (L˚ \ K) ∪ {p j , q j } intersecting bK and bL transversely at p j and q j , respectively, such that the arcs γ j for j ∈ Zl are pairwise disjoint (see Figure 3.1). Hence, S=K∪

!

γj

j∈Zl

is an admissible subset of M (see Definition 1.12.9). For each j ∈ Zl we denote by β j ⊂ bL the arc with the endpoints q j−1 and q j which does not contain any of the other points qi for i ∈ Zl \ { j − 1, j}. Note that !

β j = bL.

(3.59)

j∈Zl

Finally, we denote by D j ⊂ L \ K˚ the closed disc bounded by the arcs γ j−1 , α j , γ j , and β j (see Figure 3.1). It follows that L \ K˚ =

! j∈Zl

Fig. 3.1 Sets in the proof of Lemma 3.11.1

D j.

(3.60)

3.11 Proof of Theorem 3.10.3

181

The proof of the lemma consists of two different procedures. In the first one we approximate x by a conformal minimal immersion from a neighbourhood of L which satisfies conditions (i), (ii), and (iii) in the statement of the lemma, but only meets conditions (iv) and (v) at points of j∈Zl γ j . In the second step we recursively apply deformations which ensure conditions (iv) and (v) at all points in the discs D j . Step 1: Pushing the image of the points q j far away. In view of condition (A4) we may, by virtue of Lemma 3.5.4, extend x to a generalized conformal minimal immersion (x, f θ ) ∈ GCMIr (S) satisfying the following two conditions. (B1) If j ∈ Ia for some a ∈ {1, . . . , n}, then |xa (p)| > δ for all p ∈ γ j−1 ∪ γ j . (B2) If j ∈ Ia for some a ∈ {1, . . . , n}, then |xa (p)| > 1/ε for p ∈ {q j−1 , q j }. By Theorem 3.6.1 we may assume that x with these properties is a conformal minimal immersion on a neighbourhood of L in M. Step 2: Pushing image of the discs D j far away. Set x0 = (x10 , . . . , xn0 ) = x and I0 = ∅. We shall inductively construct conformal minimal immersions x b = (x1b , . . . , xnb ), b = 1, . . . , n, from a neighbourhood of L to Rn such that the following conditions hold for each b. (C1b ) (C2b ) (C3b ) (C4b ) (C5b ) (C6b ) (C7b )



x b and x b−1 are ε/n close in the C r topology on K ∪ ( j∈I\Ib D j ). The difference x b − x b−1 vanishes to order k at every point in Λ . Fluxx b = Fluxx b−1 . If j ∈ bi=1 Ii then x b (p)∞ > δ for all p ∈ D j . If j ∈ bi=1 Ii then x b (p)∞ > 1/ε for all p ∈ β j . If j ∈ Ia for some a ∈ {1, . . . , n} then |xab (p)| > δ for all p ∈ γ j−1 ∪ α j ∪ γ j . If j ∈ Ia for some a ∈ {1, . . . , n} then |xab (p)| > 1/ε for p ∈ {q j−1 , q j }.

Note that condition (C60 ) is implied by (A4) and (B1), whereas (C70 ) coincides with (B2). The other conditions are void for b = 0. To explain the induction, assume that b ∈ {1, . . . , n} and we have found conformal minimal immersions x0 , . . . , x b−1 satisfying the required properties. By conditions (C6b−1 ) and (C7b−1 ) there is for each j ∈ Ib a closed disc Ω j ⊂ D j \ (γ j−1 ∪ α j ∪ γ j ) for which the following conditions hold (see Figure 3.1). (I) Ω j ∩ β j is a compact connected Jordan arc. (II) |xbb−1 (p)| > δ for all p ∈ D j \ Ω j . (III) |xbb−1 (p)| > 1/ε for all p ∈ β j \ Ω j . Choose any b ∈ {1, . . . , n} \ {b} and a number λ > 0 so big that   b−1 x  (p) + λ  > 1 b ε

for all p in the compact set

Consider the map xˆ = (xˆ1 , . . . , xˆn ) : L \

!

Ω j.

j∈Ib



j∈Ib (D j \ Ω j )

→ Rn defined by

(3.61)

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3 Approximation and Interpolation Theorems for Minimal Surfaces

 xˆa =

xab−1



for all a = b ,

xˆb =

xbb−1 

on L \

λ

on

+ xbb−1 





j∈Ib D j ,

j∈Ib Ω j .

(3.62)

Clearly, xˆ is a conformal minimal immersion on a neighbourhood of the compact Runge domain L \ j∈Ib (D j \ Ω j ) ⊂ M all whose components except the b -th extend harmonically to L. In this situation, Theorem 3.7.1 (the Mergelyan theorem for conformal minimal immersions with fixed components) furnishes a conformal minimal immersion x b = (x1b , . . . , xnb ) from a neighbourhood of L to Rn satisfying xbb = xbb−1

(3.63)

and conditions (C1b ), (C2b ), (C3b ), (C6b ), and (C7b ). We claim that if the close, then (C4b ) and (C5b ) are also satisfied approximation of xˆ by x b is sufficiently / Ib then for this map x b . Indeed, take j ∈ bi=1 Ii and p ∈ D j . If j ∈ (3.62)

b−1 ˆ (p)∞ x b (p)∞ ≈ x(p) ∞ = x

(C4b−1 )

>

δ.

If j ∈ Ib and p ∈ Ω j , then (3.61)

(3.62)

b−1 x b (p)∞ ≈ x(p) ˆ ∞ ≥ |xˆb (p)| = |λ + xb (p)| >

1 (3.58) > δ. ε

Finally, if j ∈ Ib and p ∈ D j \ Ω j , then (3.63)

(II)

x b (p)∞ ≥ |xbb (p)| = |xbb−1 (p)| > δ . This shows (C4b ). The same argument but using (C5b−1 ) and (III) instead of (C4b−1 ) and (II), respectively, enables us to verify condition (C5b ). Completion of the proof. It is clearly seen from (3.59), (3.60), and conditions (C4n ), (C5n ), and (C1b )–(C3b ) for b = 1, . . . , n, that x˜ = xn satisfies the lemma.   Proof of Theorem 3.10.3. We shall prove the first assertion in the theorem, which is equivalent to the special case m = n in the second assertion. Thus, we assume that x|Λ : Λ → Rn is a proper map. By Theorem 3.6.1 we may also assume that x = (x1 , . . . , xn ) is a full conformal minimal immersion M → Rn . Choose a strongly subharmonic Morse exhaustion function ρ : M → R such that every level set {ρ = c} contains at most one critical point of ρ or at most one point of Λ , but not both. We also choose an increasing sequence of real numbers c0 < c1 < c2 < · · · with limi→∞ ci = +∞ such that conditions (i), (ii), and (iii) in the proof of Theorem 3.6.1 (p. 157) hold true. Set Mi = {ρ ≤ ci }, i ∈ Z+ . Since x|Λ is proper, we may choose c0 so big that x has no zeros on Λ \ M0 .

(3.64)

3.11 Proof of Theorem 3.10.3

183

By a generic choice of c0 we may assume that x has no zeros on bM0 = {ρ = c0 }. Set x0 = x and fix a pair of numbers 0 < δ0 < min{x(p)∞ : p ∈ bM0 } and 0 < ε0 < ε/2, where ε > 0 is the given number in Theorem 3.6.1. We shall inductively construct a sequence of conformal minimal immersions x i from a neighbourhood of Mi to Rn , a sequence δi > 0 with limi→∞ δi = +∞, and a decreasing sequence εi > 0 such that conditions (ai )–(fi ) in the proof of Theorem 3.6.1 (see p. 157) and also the following ones are satisfied for all i ∈ N. (gi ) x i (p)∞ > δi for all p ∈ bMi . (hi ) x i (p)∞ > min{δi−1 , δi } > 0 for all p ∈ Mi \ M˚ i−1 . If such sequences exist then, as in the proof of Theorem 3.6.1, there is a limit map x˜ = lim x i : M → Rn i→∞

satisfying the conclusion of that theorem. We claim that x˜ is a proper map. Indeed, fix a number T > 0 and take any divergent sequence pm ∈ M, m ∈ N. By conditions (ai ) and (fi ) on p. 157 we have that ∞

|x(p) ˜ − x i (p)| < ∑ ε j < 2εi < ε,

p ∈ Mi , i ∈ Z+ .

(3.65)

j=i

Since limi→∞ δi = +∞, there is an i0 ∈ N so big that δi−1 > T + ε

for all i ≥ i0 .

(3.66)

Since the sequence pm ∈ M diverges, there is an m0 ∈ N so large that for every / Mi0 , and hence there is an im ≥ i0 with pm ∈ Mim \ M˚ im −1 . It m ≥ m0 we have pm ∈ follows that for any m ≥ m0 we have that x(p ˜ m )∞

≥ (him ),(3.65)

>

x im (pm )∞ − x im (pm ) − x(p ˜ m )∞ min{δim −1 , δim } − ε

(3.66)

> T.

Since T > 0 was arbitrary, this shows that x(p ˜ m ) is a divergent sequence in Rn . Since pm ∈ M was an arbitrary divergent sequence, x˜ is a proper map. This concludes the proof of the first assertion of the theorem under the assumption that there exist sequences of maps x i and numbers δi , εi (i ∈ N) satisfying conditions (ai )–(hi ). In order to complete the proof, we now explain the induction. The basis is given by the immersion x0 = x and the numbers δ0 and ε0 > 0 chosen at the beginning of the proof. Assume inductively that for some i ∈ N we have found maps x j and numbers δ j and ε j satisfying the required conditions for all j ∈ {0, . . . , i − 1}. We now explain how to find the next triple x i , δi , εi . We look at three different cases. Case 1: the domain M˚ i \ Mi−1 = {ci−1 < ρ < ci } does not contain any critical point of ρ or point from Λ . In this case we choose

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3 Approximation and Interpolation Theorems for Minimal Surfaces

δi = i.

(3.67)

Since Mi−1 is a deformation retract of Mi , it is clear in view of condition (gi−1 ) that Lemma 3.11.1 gives a conformal minimal immersion x i from a neighbourhood of Mi to Rn satisfying conditions (ai ), (bi ), (ci ), (gi ), and (hi ). Moreover, by the general position Theorem 3.4.1 we may assume that (di ) and (ei ) are also satisfied. Finally, we choose εi > 0 so small that (fi ) holds true for this immersion x i . Case 2: the domain M˚ i \ Mi−1 = {ci−1 < ρ < ci } contains a (unique) point p ∈ Λ . In this case we choose 1 (3.68) δi = x(p)∞ > 0; 2 see (3.64). We have that Λ ∩ Mi \ M˚ i−1 = {p} and Mi−1 is a deformation retract of Mi . Reasoning as in step 2 in the proof of Theorem 3.6.1 (p. 157), we obtain a Runge admissible set Si = Mi−1 ∪ E, where E is a smooth embedded arc in M˚ i \ M˚ i−1 connecting p to bMi−1 , and a generalized conformal minimal immersion (x i−1 , f i−1 θ ) ∈ GCMIr (Si ) extending x i−1 such that x i−1 (p) = x(p) and x i−1 (q)∞ > min{δi−1 , δi } for all q ∈ E (see (gi−1 ) and (3.68)). Further, by Theorem 3.6.1 we may assume that x i−1 is a conformal minimal immersion on a   of Si in M˚ i such that Mi−1 is a smoothly bounded compact neighbourhood Mi−1 i−1  ˚ deformation retract of Mi and x (q)∞ > min{δi−1 , δi } for all q ∈ Mi−1 \ Mi−1 .  Since Mi \ M˚ i−1 does not contain any critical point of ρ or point from Λ , this reduces the proof to case 1. Case 3: the domain M˚ i \ Mi−1 = {ci−1 < ρ < ci } contains a (unique) critical point p of ρ. In this case we again choose δi = i.

(3.69)

Reasoning as in the previous case, we may easily adapt the arguments in case 3 in the proof of Theorem 3.6.1 to find a smoothly bounded compact neighbourhood   Mi−1 ⊂ M˚ i of Mi−1 such that Mi−1 is a deformation retract of Mi and x i−1 is a  satisfying x i−1 (q)∞ > conformal minimal immersion on a neighbourhood of Mi−1 δi−1 . Again, this reduces the proof to case 1. The fact that limi→∞ δi = +∞ is clearly implied by (3.67), (3.68), (3.69), and the assumption that the map x|Λ : Λ → Rn is proper. The latter condition ensures that limi→∞ x(pi )∞ = +∞ for any ordering {p1 , p2 , p3 , . . .} of the closed discrete set Λ . This completes the proof of the first assertion in the statement of the theorem. It is a trivial task to modify the proof of Lemma 3.11.1 and the argument given above to prove the second and more general assertion of the theorem as well as Corollary 3.10.4 (i); note that the choice of b ∈ {1, . . . , n} \ {b} in (3.61) is free. We leave the details to the reader.  

3.12 The h-Principle for Minimal Surfaces and Null Curves

185

3.12 The h-Principle for Minimal Surfaces and Null Curves In this section we describe what we know about the topological shape of the spaces CMInf (M, Rn ) ⊂ CMI(M, Rn ) of (nonflat) conformal minimal immersions from a given connected open Riemann surface M. The analogous question is considered for the spaces of null curves NCnf (M, Cn ) ⊂ NC(M, Cn ). These results depend on Oka theory, i.e., the homotopy theory for holomorphic maps from Stein manifolds to Oka manifolds; see Sect. 1.13. The main point behind this connection is that the punctured null quadric is an Oka manifold; see Example (1.13.8). Elements of Gromov’s convex integration theory (see [173, 170, 320]) also enter the picture. Initial results in this direction were obtained in the paper [24] by the first two authors who showed the following (see [24, Theorems 1.1 and 1.2]). Theorem 3.12.1. For every conformal minimal immersion x : M → Rn (n ≥ 3) there exists a smooth isotopy xt : M → Rn (t ∈ [0, 1]) of conformal minimal immersions such that x0 = x and the immersion x1 has vanishing flux, i.e., it is the real part x1 = ℜz of a holomorphic null curve z : M → Cn . In addition, x1 may be chosen to be complete, and if x is complete and nonflat then the isotopy xt can be chosen to consist of complete nonflat conformal minimal immersions. The term isotopy of immersions is understood to mean regular homotopy. Although the proofs in [24] are written in the case n = 3, they hold for any n > 3. This direction was continued and substantially expanded by F. Forstneriˇc and F. L´arusson [146]. We consider the spaces NCnf (M, Cn ) and CMInf (M, Rn ) of nonflat holomorphic null curves and nonflat conformal minimal immersions, respectively. The reason for this restriction is that we do not know how to properly deal with the flat ones; the main difficulty is indicated in Remarks 3.2.2 (C) and 3.2.4 (A). For any fixed integer n ≥ 3 we consider the following diagram of maps: φ

NCnf (M, Cn )

j1

/ O(M, A∗ )  

j2

/ C (M, A∗ ) (3.70)

ψ

ℜ

  ℜNCnf (M, Cn )

/ Onf (M, A∗ )   O

i

/ CMInf (M, Rn )

The notation was introduced in Section 3.1, except for Onf (M, A∗ ) which denotes the space of nonflat holomorphic maps M → A∗ , i.e., maps whose range is not contained in a ray of the null quadric. The map ℜ : NCnf (M, Cn ) → ℜNCnf (M, Cn ) is the real part projection taking null curves to conformal minimal immersions with vanishing flux. The maps j1 and j2 in the top row, and the map i in the bottom row, are the natural inclusion. The maps φ and ψ are defined by φ : NCnf (M, Cn ) −→ Onf (M, A∗ ), ψ : CMInf (M, Rn ) −→ Onf (M, A∗ ),

z → dz/θ , x → 2∂ x/θ ,

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3 Approximation and Interpolation Theorems for Minimal Surfaces

where θ is a fixed nowhere vanishing holomorphic 1-form on M. All spaces carry the compact-open topology. The main result of the paper [146] is the following. The notions of the parametric h-principle and the (weak) homotopy equivalence were introduced in Sect. 1.13. Theorem 3.12.2. For any open Riemann surface, M, all maps in the diagram (3.70) satisfy the parametric h-principle, and hence are weak homotopy equivalences. If M is of finite topological type then these maps are genuine homotopy equivalences. The proof relies on the parametric version of Lemma 3.2.1 (see Remark 3.2.2 (A)), the parametric version of Lemma 3.5.4 on convex integration, and the parametric version of Theorem 1.13.3 provided by [140, Theorem 5.4.4]. We do not give a proof here, but only indicate where the individual parts and their proofs can be found. We shall discuss weak homotopy equivalences, although all instances are proved by actually establishing the parametric h-principle. The left vertical map is the projection of a null curve to its real part. By continuity in the compact-open topology of the Hilbert transform that takes x ∈ ℜNC(M, Cn ) to its harmonic conjugate y : M → Rn with y(p0 ) = 0, where p0 ∈ M is any chosen base point, this projection is a homotopy equivalence. The inclusion i in (3.70) is a weak homotopy equivalence by [146, Theorem 1.1]. That φ is a weak homotopy equivalence is proved in [146, Theorem 5.6]. The same proof shows that the map ψ is a weak homotopy equivalence, but this already follows from what has been said since a composition of weak homotopy equivalences is again a weak homotopy equivalence. Finally, the inclusion j1 is a weak homotopy equivalence by [146, Corollary 5.5], and j2 is a weak homotopy equivalence by the Oka principle (Corollary 1.13.9). Theorem 3.12.2 describes the rough topological shape of the spaces NCnf (M, Cn ) and CMInf (M, Rn ) in terms of the space of continuous maps M → A∗ into the punctured null quadric. We can go a step further by observing that A∗ ⊂ Cn admits a deformation retraction onto the real analytic submanifold L(A) = A ∩ S2n−1 ⊂ Cn ,

(3.71)

the intersection of the null quadric A ⊂ Cn with the unit sphere in Cn . Hence, the space C (M, A∗ ) is homotopy equivalent to C (M, L(A)). The manifold L(A) is the Milnor link of the isolated normal singularity of A at the origin. (This terminology originates in the influential book of J. Milnor [261] on the topology of isolated complex hypersurface singularities.) Denoting by π : Cn∗ → CPn−1 the canonical projection (2.74), we have a circle bundle S1



/ L(An−1 ) π

 Q n−2

(3.72)

3.12 The h-Principle for Minimal Surfaces and Null Curves

187

where π(An−1 ) = Q n−2 ⊂ CPn−1 (see (2.72)). This gives the following corollary. Corollary 3.12.3. Let M be an open Riemann surface, and let L(A) (3.71) be the Milnor link of the null quadric A ⊂ Cn for some n ≥ 3. Then, the maps NCnf (M, Cn ) −→ C (M, L(A)),

CMInf (M, Rn ) −→ C (M, L(A))

arising from the diagram (3.70) are weak homotopy equivalences, and are homotopy equivalences if M is of finite topological type. The homotopy groups of mapping spaces such as C (M, L(A)) can be calculated from homotopy groups of spheres. We shall not pursue this issue here, but will look at the simplest question of determining the connected components of these spaces. Recall that the first homology group H1 (M, Z) of an open Riemann surface is a free abelian group on at most countably many generators: H1 (M, Z) ∼ = Zl with n−1 n l ∈ Z+ ∪ {∞}. The punctured null quadric A∗ ⊂ C is simply connected when n ≥ 4, while π1 (A2∗ ) ∼ = Z2 (see [24, Equation (8.3)]). The latter fact is also seen by noting that A2∗ admits the unbranched two-sheeted holomorphic covering map   ι : C2∗ −→ A2∗ , ι(u, v) = u2 − v2 , i(u2 + v2 ), 2uv (3.73) from the simply connected space C2∗ = C2 \ {0}. This map is called the spinorial representation of the null quadric in C3 . It follows that the path connected components of the space C (M, A2∗ ) are in one-to-one correspondence with the elements of the abelian group (Z2 )l (see [24, Proposition 8.4]), and C (M, An−1 ∗ ) is path connected if n ≥ 4. Hence, Corollary 3.12.3 implies the following result (see [146, Corollary 1.6]). Corollary 3.12.4. Let M be a connected open Riemann surface with H1 (M, Z) ∼ = Zl 3 for some l ∈ Z+ ∪ {∞}. Then, each of the spaces CMInf (M, R ) and NCnf (M, C3 ) has 2l path connected components, and a path component is determined by a group homomorphism H1 (M, Z) → Z2 (hence by an element of H 1 (M, Z2 ) = (Z2 )l ). The spaces CMInf (M, Rn ) and NCnf (M, Cn ) are path connected for every n ≥ 4. We shall see in Section 5.7 (see Theorem 5.7.6) that the natural inclusions CMInf (M, Rn ) → CMI(M, Rn ) and NCnf (M, Cn ) → NC(M, Cn ) induce bijections of their path components, and hence the analogue of Corollary 3.12.4 also holds for the spaces CMI(M, Cn ) and NC(M, Cn ) (see Corollary 5.7.7). We illustrate Corollary 3.12.4 by a few simple examples in dimension n = 3 in which M is C∗ or an annulus. Since A∗ is an Oka manifold and π1 (A∗ ) = Z2 , there are precisely two homotopy classes of holomorphic maps f : M → A∗ . Note that f is nullhomotopic if and only if it factors through the universal covering map ι : C2∗ → A∗ (3.73) (by a continuous or, equivalently, a holomorphic map). Assume now that f = ( f1 , f2 , f3 ) = z is the derivative of a holomorphic null curve z : M → C3 and f1 = i f2 . Consider the Weierstrass representation: f1 = (1 − g2 )η,

f2 = i(1 + g2 )η,

f3 = 2gη,

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3 Approximation and Interpolation Theorems for Minimal Surfaces

where g is a meromorphic function and η is a holomorphic function on M. Assume for simplicity that g is holomorphic or, equivalently, that η has no zeros. Then f factors through ι if and only if η has a square root on M. Indeed, if η has a square √ √ root then f = ι( η, g η); conversely, if f = ι(σ , τ) for some holomorphic map (σ , τ) : M → C2∗ , then σ 2 = η. Example 3.12.5. 1. A flat null curve: Let M = C∗ , and let f : C∗ → A∗ ⊂ C3 be the holomorphic map f (ζ ) = 2ζ (1, i, 0) onto the ray of A∗ spanned by the null vector (1, i, 0). In this case, g = 0 and η(ζ ) = 2ζ does not have a square root on M. Thus, the flat null curve Z(ζ ) = (ζ 2 , iζ 2 , 0) (ζ ∈ C∗ ) lies in the nontrivial isotopy class. 2. The catenoid: M = C∗ , g(ζ ) = ζ , and η(ζ ) = 1/ζ 2 . Since η has a square root on M, we are in the trivial isotopy class. 3. Two-sheeted covering of Meeks’s minimal M¨obius strip (see Meeks [248, Theorem 2] or Example 2.8.10): M = C∗ ,

g(ζ ) = ζ 2

ζ +1 , ζ −1

η(ζ ) = i

(ζ − 1)2 . ζ4

Note that η has a square root on M. Despite the pole of g at 1, we get a holomorphic factorization through ι, so we are in the trivial isotopy class. Let z = x+iy : C∗ → C3 be the null curve with this Weierstrass data. Then, x is invariant with respect to the antiholomorphic involution I(ζ ) = −1/ζ¯ on C∗ , and hence it induces a conformal minimal immersion x˜ : C∗ /I → R3 . This is Meeks’s complete minimal M¨obius strip  in R3 with total Gaussian curvature −6π (see Subsection 2.8.10).  Remark 3.12.6. A nonconstant harmonic map x : M → Rn is called a branched conformal minimal surface if f = 2∂ x/θ : M → An−1 is a holomorphic map with isolated zeros; see Remark 2.3.7. A branch point p0 ∈ M (i.e., a zero of f ) is removable by a small deformation of x through conformal minimal surfaces in a neighbourhood of p0 if an only if f admits a local holomorphic deformation near p0 through holomorphic maps with range in An−1 to a map with image in An−1 ∗ . Let us look at this problem in dimension n = 3. Let π : C3∗ → CP2 denote the canonical projection (2.74). We have that π(A2∗ ) = Q1 ∼ = CP1 (see Sect. 2.5) and the circle bundle (3.72) is  / L(A2 ) (3.74) S1 π

 CP1 . The fibre S1 → L(A2 ) is a generator of the fundamental group π1 (A2∗ ) = π1 (L(A2 )) = Z2 . Choose a closed coordinate disc D ⊂ M around a branch point p0 so that f (p0 ) = 0 and p0 is the only zero of f on D. Let z be a holomorphic coordinate on D with z(p0 ) = 0. In this coordinate we have that f (z) = zm g(z) where m ∈ N and g : D → A∗2 .

(3.75)

3.12 The h-Principle for Minimal Surfaces and Null Curves

189

The curve f (bD) ⊂ A2∗ equals m times the generator of π1 (A2∗ ). Let ι : C2 → A2 be the map (3.73). Note that g lifts to a map g˜ : D → C2∗ with ι ◦ g˜ = g. Hence, f factors through ι if and only if m = 2k is even, and in this case the map D z → f˜(z) = zk g(z) ˜ ∈ C2 satisfies ι ◦ f˜ = f . Obviously we can deform f˜ to a 2 map D → C∗ avoiding the origin, and hence we can deform f to a map D → A∗ . Conversely, if f admits a deformation to a map D → A2∗ then f (bD) is the zero  element in π1 (A2∗ ), which means that the integer m in (3.75) is even.  The weak homotopy equivalence principle in Theorem 3.12.2 was generalized in [146] to holomorphic immersions Z = (Z1 , . . . , Zn ) : M → Cn from open Riemann surfaces M whose derivative map f = dZ/θ : M → A∗ lies in the nonsingular part of a closed conical complex subvariety A ⊂ Cn with the only singularity at the origin such that A∗ = A \ {0} is an Oka manifold (see Def. 1.13.2). Here, θ is a nowhere vanishing holomorphic 1-form on M. By conical, we mean that A is a union of complex lines passing through the origin. Such immersions are said to be directed by A, or A-immersions. For example, taking A = A gives null immersions, while A = Cn gives usual holomorphic immersions. Clearly, a holomorphic map dZ = f θ if and only if f : M → A∗ determines an A-immersion Z : M → Cn with  the holomorphic 1-form f θ has vanishing periods, i.e., γ f θ = 0 for every closed  curve γ ⊂ M. In this case, Z(p) = p f θ for p ∈ M. An A-immersion is said to be nondegenerate if the tangent spaces T f (p) A ⊂ T f (p) Cn ∼ = Cn over all points p ∈ M n span C . The following is [146, Theorem 5.6]. Theorem 3.12.7 (Weak homotopy equivalence principle for A-immersions). Let M be an open Riemann surface and θ be a nowhere vanishing holomorphic 1form on M. Assume that A is a closed irreducible conical complex subvariety of Cn such that A∗ = A \ {0} is an Oka manifold. Denote by IA,∗ (M, Cn ) the space of all nondegenerate A-immersions M → Cn with the compact-open topology. The map IA,∗ (M, Cn ) −→ C (M, A∗ ),

Z −→ dZ/θ

is a weak homotopy equivalence, and is a genuine homotopy equivalence if M is of finite topological type. This is based on a parametric h-principle in [146, Theorem 5.4] and the fact that the natural inclusion map O(M, A∗ ) → C (M, A∗ ) is a weak homotopy equivalence provided that A∗ is an Oka manifold (see Corollary 1.13.6). Taking A = Cn we see that the natural map I(M, Cn ) → C (M, S2n−1 ) from the space I(M, Cn ) of all holomorphic immersions Z : M → Cn , obtained by composing the derivative dZ/θ : M → Cn∗ with the retraction of Cn∗ onto the sphere S2n−1 , is a weak homotopy equivalence (cf. [146, Theorem 1.4]). It follows that the path components of the space I(M, C) are in bijective correspondence with the elements of H 1 (M, Z), and I(M, Cn ) is (2n − 3)-connected if n > 1 (cf. [146, Corollary 1.5]). A further unification of the results presented in this section was made by A. Alarc´on and F. L´arusson in [36]. Assume that M is a connected open Riemann surface. Let X be a connected Oka domain in the smooth locus of an analytic

190

3 Approximation and Interpolation Theorems for Minimal Surfaces

subvariety of Cn , n ≥ 1, such that the convex hull of X is all of Cn . Let O∗ (M, X) be the space of nondegenerate holomorphic maps f : M → X, where we use the same definition of nondegeneracy as above (i.e., the tangent spaces T f (p) X ⊂ T f (p) Cn ∼ = Cn over all p ∈ M span Cn ). Take a holomorphic 1-form θ on M, not identically zero, and let π : O∗ (M, X) −→ H 1 (M, Cn ) send a map f to the cohomology class of the Cn -valued holomorphic 1-form f θ . Their main result (see [36, Theorem 1]) states that the map π is a Serre fibration, i.e., it satisfies the homotopy lifting property. Their proof follows very closely the proofs of the results mentioned above, given in the paper [146]. The authors show that this implies the results presented in this section by standard homotopy-theoretic arguments.

Chapter 4

Complete Minimal Surfaces of Finite Total Curvature

Complete minimal surfaces of finite total curvature in Rn (n ≥ 3) are in many ways the simplest minimal surfaces. The underlying complex structure of any such (orientable) surface is a finitely punctured compact Riemann surface; in particular, the surface has finite topology. Furthermore, by the Chern–Osserman theorem (see Theorem 4.1.1) the Weierstrass datum of the surface extends meromorphically to the punctures, and the surface itself has a fairly simple and well understood asymptotic behaviour at infinity, described by the Jorge–Meeks theorem (see Theorem 4.1.3). It follows in particular that the surface is proper in Rn and its Gauss map is algebraic and meets important restrictions (see Theorems 4.1.7 and 4.1.8). Concerning the construction of examples, on the positive side the simplicity of the Weierstrass data enables one to give explicit examples in a rather noninvolved way, and it is often possible to make fairly explicit illustrations of them; see Section 2.8 where several classical examples are described. On the negative side, algebraicity of the Weierstrass data of such surfaces makes it more difficult to obtain general existence and approximation results. Complete minimal surfaces of finite total curvature have been an important, and perhaps even the main focus of research in the global theory of minimal surfaces during the last six decades. This justifies the inclusion of the present chapter, in particular since we present not only some classical results, but also several recent ones which were obtained during the preparation of the book. In Section 4.1 we present a brief survey of the classical theory of complete minimal surfaces of finite total curvature in Rn for any n ≥ 3. We do not intend to be exhaustive, but merely discuss a few fundamental results which are obtained by means of complex analytic methods. Special emphasis is placed on those classical theorems which are used in the proofs of our new results on this subject, given in the subsequent sections and in the remainder of the book. Highlights include the already mentioned Chern–Osserman Theorem 4.1.1 and the Jorge–Meeks Theorem 4.1.3, the two basic pillars of the theory of complete minimal surfaces of finite total curvature in Euclidean spaces Rn .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_4

191

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4 Complete Minimal Surfaces of Finite Total Curvature

After introducing some additional notation and function spaces in Section 4.2, we proceed to develop preliminary results in Sections 4.3 and 4.4. In particular, we present an extension of Runge’s theorem for maps to CP1 with additional control on the divisor of the approximating maps; see Theorem 4.3.1. We also construct special period dominating sprays, similar to those in Section 3.2 but adapted to the particular features of complete minimal surfaces of finite total curvature. Our main new results on the subject come in Section 4.5 where we obtain general approximation and interpolation theorems of Runge–Weierstrass type for complete minimal surfaces of finite total curvature in Rn , and null curves in Cn , for arbitrary n ≥ 3; see Theorems 4.5.1 and 4.5.4. The case n = 3 is simpler since one can use the spinor representation formula (3.73) available only for minimal surfaces in R3 , as was done in [230, 13]. The approach that we present here, based on a recent paper by Alarc´on and L´opez [37], works in all dimensions n ≥ 3. The main new ingredient is that one deals with maps into a certain complex hyperquadric Sn−1 of Cn (see ∗ n−1 n (4.18)) which is linearly biholomorphic to the null quadric A∗ ⊂ C and has a kind of spinorial nature. Therefore, this procedure may be understood as a spinor representation formula for minimal surfaces in Rn , as explained in Section 4.2. We show in particular that, given a pair of disjoint finite subsets E and Λ of a compact Riemann surface Σ , every map Λ → Rn extends to a complete conformal minimal immersion Σ \ E → Rn of finite total curvature (see Corollary 4.5.2). We expect that these results could also be proved by using Theorem 1.15.1 on algebraic approximation of holomorphic maps from affine algebraic manifolds into the punctured null quadric An−1 ∗ . However, this result by itself does not guarantee interpolation at finitely many points and the existence of effective poles of the Weierstrass data at the point ends of the surface. Dealing with these issues would likely bring complications similar to those encountered in the chosen approach. Theorem 4.5.1 opens the door to the study of optimal hitting problems for complete minimal surfaces of finite total curvature. Indeed, once it is known that there exist surfaces of this kind passing through any given finite set A in Rn , it is natural to wonder what the simplest ones are from a geometric viewpoint; i.e., to determine the minimum absolute value of the total curvature among all complete minimal surfaces in Rn containing the set A (see Question 4.7.1). In a dual direction, one may ask about the cardinality of the smallest sets which are not contained in any complete minimal surface with a given value of the total curvature. We discuss these questions in dimension n = 3 in Section 4.7; see Theorem 4.7.2. On the way to that, we give an upper bound obtained in [13] for the number of intersections between a complete minimal surface of finite total curvature in R3 and an affine line not contained in the surface; see Theorem 4.6.1. The bound is expressed in terms of the topology and the total curvature of the surface. The main tools to carry out these tasks are the Jorge–Meeks formula (see Theorem 4.1.3) and the Schwarz reflection principle for minimal surfaces (see Corollary 2.3.6).

4.1 A Brief Survey of the Classical Theory

193

4.1 A Brief Survey of the Classical Theory In this section we review some of the fundamental results from the classical theory of complete minimal surfaces of finite total curvature. Recall from Definition 2.6.1 that a conformal minimal immersion x : M → Rn from an open Riemann surface M is said to have finite total curvature if 

TC(x) = M

KdA = −

 M

|K|dA > −∞,

where K is the Gaussian curvature function and dA is the area element of the metric g on M induced by the Euclidean metric on Rn by the immersion x; see (2.86). In view of the formula (2.91), this means that the Gaussian image G : M → Qn−2 ⊂ CPn−1 of the immersion has finite Fubini–Study area when counting multiplicities. Until the mid 20th century, the only known complete minimal surfaces in R3 of finite topology were planes, catenoids, helicoids, and Enneper’s surface; see Section 2.8. The fact that all of them except the helicoid have finite total curvature suggests that this family of minimal surfaces lies at the very heart of the theory. It was R. Osserman who in the 1960s initiated a systematic study of minimal surfaces of finite total curvature, describing their geometry and showcasing the Enneper– Weierstrass representation as a powerful tool for constructing new examples and developing the theory. A vast family of examples of such surfaces is now available in the literature; see e.g. [57, 350, 231]. Contemporary research interests in this subject mainly lie in the study of moduli problems, classification results, Picard type theorems for the Gauss map, and the influence of topological embeddedness on their global geometry. Further information in this direction is indicated in the discussion of examples in Section 2.8. The following fundamental result by S. S. Chern and R. Osserman [87] from 1967 provides basic information about the complex structure and the Weierstrass data of complete minimal surfaces of finite total curvature. Theorem 4.1.1 (Chern–Osserman Theorem, [87]). Let M be an open Riemann surface and x : M → Rn (n ≥ 3) be a conformal minimal immersion. The following two conditions are equivalent. (i) x : M → Rn is complete and of finite total curvature. (ii) M is biholomorphic to the complement of finitely many points in a compact Riemann surface M and the 1-form ∂ x is meromorphic on M with effective poles at all punctures (that is, at points of M \ M). In particular, the Gauss map G : M → CPn−1 of x extends to an algebraic map G : M → CPn−1 . If these equivalent conditions hold then TC(x) = −2π deg(G ). The proof relies on the following result about flat metrics on the punctured disc, due to G. R. MacLane [236] and K. Voss [335]; see also R. Osserman [282, Lemma 9.6] and J. C. C. Nitsche [275, p. 134]. We give a somewhat different proof.

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4 Complete Minimal Surfaces of Finite Total Curvature

Lemma 4.1.2. If f is a nowhere vanishing holomorphic function on D∗ = D \ {0}  such that γ | f (z)||dz| = +∞ for any path γ : [0, 1) → D∗ with limt→1 γ(t) = 0, then f extends meromorphically to D with an effective pole at 0. Proof. Since the only divergent curves in the punctured disc D∗ = D \ {0} are those converging to 0, the condition in the lemma means that the flat metric | f |2 |dz|2 is complete on D∗ . It is elementary to find a pair (λ , μ) ∈ C2 \ {0} such that the function F = (λ + μ/z) f has vanishing residue at 0, and hence Fdz is exact on D∗ . Replacing D by a smaller disc centred at the origin, we may assume that F is holomorphic and vanishes nowhere on D∗ and the flat metric |F|2 |dz|2 is complete  on D∗ as well. Let H = Fdz : D∗ → C be a primitive of Fdz. Let L ⊂ C be an affine real line and α be a connected component of H −1 (L). Since H is a local biholomorphism, α is a proper arc in D∗ whose endpoints (if any) lie in the circle T = bD and H|α : α → L is injective. We claim that H(α) is a closed subset of L. This is obvious if α is compact. Otherwise, α approaches the origin 0 ∈ C at one or both of its ends, and since the metric |dH|2 = |F|2 |dz|2 is complete, H(α) is infinitely long at the corresponding ends. Since H is a local diffeomorphism and H(α) ⊂ L, it follows that H(α) is proper in C, hence closed. Let T q → nq be a unit normal vector field orienting the closed immersed curve H(T) ⊂ C. For each q ∈ T consider the affine line Lq = {H(q) + tnq : t ∈ R} ⊂ C and denote by αq the connected component of H −1 (Lq ) containing q. We claim that A :=

!

αq = D∗ .

(4.1)

q∈T

Since T ⊂ A ⊂ D∗ , it suffices to show that D∗ ⊂ A. Fix a point p ∈ D∗ and denote by L p the pencil of closed halflines in C with vertex H(p). For each Λ ∈ L p let βΛ denote the connected component of H −1 (Λ ) containing p and set Ap =

!

βΛ ⊂ D∗ .

Λ ∈L p

Since H is a local biholomorphism, βΛ is a regular arc in D∗ with one endpoint at p and H : βΛ → Λ is injective. If βΛ is not compact, it must approach the origin, and since the metric |dH|2 = |F|2 |dz|2 is complete, we infer that H(βΛ ) = Λ . Hence, if none of the curves βΛ is compact then A p is an open subset of D∗ and H|A p : A p → C is a biholomorphism onto C, which is impossible. This shows that there exists a Λ p ∈ L p such that βΛ p is a compact Jordan arc with the endpoints p and q p ∈ T. We choose Λ p with the additional property that |H(q p ) − H(p)| = length(H(βΛ p )) ≤ length(H(βΛ )) for all Λ ∈ L p . This implies that H(βΛ p ) and H(T) meet orthogonally at the point H(q p ) and therefore p ∈ βΛ p ⊂ αq p ⊂ A, thereby proving (4.1). We next claim that H −1 (a) is infinite for at most one point a ∈ C.

(4.2)

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195

Assuming this, Picard’s theorem shows that H is meromorphic at 0, and hence so are F and f . Since | f |2 |dz|2 is complete on D∗ , we infer that f has an effective pole at 0, which proves the lemma. So, it remains to see that (4.2) holds. Take a ∈ H(D∗ ). By (4.1), for each p ∈ H −1 (a) ⊂ D∗ there is a point q p ∈ T such that p ∈ αq p . Note that H −1 (a) ∩ αq p = {p} since H|αq : αq → Lq is injective. If H −1 (a) is infinite then so is {q p : p ∈ H −1 (a)}. Since {q p : p ∈ H −1 (a)} ⊂ {q ∈ T : (H(q) − a) · (inq ) = 0}, the identity principle shows that H(q) − a is parallel to nq for all q ∈ T. Thus, by compactness H(T) is a round circle with centre a. This establishes (4.2).   Proof of Theorem 4.1.1. That (ii) implies (i) follows from Corollary 2.6.6 and the observation that ∂ x having an effective pole at every point in M \ M implies that x is complete. Assume now that x is complete and of finite total curvature, i.e., condition (i) holds. Since its Gauss curvature is nowhere positive, Huber’s Theorem 2.6.4 shows that M = M \ {p1 , . . . , pr } where M is a compact Riemann surface. This gives the former assertion in (ii). For each j ∈ {1, . . . , r} let D j be an open disc in M with D j \ M = {p j }. Write ∂ x = ( f1 , . . . , fn )dz, where z is a holomorphic coordinate on D j with z(p j ) = 0 and f = ( f1 , . . . , fn ) is holomorphic on the punctured disc D∗j := D j \ {p j }. It suffices to see that f extends meromorphically to each D j with an effective pole at p j . Let us first show that the Gauss map G : M → CPn−1 is algebraic. A point v = [v1 : · · · : vn ] ∈ CPn−1 determines the hyperplane  Zv = [z1 : · · · : zn ] ∈ CPn−1 :

 ∼ v z = 0 = CPn−2 . ∑ jj n

j=1

The assumption TC(x) > −∞ implies that the set {v ∈ CPn−1 : # G −1 (Zv ) < ∞} has positive measure, where # denotes the cardinal number of a set. Otherwise, # G −1 (Zv ) = ∞ for almost every Zv and Santal´o’s formula [309] then shows that the area of G (M) is infinite, hence TC(x) = − 12 Area(G (M)) = −∞ by (2.91), a contradiction. Therefore, up to an orthogonal transformation of Rn and shrinking the disc D j around p j if necessary, we may assume that f1 has no zeros on D∗j and hence G |D∗j = [ f1 : f2 : · · · : fn ] = [1 : f2 / f1 : · · · : fn / f1 ]. If fk / f1 has an essential singularity at p j for some k, then for almost every v = [v1 : · · · : vn ] ∈ CPn−1 the function ∑ni=1 vi fi / f1 has an essential singularity at p j , and hence it assumes the value 0 infinitely often in any neighbourhood of p j by the big Picard Theorem, contradicting what has been said above. This shows that fk / f1 is meromorphic at p j for each k ∈ {2, . . . , n}. Since this holds on every disc D j , G is algebraic. Since fk / f1 is meromorphic at p j for each k ∈ {2, . . . , n}, it remains to see that f1 is meromorphic at p j . Since the Gauss map assumes values in Qn−2 , we have that ∑nk=2 ( fk / f1 )2 = −1, so at least one of the meromorphic functions fk / f1

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4 Complete Minimal Surfaces of Finite Total Curvature

(k = 2, . . . , n) does not vanish at p j . Let m ≥ 0 be the unique integer such that zm ( f2 / f1 , · · · , fn / f1 ) is holomorphic on D j and does not vanish at p j . Recall that ∂ x = ( f1 , . . . , fn )dz on D∗j . In light of the formula (2.55) we have on D∗j that

g = x∗ (ds2 ) = 2



n 2 2 −m 2 2m m 2 |dz| |z | f | = 2|z f | | + |z f / f | 1 ∑ k ∑ k 1 |dz|2 . n

k=1

k=2

Since the function in the parenthesis on the right-hand side is smooth and nowhere vanishing on D j and the metric g is complete on M, we infer that the flat metric |z−m f1 |2 |dz|2 on D∗j is complete at the puncture p j . It then follows from Lemma 4.1.2 that the nonvanishing holomorphic function z−m f1 (z) on D∗j has at most a pole at z = 0 (which corresponds to p j ), so f1 also has at most a pole at p j as claimed. The final assertion in the theorem follows from the formula (2.95).   The following important theorem of L. P. Jorge and W. H. Meeks [202] from 1983 describes the asymptotic behaviour of complete minimal surfaces of finite total curvature in Rn at their (point) ends. Recall that BnR denotes the open unit ball in Rn . By the order of pole of a vector valued meromorphic 1-form at a point p, we mean the maximum order of poles of its components at p. Theorem 4.1.3 (Jorge–Meeks Theorem, [202]). Let x = (x1 , . . . , xn ) : M → Rn (n ≥ 3) be a complete conformal minimal immersion of finite total curvature, where M = M \ {p1 , . . . , pr } for a compact Riemann surface M. Denote by Ik + 1 the order of the pole of ∂ x at the point pk for k = 1, . . . , r. (Recall that ∂ x is meromorphic on M and the Gauss map G : M → CPn−1 is algebraic by Theorem 4.1.1.) There exists a ρ > 0 (big) for which the following conditions hold.

(i) x−1 (Rn \ ρBnR ) = rk=1 (Dk \ {pk }), where D1 , . . . , Dr are pairwise disjoint closed discs in M with pk ∈ Dk \ bDk for all k = 1, . . . , r. (ii) Fix k ∈ {1, . . . , r} and write G (pk ) = [ak + ibk ] ∈ Qn−2 for a pair of unit orthogonal vectors ak , bk ∈ Rn . We say that Πk := {λ ak + μbk : λ , μ ∈ R} ⊂ Rn is the tangent plane to x at the end pk ∈ M \ M. If πk : Rn → Πk denotes the orthogonal projection, then (πk ◦ x)|Dk \{pk } : Dk \ {pk } → Πk \ ρBnR is an Ik -sheeted covering with sublinear asymptotic growth, in the sense that |x(p) − πk ◦ x(p)| = 0. p→pk |πk ◦ x(p)| lim

(4.3)

In particular, we have that Ik ≥ 1 (i.e., ∂ x has a pole of order ≥ 2 at each end pk ) and x : M → Rn is a proper map. (iii) The degree of the Gauss map G is given by the following Jorge–Meeks formula: r

r

k=1

k=1

deg(G ) = −χ(M) + ∑ Ik = −χ(M) + ∑ (Ik + 1) ≥ −χ(M) + 2r.

(4.4)

4.1 A Brief Survey of the Classical Theory

197

In (4.4), χ denotes the Euler number (1.9), (1.10). Since TC(x) = −2πdeg(G ) (see (2.95)), the Jorge–Meeks formula shows that the asymptotic behaviour of a complete minimal surface of finite total curvature is controlled by its topology and the total Gaussian curvature. This formula is related to the classical Cohn– Vossen inequality [90] and the Gauss–Bonnet theorem. In the case n = 3 we have deg(G ) = 2 deg(g) where g : M → CP1 is the complex Gauss map of x (see Remark 2.6.3), and hence the formula (4.4) reads −

r r TC(x) = 2 deg(g) = −χ(M) + ∑ Ik = −χ(M) + ∑ (Ik + 1). 2π k=1 k=1

(4.5)

The following corollary of Theorem 4.1.3 is of independent interest, and it will also be used in the proof of Theorem 4.7.2 in the last section of this chapter. Corollary 4.1.4. Let M be a compact Riemann surface, E be a finite subset of M, and M = M \ E. If x : M → Rn (n ≥ 3) is a complete conformal minimal immersion of finite total curvature, then the Gaussian curvature function K : M → (−∞, 0] of x satisfies lim p→E K(p) = 0. Proof. This is an elementary consequence of the formulas (2.87), (2.93) for the Gaussian curvature K and the fact that ∂ x has a pole of order at least two at each point p ∈ E (see the last sentence in Theorem 4.1.3 (ii)).   We now present a sketch of proof of Theorem 4.1.3, referring to [202] for further details and a more general result. Proof. Since x is complete, ∂ x has an effective pole at each pk , so Ik ≥ 0. We claim that Ik ≥ 1 for k = 1, . . . , r, so ∂ x has a pole of order at least two at each point pk . Since ∂ x has vanishing real periods on M, the residue Res pk (∂ x) (1.131) lies in Rn . If Ik = 0, the nullity condition (2.53) implies Res pk (∂ x) = 0, so ∂ x is holomorphic at pk . This contradiction proves the claim. Choose pairwise disjoint closed discs D1 , . . . , Dr in M such that pk ∈ Dk \ bDk and Dk \ M = {pk } for each k = 1, . . . , r. Fix k and choose a holomorphic coordinate z : Dk → C with z(pk ) = 0. Since the Gauss map G = [∂ x] : M → Qn−2 extends holomorphically to pk and the 1-form ∂ x has a pole of order Ik + 1 at pk , it is easily seen that in the holomorphic coordinate z we have 2∂ x = −

 ck Ik  ak + ibk + O(z) dz, zIk +1

z → 0,

(4.6)

where ak , bk ∈ Rn are unit orthogonal vectors such that G (pk ) = [ak + ibk ] ∈ Qn−2 and ck> 0. Up to rescaling the coordinate z, we may assume that ck = 1. Since x = ℜ 2∂ x, a calculation using (4.6) shows that     x(p) = ℜ z(p)−Ik ak − ℑ z(p)−Ik bk + O(|z(p)|1−Ik ) (4.7) as p → pk , and hence

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4 Complete Minimal Surfaces of Finite Total Curvature

lim

p→pk

x(p)· ak − i x(p)· bk =1 z(p)−Ik

and

lim sup p→pk

|x(p) − πk (x(p))| < +∞. |z(p)|1−Ik

For the last limit, use the definition of πk , (4.7), and note that     ℜ z(p)−Ik ak − ℑ z(p)−Ik bk ∈ Πk .

(4.8)

This gives (4.3). It also follows that lim p→pk |z(p)|Ik |πk ◦ x(p)| = 0, so the map πk ◦ x = (x · ak )ak + (x · bk )bk : Dk \ {pk } → Πk ⊂ Rn is proper. Furthermore, by (4.6) and (4.8), and up to shrinking Dk if necessary, the proper map πk ◦x : Dk \ {pk } → Πk is a local diffeomorphism. Thus, for any cylinder CRk = {q ∈ Rn : |πk (q)| < Rk }, Rk > 0, with x(bDk ) ⊂ Ck , the map   πk ◦ x : Dk \ x−1 (Ck ) ∪ {pk } → Πk \Ck is a topological covering. Relabelling Dk \ x−1 (Ck ) by Dk and taking into account (4.7), we infer that the map (πk ◦ x)|Dk \{pk } : Dk \ {pk } → Πk \CRk is an Ik -sheeted covering. Choosing ρ > max{Rk : k = 1, . . . , r} big enough, this implies (i) and (ii). To verify (iii), note that for almost every v = [v1 : · · · : vn ] ∈ CPn−1 the meromorphic 1-form φv = ∑nj=1 v j ∂ x j on M has a pole of order Ik + 1 ≥ 2 at pk for k = 1, . . . , r. The Riemann formula for canonical divisors (1.139) gives Zv − Pv = −χ(M),

(4.9)

where Zv and Pv are the number of zeros and poles of φv , respectively. Since deg(G ) equals the intersection number of G with a generic hyperplane in CPn−1 (see Remark 1.5.5), we have Zv = deg(G ). Hence, (4.9) and Ik ≥ 1 for all k imply r

r

k=1

k=1

deg(G ) = −χ(M) + Pv = −χ(M) + ∑ (Ik + 1) = −χ(M) + ∑ Ik , thereby proving the Jorge–Meeks formula (4.4).   Theorems 4.1.1 and 4.1.3 are the two basic pillars of the theory of complete minimal surfaces of finite total curvature in Euclidean spaces. They play a fundamental role in the construction and classification of examples, the analysis of moduli spaces, and the study of the Gauss map; for an overview of the subject we refer to the publications [282, 350, 231, 289], among many others. There has been a considerable emphasis on the study of embedded examples in R3 , some of which are discussed in Section 2.8. Another family in the focus of interest are complete minimal surfaces having the smallest possible absolute total curvature for each given topological type. In this direction, we mention the following consequence of Theorem 4.1.3. Recall that a conformal minimal immersion into Rn is said to be nonflat if its image does not lie in any affine plane; see Definition 2.5.2.

4.1 A Brief Survey of the Classical Theory

199

Corollary 4.1.5. If x : M = M \ {p1 , . . . , pr } → Rn (n ≥ 3) is a complete conformal nonflat minimal immersion of finite total curvature, then TC(x) ≤ −2π(2gen(M) + 1).

(4.10)

In case n = 3 we have the sharper bound TC(x) ≤ −4π(gen(M) + 1).

(4.11)

For each nonnegative integer k ∈ Z+ there are complete nonflat conformal minimal surfaces x : M → R3 with gen(M) = k achieving the critical total curvature TC(x) = −4π(k +1) (see K. Sato [310]), so the bound in (4.11) is sharp. Concerning higher dimensions, the Weierstrass data on z ∈ C4 given by   Φ = a(z + 1), ia(z − 1), i(z + a2 ), (z − a2 ) dz, a ∈ C \ {0, 1, −1} show that the bound in (4.10) is sharp for gen(M) = 0 and n = 4 (and hence for gen(M) = 0 and any n ≥ 4 by choosing the extra coordinate functions for n > 4 to be constant). It is not known whether this bound is sharp for gen(M) > 0 and n ≥ 4. Proof. We explain the case n = 3; the general one follows by an analogous argument. In view of the Jorge–Meeks formula (4.5) it suffices to show that ∑rk=1 (Ik + 1) ≥ 4 with the notation of that formula. Assume on the contrary that ∑rk=1 (Ik + 1) < 4. We see from (4.5) that the number ∑rk=1 (Ik + 1) is even and hence it equals 2. Since Ik ≥ 1 for each k, this forces that r = 1 and I1 = 1, so M = M \{p1 }. Up to a rigid motion we may assume that the limit unit normal of x at p1 is vertical. Writing x = (x1 , x2 , x3 ) and following the proof of Theorems 4.1.1 and 4.1.3, the 1form ∂ x3 is either holomorphic or has a pole of order one at p1 . Hence, we infer that x3 is either bounded or a graph around p1 with logarithmic asymptotic growth over the (x1 , x2 )-plane. In either case, x3 is bounded from above or bounded from below. Since M = M \ {p1 } is a parabolic Riemann surface, it follows that x3 is constant and hence x(M) is a plane. This contradiction proves the corollary.   Remark 4.1.6. In view of the formula (2.96) for TC(x), the inequality (4.11) is equivalent to deg(g) ≥ gen(M) + 1, where g : M → CP1 is the Gauss map of x. This inequality does not hold for an arbitrary holomorphic map. Indeed, if p is a Weierstrass point of a compact Riemann surface M of genus gen(M) > 1, there exists a meromorphic function with a pole of order at most gen(M) at p and no other poles (see [123, p. 88]). Furthermore, the Brill–Noether theory shows that any compact Riemann surface M carries nonconstant meromorphic functions of degree d ≤ [(gen(M) + 1)/2] + 1, where the bracket indicates the integer part; see [52, p. 159]. Hence, Corollary 4.1.5 says that Gauss maps of complete nonflat minimal  surfaces of finite total curvature in R3 are special.  It is a long-standing open problem whether the moduli space of complete nonflat minimal surfaces in R3 with the critical total curvature TC(x) = −4π(gen(M) + 1) (4.11) is discrete. This is true if M has genus 0 or 1. Indeed, by [282, Theorem 9.4]

200

4 Complete Minimal Surfaces of Finite Total Curvature

the catenoid and the Enneper surface are the only complete minimal surfaces in R3 of genus zero and total curvature −4π, whereas L´opez proved in [225] that the only such surface of genus one and total curvature −8π is the Chen–Gackstatter surface (see Example 2.8.7). The Gaussian image of complete minimal surfaces in Rn of finite total curvature was studied by Chern and Osserman in the 1960s. The following main result on this subject can be found in their paper [87] from 1967. Theorem 4.1.7. If x : M → Rn (n ≥ 3) is a full complete conformal minimal immersion of finite total curvature from an open Riemann surface M, then the image of the Gauss map G : M → CPn−1 of x can fail to intersect at most 21 (n − 1)(n + 2) hyperplanes in general position in CPn−1 . Recall that a conformal minimal immersion M → Rn is full if its Gauss map M → CPn−1 is full; see Definition 2.5.2. The notion of a set of hyperplanes of CPn−1 in general position is given by Definition 5.5.1. We shall not prove Theorem 4.1.7 here. Instead, we give the proof of the following analogous result for n = 3, due to Osserman (see [282, p. 89]). Theorem 4.1.8. Let x : M → R3 be a complete conformal minimal immersion of finite total curvature from an open Riemann surface M, and let g : M → CP1 denote its complex Gauss map. If either of the following two conditions hold then x(M) is a plane. (i) g(M) omits four points of CP1 . (ii) M is a planar domain (i.e. gen(M) = 0) and g(M) omits three points of CP1 . Proof. Since x is complete and of finite total curvature, Theorem 4.1.1 shows that M = M \ {p1 , . . . , pr }, r ≥ 1, where M is a compact Riemann surface, and the Weierstrass data (φ3 , g) of x (see (2.83)) extend meromorphically to M. Since x is nonflat, we may assume up to a rigid motion that g(p j ) ∈ / {0, ∞} for all j = 1, . . . , r, and g has only simple zeros and poles in M. Let b(g) denote the branching number (1.133) of the Gauss map g : M → CP1 . Since g has only simple poles, those are not branch points and hence b(g) is the number of zeros of dg counted with multiplicities. The Riemann–Hurwitz formula (1.134) gives b(g) = deg(g)χ(CP1 ) − χ(M) = 2(deg(g) + gen(M) − 1).

(4.12)

Assume that g(M) omits k distinct points of CP1 , say q1 , . . . , qk . Then we have that g−1 ({q1 , . . . , qk }) ⊂ {p1 , . . . , pr }. Let bi ≥ 0 denote the order of zero of dg at pi for i = 1, . . . , r. For each j = 1, . . . , k we clearly have deg(g) = ∑g(pi )=q j (1 + bi ), and summing over j = 1, . . . , k gives r

r

i=1

i=1

k deg(g) ≤ ∑ (1 + bi ) = r + ∑ bi ≤ r + b(g). Combining with (4.12) gives the following estimate on deg(g):

4.2 Spaces of Functions and Conformal Minimal Immersions

(k − 2)deg(g) ≤ 2gen(M) + r − 2.

201

(4.13)

We can also obtain a lower bound on deg(g) from the Jorge–Meeks formula (4.5): 2deg(g) = 2gen(M) − 2 + ∑rj=1 (I j + 1), where I j + 1 is the pole order of φ3 at p j for j = 1, . . . , r. Since I j ≥ 1 for all j by Theorem 4.1.3 (ii), it follows that deg(g) ≥ gen(M) + r − 1. Multiplying this inequality by −1 or −2 and adding to (4.13) gives the inequalities (k − 3)deg(g) ≤ gen(M) − 1,

r ≤ (4 − k)deg(g).

Since r ≥ 1 and deg(g) ≥ 1 by nonflatness of x, we obtain that k < 4 from the second inequality, and that k < 3 if gen(M) = 0 from the first one.   One of the most challenging open problems in the theory is to determine whether there is a complete nonflat minimal surface of finite total curvature in R3 whose Gauss map omits three points of the sphere (see Osserman [282, p. 90]). If such a surface exists then the value three is the maximum; otherwise, the largest possible number of omitted points is two and is reached by the catenoid.

4.2 Spaces of Functions and Conformal Minimal Immersions In this section we introduce some spaces of functions and conformal minimal immersions which are used in the following sections and complement the ones already introduced in Sections 1.1, 1.13, and 3.1. These new spaces are particularly useful when dealing with complete minimal surfaces of finite total curvature. Let M be a Riemann surface, either open or compact. Recall that for any subset S ⊂ M, O(S) = O(S, C) and M (S) denote, respectively, the spaces of holomorphic and meromorphic functions on unspecified open neighbourhoods of S in M. For a finite subset E of S˚ = S \ bS we denote by M (S | E) = M (S) ∩ O(S \ E)

(4.14)

the space of meromorphic functions on S having poles (if any) only at points in E. Likewise, we denote by Ω (S) and M 1 (S) the spaces of holomorphic and meromorphic 1-forms, respectively, on neighbourhoods of S in M, and we set M 1 (S | E) = M 1 (S) ∩ Ω (S \ E).

(4.15)

We denote by Div(S) the space of divisors on open neighbourhoods of S in M having support in S; cf. Definition 1.11.1. In general, Div(S) need not be a subset of Div(M). Indeed, if S  M is open then Div(S) is the set of all divisors on S as an open Riemann surface, so their supports may have limits points on bS and hence need not be divisors on M. On the other hand, if S is compact then every divisor in

202

4 Complete Minimal Surfaces of Finite Total Curvature

Div(S) has finite support, and we consider Div(S) in a natural way as the subset of Div(M) consisting of divisors on M having support in S. Given a function f ∈ M (S) different from the zero function, we define its divisor of zeros ( f )0 ∈ Div(S) and divisor of poles ( f )∞ ∈ Div(S) in the usual way (see Example 1.11.2), and the divisor of f is ( f ) = ( f )0 − ( f )∞ . Similarly we define the corresponding divisors of a nonzero meromorphic 1-form in M 1 (S); cf. Example 1.11.3. For each divisor D ∈ Div(S) we define OD (S) = { f ∈ O(S) : ( f ) ≥ D}.

(4.16)

We adopt the convention that the divisor of the zero function is greater than every ˚ we set divisor D ∈ Div(S), and hence 0 ∈ OD (S). If S is compact and D ∈ Div(S), AD (S) = { f ∈ A (S) : ( f |S˚ ) ≥ D}.

(4.17)

If f ∈ AD (S) then f −1 (0) ∩ S˚ may be infinite since f is merely continuous on bS. Let n ≥ 3 be an integer. We shall consider the complex hyperquadric  = (u1 . . . , un ) ∈ Cn \ {0} : u1 u2 = Sn−1 ∗

n

∑ u2j



.

(4.18)

j=3

A calculation shows that the linear automorphism Ξ ∈ Aut(Cn ) given by   Ξ (z1 , z2 , . . . , zn ) = − z1 + iz2 , z1 + iz2 , z3 , . . . , zn

(4.19)

maps the punctured null quadric A∗ onto S∗ = Sn−1 ∗ . The map Ξ is reminiscent of the spinorial representation of the null quadric in C3 in (3.73), and is particularly useful for algebraic approximation of maps into the null quadric in Cn . Given a compact Riemann surface Σ , a finite set E ⊂ Σ , and a Runge compact set S in Σ \ E, one may easily approximate uniformly on S any map u = (u1 , . . . , un ) ∈ M (S)n assuming its n finite values in S∗ by maps uˆ = (uˆ1 , . . . , uˆn ) ∈ M (Σ )∩O(Σ \(E ∪S)) with values in S∗ as well. Indeed, we first approximate u1 by some uˆ1 and then approximate the (n − 2)-tuple (u3 , . . . , un ) by a suitable meromorphic map (uˆ3 , . . . , uˆn ); doing this in the right way, the meromorphic function uˆ2 = uˆ11 ∑nj=3 uˆ2j on Σ then completes the task. We develop this approach in Section 4.4. Let S be a subset of a Riemann surface M. Given a map f = ( f1 , . . . , fn ) ∈ C (S, CP1 )n we write f −1 (∞) = nj=1 f j−1 (∞). Consider the space   M (S, S∗ ) = { f ∈ M (S)n : f S \ f −1 (∞) ⊂ S∗ } ˚ its subspace and, for a finite subset E ⊂ S, M (S | E, S∗ ) = { f ∈ M (S, S∗ ) : f −1 (∞) ⊂ E}.

(4.20)

The corresponding spaces M (S, A∗ ) and M (S | E, A∗ ) for the punctured null quadric A∗ (1.116) are defined in the same way.

4.3 A Runge Theorem for Maps to CP1

203

˚ we let If S is a compact subset of M and E is a finite subset of its interior S, CMI∞ (S | E, Rn )

(4.21)

denote the space of conformal minimal immersions x : S \ E → Rn (i.e., extending as a conformal minimal immersion to an unspecified neighbourhood of S \ E in M) which are complete and of finite total curvature, meaning that ∂ x has an effective pole at every point of E; cf. Definition 3.8.1. By Theorem 4.1.1, it turns out that x ◦ γ has infinite length for any path γ : [0, 1) → S \ E with limt→1 γ(t) ∈ E and W \E KdA > −∞ for any smoothly bounded compact neighbourhood W of S such that x is a conformal minimal immersion on W \ E.

4.3 A Runge Theorem for Maps to CP1 The Weierstrass data (g, φ3 ) of a conformal minimal immersion x : M → R3 (see (2.79) and (2.83)) have spinorial nature, in the sense that all zeros of the holomorphic 1-forms gφ3 and φ3 /g are of even order. It is therefore convenient to have a Runge type approximation theorem for functions with zeros of even order. In this direction, we provide here an extension of the Runge–Behnke–Stein–Royden theorem for maps to CP1 (see Corollary 1.12.2), ensuring additional control on the divisor of the approximating map. The following result is [37, Proposition 4.1]. Theorem 4.3.1. Let Σ be a compact Riemann surface, E be a nonempty finite subset of Σ , and S be a compact Runge admissible subset of the open Riemann surface Σ \ E (see Definition 1.12.9). Given a map f ∈ A (S, CP1 ) with f −1 (∞) ∩ bS = ˚ of finite support, a number δ > 0, and an ∅, an effective divisor D1 ∈ Div(S) integer k ≥ 1 there is a meromorphic function f˜ ∈ M (Σ ) which is holomorphic on Σ \ (E ∪ f −1 (∞)) and satisfies the following conditions. (i) f˜ − f ∈ AD1 (S) (see (4.17)) and  f˜ − f C (S) < δ . (ii) ( f˜|Σ \(S∪E) ) = 2D0 for some effective divisor D0 ∈ Div(Σ \ (S ∪ E)). (iii) ( f˜)∞ ≥ ∑ p∈E kp. These conditions imply that f˜ has the same poles and principal parts as f on ˚ it has a pole of order at least k at every point Σ \ E (so they are all contained in S), of E, and the zeros of f˜ on Σ \ (S ∪ E) are of even order. Proof. By the Weierstrass interpolation theorem (see Theorem 1.12.14) there exists a meromorphic function g on Σ \ E with the same poles and principal parts as f , so f = g + h with h ∈ A (S). By the Mergelyan theorem (see Theorem 1.12.11) we can approximate h uniformly on S by a function h˜ ∈ O(S) such that h˜ − h ∈ AD1 (S), where D1 is the effective divisor with finite support in S˚ given in the statement of the theorem. Replacing f by f˜ = g + h˜ we see that it suffices to prove the theorem for functions which are holomorphic on a neighbourhood of S except for the poles

204

4 Complete Minimal Surfaces of Finite Total Curvature

˚ Assume from now on that f is such. Runge’s theorem for maps to CP1 (see in S. Corollary 1.12.2) provides a function 0 ≡ f0 ∈ M (Σ ) which is holomorphic on Σ \ (S ∪ E) and satisfies the following two conditions. (a) f0 − f ∈ OD1 (S). In particular, ( f0 |S )∞ = ( f )∞ . (b)  f0 − f C (S) < δ . Clearly f0 satisfies condition (i) in the theorem, but it need not satisfy (ii) or (iii). The next step is to find a function h0 ∈ M (Σ | E) (see (4.14)) such that h0 f0 meets conditions (i) and (ii). If all zeros of f0 in Σ \ (S ∪ E) are of even order (or there are no zeros), we may choose h0 ≡ 1. Otherwise, write ( f0 |Σ \(S∪E) ) =

s

∑ m j p j ∈ Div(Σ \ (S ∪ E)),

j=1

where p1 , . . . , ps are pairwise distinct points in Σ \ (S ∪ E), m j ∈ N for all j ∈ {1, . . . , s}, and m1 is odd. Define E1 = {p j : m j is odd} ⊂ Σ \ (S ∪ E)

(4.22)

and note that p1 ∈ E1 . The Weierstrass–Florack interpolation theorem (see Theorem 1.12.14) gives a function g ∈ O(Σ \ E) with the divisor (g) =

∑

p.

(4.23)

p∈E1

Since g vanishes nowhere on S and C∗ is Oka, the Runge theorem for maps to Oka manifolds (see Theorem 1.13.3) allows us to assume in addition that 1 g − 1C (S) < . 2

(4.24)

Indeed, it suffices to replace g by g0 g where g0 ∈ O(Σ \ E, C∗ ) is sufficiently close to 1/g uniformly on the compact set S. Consider the open Riemann surface R of the multivalued holomorphic function √ g on Σ \ (E ∪ E1 ), which is represented by the smooth complex curve   R = (p, u) ∈ (Σ \ (E ∪ E1 )) × C : u2 = g(p) . √ Note that g is a well defined holomorphic function on R. Let π : R → Σ \ (E ∪ E1 ) denote the projection given by π(p, u) = p. The Riemann surface R admits an analytic compactification R which is a compact Riemann surface, the projection π extends to a 2-sheeted branched holomorphic covering π : R → Σ with R = R \ π −1 (E ∪ E1 ), and condition (4.23) implies that π −1 (E1 ) is the ramification locus of π|R\π  −1 (E) . This construction is classical, see e.g. [123] or [321].

(4.25)

4.3 A Runge Theorem for Maps to CP1

205

Let A : R → R be the nontrivial deck transformation of π, i.e., the holomorphic automorphism of R satisfying A(p, u) = (p, −u) for all (p, u) ∈ R. By (4.25), π −1 (E1 ) is the fixed point set of A in the A-invariant set R \ π −1 (E). Moreover, (4.24) shows that π −1 (S) = S+ ∪ S− where S+ and S− are pairwise disjoint Runge compact subsets of R, A(S+ ) = S− , and each of the projections π|S± : S± → S is a biholomorphism. ˚ be the effective divisor of finite support in the statement of Let D1 ∈ Div(S) − + − the theorem. Let D+ 1 ∈ Div(S ) and D1 ∈ Div(S ) be the divisors determined ± ± by the condition π(D1 ) = D1 , where π : Div(S ) → Div(S) is the natural group isomorphism induced by π. Given a number β > 0, the Runge theorem for maps to CP1 (see Corollary 1.12.2) yields a function h ∈ M (R | π −1 (E)) satisfying the following conditions. (c) h − 1C (S+ ) < β and h + 1C (S− ) < β . (d) h has simple zeros at all points in π −1 (E1 ). − (e) (h|S+ − 1) ≥ D+ 1 + (( f 0 ◦ π)|S+ )∞ and (h|S− + 1) ≥ D1 + (( f 0 ◦ π)|S− )∞ . Up to replacing h by (h − h ◦ A)/2 if necessary, we can also assume that h ◦ A = −h, and hence there is a function h0 ∈ M (Σ | E) such that h2 = h0 ◦ π.

(4.26)

The following conditions are clearly satisfied. (f) h0 − 1C (S) < β 2 + 2β ; see (c). (g) (h0 |Σ \E )0 = 2D + ∑ p∈E1 p for some effective divisor D ∈ Div(Σ \ (S ∪ E)); see (d), (4.25), (4.26) and note that each point of π −1 (E1 ) is a ramification point of order 2, so h0 has a simple zero at every point of E1 . (h) (h0 − 1)0 ≥ D1 + ( f0 |S )∞ ; see (c) and (e) and assume that β > 0 is chosen sufficiently small. Define f1 = h0 f0 . Since f1 − f = (h0 − 1) f0 + ( f0 − f ), conditions (a) and (h) guarantee that f1 − f ∈ OD1 (S). Moreover, (b) and (f) ensure that  f1 − f C (S) < δ

(4.27)

provided that the number β > 0 in (c) is chosen sufficiently small, and hence f1 satisfies condition (i) in the statement of the theorem. By the definition of E1 in (4.22) we have that ( f0 |Σ \(S∪E) ) = 2D + ∑ p∈E1 p for some effective divisor   D ∈ Div Σ \ (S ∪ E) of finite support, and hence the property (g) yields that ( f1 |Σ \(S∪E) ) = 2D + 2D + ∑ p∈E1 2p. This shows that the function f1 = h0 f0 satisfies condition (ii) with D0 = D + D + ∑ p∈E1 p. To complete the proof, it suffices to find a function h1 ∈ M (Σ | E) such that the function f˜ = h21 f1 satisfies the theorem. By Corollary 1.12.2 there exists for any p ∈ E and ε > 0 a function 0 ≡ Fp ∈ M (Σ | {p}) such that (Fp )0 ≥ D1 + ( f1 |S )∞

and

Fp C (S) < ε.

(4.28)

206

4 Complete Minimal Surfaces of Finite Total Curvature

Note that Fp has an effective pole at p. Choose an integer m > k + k0 , where k is the number in the statement of the theorem and k0 ≥ 0 is the maximum order of vanishing of f1 at the points of E. In view of (4.28), the function h1 = 1 + ∑ p∈E Fpm satisfies the inequality h21 − 1C (S) < l 2 ε 2 + 2lε, where l is the cardinal of E, and the condition (h21 − 1)0 ≥ D1 + ( f1 |S )∞ . Reasoning as above, we infer that the function f˜ = h21 f1 satisfies conditions (i) and (ii) if the number ε > 0 is chosen sufficiently small. Finally, since Fp has an effective pole at each p ∈ E, h1 has a pole of order at least m at each point p ∈ E, and hence f˜ has a pole of order at least  2m − k0 > 2k + k0 ≥ k at every such p. This shows (iii). 

4.4 Period Dominating Multiplicative Sprays In this section, we construct period dominating sprays that will be used in the proof of the Runge theorem for complete minimal surfaces of finite total curvature (see Theorem 4.5.1 in the following section). In contrast to Section 3.2, where we did not take care of the total curvature, we now deal with meromorphic maps ⊂ Cn (see (4.18)) instead assuming values in the punctured hyperquadric Sn−1 ∗ n−1 of the null quadric A∗ . This enables us to express the computations in a more friendly way, although the proofs can easily be reproduced by using maps into An−1 ∗ . Recall that these two hyperquadrics are biholomorphic under a linear automorphism Ξ ∈ Aut(Cn ) (see (4.19)), and hence we shall also obtain conformal minimal immersions in this way. Such sprays will enable us to control the periods of the algebraic Weierstrass data in the proof of Theorem 4.5.1. Let M be an open Riemann surface, K ⊂ M be a compact connected smoothly ˚ Let n ≥ 3. Fix a bounded domain, and let E0 and Λ be disjoint finite subsets of K. n−1 full map f ∈ M (K | E0 , A∗ ) (see Definition 2.5.2) and define u = (u1 , . . . , un ) := Ξ ◦ f ∈ M (K | E0 , Sn−1 ∗ );

(4.29)

see Section 4.2 and in particular (4.20) for the notation. Fix an integer r ≥ 0 and consider the following effective divisors of finite support contained in E0 ∪ Λ : n

D=

∑

p∈E0 ∪Λ

rp,

D0 = D + ∑ (ui )∞ .

(4.30)

i=1

Recall that the space OD (K) was defined by (4.16). In this framework, we define the following space of maps:     : v j − u j ∈ OD (K), j = 1, . . . , n , OD (K, u) = v ∈ M K | E0 , Sn−1 (4.31) ∗ where v = (v1 , . . . , vn ). Obviously, u ∈ OD (K, u) and hence this space is nonempty. Actually, the space OD (K, u) is really large, as shown in Remark 4.4.1.

4.4 Period Dominating Multiplicative Sprays

207

Recall that D is the unit disc in C. We denote by S2 (D) the space of all holomorphic maps (s1 , s2 , s3 ) ∈ O(D, C∗ )3 satisfying the following conditions. • s j (0) = 1 for j = 1, 2, 3. • s1 s2 = s23 on D. (This means that (s1 , s2 , s3 ) ∈ O(D, S2∗ ), see (4.18).) • s1 (0)s2 (0)s3 (0) = 0. The first condition in the definition of S2 (D) is motivated by the following remark. The utility of the third condition will become clear later. Remark 4.4.1. If g ∈ OD0 (K) is a function with values in D, (s1 , s2 , s3 ) ∈ S2 (D), and v = (v1 , . . . , vn ) ∈ OD (K, u), then the map     (s1 ◦ g)v1 , (s2 ◦ g)v2 , (s3 ◦ g v3 , . . . , (s3 ◦ g vn belongs to OD (K, u). Note that the function appearing in the last n − 2 components  is always s3 ◦ g.  Fix a point p0 ∈ K˚ \ (E0 ∪ Λ ). Set l = l1 + l2 , where l1 = dim H1 (K, Z) and l2 = # (E0 ∪ Λ ). Let {C1 , . . . ,Cl } be a family of smooth closed oriented Jordan curves {C1 , . . . ,Cl1 } and arcs {Cl1 +1 , . . . ,Cl } in K˚ such that Ci ∩ C j = {p0 } for all i = j, {C1 , . . . ,Cl1 } form a basis of H1 (K, Z), C j connects p0 with a point of E0 ∪ Λ for each j ∈ {l1 + 1, . . . , l}, and C = li=1 C j is a deformation retract of K. (See Lemma 1.12.10 and the proof of Proposition 3.3.2 for the details.) Consider the space   n C (C, u) = h ∈ C (C \ E0 , Sn−1 ∗ ) : h − u ∈ C (C, C ) . We have that OD (K, u) ⊂ C (C, u). Fix a nowhere vanishing holomorphic 1-form θ on M and consider the period map P : C (C, u) → (Cn )l given by 

(h − u)θ . (4.32) C (C, u) h = (h1 , . . . , hn ) −→ P(h) = Cj

j=1,...,l

  The following lemma shows that we can embed the map u ∈ M K | E0 ,Sn−1 ∗ . This in (4.29) into a special period dominating spray of maps in M K | E0 , Sn−1 ∗ result follows the spirit of Section 3.2; see in particular Lemma 3.2.1. Lemma 4.4.2. Let M, K, E0 , Λ , u, θ , C, D, D0 , and l be as above;  the map  recall that u given by (4.29) is full since f is full. There is a map h = (hi, j )i=1,...,n j=1,...,l ∈ l  OD0 (K)n , h ≡ 0, satisfying the following conditions. (i) For any number ρ0 with 0 < ρ0 < 1/hC (K) and map s = (s1 , s2 , s3 ) ∈ S2 (D),   nl given by the map Ψs : ρ0 B → M K | E0 , Sn−1 ∗   Ψs (ζ ) = s1 (ζ · h)u1 , s2 (ζ · h)u2 , s3 (ζ · h)u3 , . . . , s3 (ζ · h)un (4.33)   nl (here, ζ = (ζi, j )i=1,...,n j=1,...,l ∈ ρ0 B and ζ · h = ∑ni=1 ∑lj=1 ζi, j hi, j ) assumes values in OD (K, u) (4.31) and is period dominating, meaning that

208

4 Complete Minimal Surfaces of Finite Total Curvature

∂  nl  (P ◦Ψs (ζ )) : T0 (ρ0 B ) = (Cn )l → (Cn )l is an isomorphism. ∂ ζ ζ =0 Therefore, for any ε > 0 and s ∈ S2 (D) there is a number ρ with 0 < ρ < 1/hC (K) so small that the following conditions hold. nl

(ii) Ψs (ζ ) : K \ E0 → Sn−1 ⊂ Cn is full and Ψs (ζ ) − uC (K) < ε for all ζ ∈ ρB . ∗ nl

nl

(iii) P ◦Ψs : ρB → (P ◦Ψs )(ρB ) is a biholomorphism with P(Ψs (0)) = 0. Note that u = Ψs (0) is the core map of the spray Ψs . Proof. Consider a nonconstant continuous map   F = ( fi, j )i=1,...,n j=1,...,l : C → (Cn )l

(4.34)

satisfying the following conditions. • The support supp( fi, j ) of fi, j is connected and lies in C j \ ({p0 } ∪ E0 ∪ Λ ) ⊂ K˚ for all i ∈ {1, . . . , n} and j ∈ {1, . . . , l}. • For each j ∈ {1, . . . , l} the sets supp( fi, j ), i ∈ {1, . . . , n}, are pairwise disjoint. The precise choice of F will be specified later. Choose a number ρ0 with 0 < ρ0 < 1/FC (C) and a map s = (s1 , s2 , s3 ) ∈ nl

S2 (D). Consider the map ΦF : ρ0 B → C (C, u) defined by   nl ρ0 B ζ −→ ΦF (ζ ) = s1 (ζ · F)u1 , s2 (ζ · F)u2 , s3 (ζ · F)u3 , . . . , s3 (ζ · F)un . Since supp( fi, j ) ⊂ K˚ \ E0 , we have that ΦF (ζ ) = u on a neighbourhood of E0 in C, nl

and hence the map ΦF assumes values in C (C, u). Write ζ = (ζ1 , . . . , ζl ) ∈ ρ0 B , where ζ j = (ζi, j )i=1,...,n for each j = 1, . . . , l. Let P = (P1 , . . . , Pl ) : C (C, u) → (Cn )l be the period map (4.32). For each pair of indices k, j ∈ {1, . . . , l} we have that  ∂ (Pk ◦ ΦF (ζ ))  Tk, j (F) := = δk, j ·W j (F) · A (4.35)  ∂ζj ζ =0 where δk, j is the Kronecker symbol,  W j (F) =

Cj

fa, j ub θ

 ,

(4.36)

a,b=1...,n

and A is the diagonal matrix of order n whose diagonal entries starting in the upper left corner are s1 (0), s2 (0), s3 (0), . . . , s3 (0). Hence, the Jacobian matrix of P ◦ ΦF has the following block diagonal structure

4.4 Period Dominating Multiplicative Sprays

209

⎛ T1,1 (F)  ∂ (P ◦ ΦF )  ⎜ .. =⎝ .  ∂ζ ζ =0 0

⎞ ··· 0 .. ⎟ .. . . ⎠ · · · Tl,l (F)

with off-diagonal blocks equal to zero. Since det T j, j (F) = detW j (F) det A and det A = s1 (0)s2 (0)(s3 (0))n−2 = 0, it follows that ∂ (P ◦ Φ )  F  = nl ⇐⇒ det(W j (F)) = 0 for all j = 1, . . . , l.  ∂ζ ζ =0

rank

(4.37)

Choose a continuous map F as in (4.34) for which the second condition in (4.37) holds; such F can be found as in the proof of Lemma 3.2.1, using the assumption that u is full. Then the first condition in (4.37) holds as well, and hence for any sufficiently small number 0 < ρ1 < 1/FC (C) the composition P ◦ ΦF |ρ

1B

nl

nl

nl

: ρ1 B −→ (P ◦ ΦF )(ρ1 B )

is a biholomorphism with P(ΦF (0)) = P(u) = 0. Choose a positive integer r0 with r0 ≥ r +

n

∑ ∑ (ui )∞ (p)

(4.38)

p∈E0 i=1

where (ui )∞ is the polar divisor of ui (so (ui )∞ (p) means the pole order of ui at p) and r is the integer in the divisor D (see (4.30)). Since C is a deformation retract of K and each fi, j vanishes on a neighbourhood of Λ ∪ E0 , the Runge–Mergelyan theorem with jet-interpolation (see Theorem 1.12.7 and Remark 1.12.12) enables us to approximate fi, j uniformly on C by a nonconstant function hi, j ∈ O(K) that vanishes to order r0 at every point of Λ ∪ E0 . Hence,  l   h := (hi, j )i=1,...,n j=1,...,l ∈ OD0 (K)n where the divisor D0 is given by (4.30) and OD0 (K) is defined by (4.16). Defining Ψs as in the statement of the lemma for this map h and any positive number ρ0 < 1/hC (S) , we obtain in view of (4.38) and Remark 4.4.1 that Ψs (ζ ) ∈ nl

OD (K, u) for all ζ ∈ ρ0 B . By (4.36), (4.37), and the fact that u = Ψs (0) is full, we see that conditions (i) and (ii) in the lemma hold true provided that the map h is close enough to F on C and the number ρ < min{ρ1 , 1/hC (K) } is sufficiently small. Likewise, the fact that P ◦ ΦF |ρ Bnl is a biholomorphism onto its image with 1

 P(ΦF (0)) = 0 ensures condition (iii) whenever ρ > 0 is taken small enough. 

210

4 Complete Minimal Surfaces of Finite Total Curvature

4.5 Approximation and Interpolation In this section we prove the following main result of this chapter — a Runge approximation theorem with jet interpolation for complete minimal surfaces of finite total curvature in Rn for any n ≥ 3. This is [37, Theorem 6.1] by Alarc´on and L´opez; earlier partial results in this direction in dimension n = 3 were obtained in [230, 13]. Theorem 4.5.1 (Runge theorem for complete minimal surfaces of finite total curvature). Assume that Σ is a compact Riemann surface without boundary, E ⊂ Σ is a nonempty finite set, and K ⊂ Σ \ E is a compact subset which is Runge in Σ \ E. Let E0 and Λ be a pair of disjoint finite sets in K˚ and x ∈ CMI∞ (K | E0 , Rn ) (n ≥ 3) be a complete conformal minimal immersion K \ E0 → Rn of finite total curvature; see (4.21) and Definition 3.8.1. Given a number ε > 0, an integer k ∈ N, and a group homomorphism p : H1 (Σ \ (E ∪ E0 ), Z) → Rn with p|H1 (K\E0 ,Z) = Fluxx there is a complete conformal minimal immersion xˆ : Σ \ (E ∪ E0 ) → Rn satisfying the following conditions. (i) xˆ is of finite total curvature. (ii) The difference map xˆ − x : K \ E0 → Rn extends harmonically to K and satisfies xˆ − xC (K) < ε. (iii) The difference xˆ − x vanishes to order at least k at each point of E0 ∪ Λ . (iv) Fluxxˆ = p on H1 (Σ \ (E ∪ E0 ), Z). Since the map x : K \ E0 → Rn in Theorem 4.5.1 is a complete conformal minimal immersion of finite total curvature, it is a proper map which means that lim p→E0 |x(p)| = ∞ (see Theorem 4.1.3). Likewise, the map xˆ : Σ \ (E ∪ E0 ) → Rn is proper as well; equivalently, lim p→E∪E0 |x(p)| ˆ = ∞. Note also that condition (iii) in ˆ − x(p)| = 0, so xˆ and x have the same asymptotic the theorem implies lim p→E0 |x(p) behaviour at their common ends (the points in E0 ). Recall that, by the Chern–Osserman theorem (see Theorem 4.1.1), every complete minimal surface of finite total curvature in Rn is conformally equivalent to a compact Riemann surface with finitely many punctures and its Weierstrass data extend meromorphically to the compact surface. Thus, minimal surfaces of finite total curvature form the natural extension of the class of affine algebraic curves in complex Euclidean spaces Cn . In this sense, Theorem 4.5.1 is an analogue for conformal minimal surfaces of the Runge theorem for holomorphic maps to CP1 (see Corollary 1.12.2), and hence also for affine algebraic curves in Cn . Before proceeding with the proof of Theorem 4.5.1, we mention a few corollaries and an open problem. A first immediate consequence is that every finitely punctured compact Riemann surface is the conformal structure of a full complete minimal surface of finite total curvature in Rn for any n ≥ 3 (see Definition 2.5.2). This was previously observed in the case n = 3 by G. P. Pirola in [290]. His examples have vanishing flux, i.e., they are real parts of algebraic null curves in C3 . Theorem 4.5.1 also enables one to prescribe the flux of the examples. The following is another immediate corollary to Theorem 4.5.1.

4.5 Approximation and Interpolation

211

Corollary 4.5.2. If E and Λ are nonempty disjoint finite subsets of a compact Riemann surface Σ , then every map Λ → Rn (n ≥ 3) extends to a complete conformal minimal immersion Σ \ E → Rn of finite total curvature. Comparing with the Mittag-Leffler theorem with approximation and interpolation for complete conformal minimal immersions of finite total curvature in Rn (see Theorems 3.8.2 and 3.9.1), the novelty of Theorem 4.5.1 is condition (i). In fact, Theorem 3.8.2 applies to punctures in an open Riemann surface. Applying Theorems 3.8.2 and 3.9.1 to the open Riemann surface Σ \ {p}, where p is any point of the finite set E in Theorem 4.5.1, we obtain a conformal minimal immersion xˆ : Σ \ E → Rn satisfying the conclusion of Theorem 4.5.1 except for condition (i); it has finite total curvature at all punctures except perhaps at p. On the other hand, Theorem 4.5.1 ensures finite total curvature at all ends. Let M be a finitely punctured compact Riemann surface. We have that H1 (M, Z) ∼ = Zl for some l ∈ Z+ . By Corollary 3.12.4, the space CMInf (M, R3 ) l has 2 path connected components, and we can realize each path component by a conformal minimal immersion on a compact domain K ⊂ M which is a deformation retract of M. Thus, Theorem 4.5.1 shows that each path component of CMInf (M, R3 ) is realized by a complete conformal minimal immersion of finite total curvature as well, thereby proving the following corollary (cf. Theorem 3.12.1 and recall that CMInf (M, Rn ) is path connected for n ≥ 4 by Corollary 3.12.4). Corollary 4.5.3. Let M be a finitely punctured compact Riemann surface. For every conformal minimal immersion x : M → Rn (n ≥ 3) there exists a smooth isotopy xt : M → Rn (t ∈ [0, 1]) of conformal minimal immersions such that x0 = x and x1 is complete and of finite total curvature. The methods in the proof of Theorem 4.5.1 easily adapt to give the following analogous approximation result for holomorphic null curves. Theorem 4.5.4. Let Σ , E, K, E0 , and Λ be as in Theorem 4.5.1. Every holomorphic null curve z : K \ E0 → Cn (n ≥ 3) extending meromorphically to K with an effective pole at each point of E0 can be approximated uniformly on K \ E0 by complete holomorphic null curves z˜ : Σ \ (E ∪ E0 ) → Cn extending meromorphically to Σ with an effective pole at each point of E ∪ E0 . Furthermore, z˜ can be chosen such that z˜ − z is holomorphic on K and it vanishes to any given finite order at every point in any given finite subset of K. The following is an immediate corollary to Theorems 4.5.1 and 4.5.4. It replaces Runge approximation by Mergelyan approximation on admissible sets, which is possible by the second part of Theorem 3.8.2. Corollary 4.5.5. Theorem 4.5.1 holds true for generalized conformal minimal immersions of finite total curvature on admissible Runge compact subsets of Σ \ E. Likewise, the corresponding version of Theorem 4.5.4 for generalized null curves on admissible sets holds true as well.

212

4 Complete Minimal Surfaces of Finite Total Curvature

Unlike Theorems 3.6.1 and 3.9.1 which pertain to conformal minimal surfaces without any restriction on their total curvature, Theorem 4.5.1 does not provide embeddings, not even when n ≥ 5. The following remains an open problem. Question 4.5.6. Does every finitely punctured compact Riemann surface admit a complete conformal minimal embedding of finite total curvature into R5 ? Since complex curves in Cn are also conformal minimal surfaces in R2n and affine algebraic curves are precisely the closed complex curves of finite total curvature, it is natural to consider the analogue of the above question in this class. It is well known that every finitely punctured compact Riemann surface, M, can be represented as a properly embedded affine complex curve in C3 , and as an immersed one in C2 . The fact that such a surface is biholomorphic to a smooth affine curve in some Cn goes back to B. Riemann; the stated conclusions then follow by applying well-chosen linear projections to lower-dimensional spaces and the transversality theorem. More generally, a smooth closed affine algebraic variety A ⊂ Cn of dimension m admits a closed algebraic embedding into C2m+1 (see A. Holme [193, Theorem 7.4]). However, a linear projection of a conformal minimal surface to a subspace need not be a minimal surface. Proof of Theorem 4.5.1. By Theorem 3.8.2, we may assume that K is a smoothly bounded compact domain which is a deformation retract of Σ \ E and that x is full. We also assume without loss of generality that the set Λ is nonempty and write ˚ Λ ∪ E0 = {p1 , . . . , pm } ⊂ K.

(4.39)

Fix a point p0 ∈ K˚ \ (Λ ∪ E0 ) and take a family of Jordan curves and arcs Recall that l = l1 + l2 {C1 , . . . ,Cl } based at p0 as in the proof of Lemma 4.4.2. ˚ Fix a 1-form where l1 = dim H1 (K, Z) and l2 = # (Λ ∪ E0 ). Set C = lj=1 C j ⊂ K. 1 ˆ θ ∈ M (Σ | E) that vanishes nowhere on K and whose polar divisor satisfies (θˆ )∞ ≥

∑ p.

(4.40)

p∈E

The existence of such θˆ follows from the Weierstrass interpolation theorem (see Theorem 1.12.13) and Theorem 4.3.1. Recall the notation (4.20) and set f = 2∂ x/θˆ ∈ M (K | E0 , A∗ ),

u = Ξ ◦ f ∈ M (K | E0 , S∗ ),

(4.41)

where A∗ is the punctured null quadric (2.54) and Ξ : A∗ → S∗ ⊂ Cn is the biholomorphism in (4.19). The proof amounts to approximating u = (u1 , . . . , un ) given by (4.41), uniformly on K \ E0 , by a meromorphic map uˆ ∈ M (Σ | E ∪ E0 ∪ supp((θˆ )0 ), S∗ ) with an effective pole at each point of E ∪ E0 such that uˆ − u is holomorphic on K, (uˆ − u)θˆ is exact on K, and uˆθˆ is holomorphic and vanishes nowhere on Σ \ (E ∪ E0 ). With this in hand, the map fˆ = Ξ −1 ◦ uˆ ∈ M (Σ )n assumes its finite values in A∗ , fˆθˆ ∈ M 1 (Σ | E ∪ E0 )n , fˆθˆ − f θˆ is exact on K, fˆθˆ is holomorphic and vanishes

4.5 Approximation and Interpolation

213

nowhere on Σ \ (E ∪ E0 ) and has an effective pole at each point of E ∪ E0 , and fˆ is uniformly close to f on K. Since K is a deformation retract of Σ \ E, it follows that the 1-form fˆθˆ has exact real part on Σ \ E. Doing this in the right way to guarantee also the interpolation condition, fˆθˆ gives via the Weierstrass formula (2.61) a complete conformal minimal immersion Σ \ (E ∪ E0 ) → Rn of finite total curvature which satisfies the conclusion of the theorem. To find a map uˆ having the required properties, we first approximate u1 by a meromorphic function uˆ1 and then approximate the (n − 2)-tuple (u3 , . . . , un ) by a suitable meromorphic map (uˆ3 , . . . , uˆn ); doing this carefully, the function uˆ2 defined on Σ \ E by uˆ2 = uˆ11 ∑nj=3 uˆ2j (see (4.53)) completes the task, except for the control of the periods that are corrected using the multiplicative sprays provided by Lemma 4.4.2. The main concern when carrying out this procedure is to have sufficient control on the divisors of all approximating functions at every step of the construction in order to avoid the appearance of branch points and to guarantee the completeness of the resulting immersion. Let us now explain the details. Consider the divisors n

D=

∑

D0 = D + ∑ (ui )∞ ,

kp,

p∈Λ ∪E0

(4.42)

i=1

where k is the integer in the statement of the theorem, and set Θ = supp((θˆ )0 ) ⊂ Σ \ (K ∪ E).

(4.43)

Recall that ui ∈ M (K | E0 ) in view of (4.41). Up to slightly enlarging K if necessary, we may assume that ui vanishes nowhere on bK = K \ K˚ for all i ∈ {1, . . . , n}. Consider the following effective divisor of finite support: Z=2

n

n

∑ (ui )0 + ∑ (ui )∞

i=1

˚ ∈ Div(K).

(4.44)

i=1

Step 1. Fix a number ε0 > 0 to be specified later and let s : C → S2∗ ⊂ C3 be the map   s := (1 + z)2 , (1 + z/2)2 , (1 + z)(1 + z/2) . Note that s|D ∈ S2 (D) (see the definition of S2 (D) on page 207). Apply Lemma 4.4.2 to the map u ∈ M (K | E0 , Sn−1 ∗ ) in (4.41), the map s|D , the divisors D and D0 in (4.42) (compare with (4.30)), the 1-form θˆ in (4.40), and the number ε0 , and let  l   h = (hi, j )i=1,...,n j=1,...,l ∈ OD0 (K)n

(4.45)

and ρ > 0 denote the map and the number obtained in this way. Also, define the   nl by spray Ψs : ρB → M K | E0 , Sn−1 ∗   Ψs (ζ ) = (1 + ζ · h)2 u1 , (1 + ζ · h/2)2 u2 , (1 + ζ · h)(1 + ζ · h/2)(ui )i=3,...,n

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4 Complete Minimal Surfaces of Finite Total Curvature

(cf. (4.33)). In particular, we have that nl

nl

P ◦Ψs : ρB → (P ◦Ψs )(ρB ) is a biholomorphism

(4.46)

with P(Ψs (0)) = 0 and Ψs (ζ ) − uC (K) < ε0

nl

for all ζ ∈ ρB .

(4.47)

Step 2. We first deal with the first component u1 of u. Fix a number ε1 > 0 to be specified later; it will depend on the fixed (but still to be specified) number ε0 > 0. For each point q in the finite set Θ (4.43) we choose a closed disc Lq ⊂ Σ \ (K ∪ E) with q ∈ L˚ q such that these discs are pairwise disjoint, and set L = q∈Θ Lq . Fix a   function u∗1 ∈ M K ∪ L | E0 ∪Θ such that u∗1 |K = u1

and (u∗1 |L ) = −(θˆ )0 .

(4.48)

Note that the compact set K ∪ L is Runge in Σ \ E. By Theorem 4.3.1 and Hurwitz’s theorem on zeros of limits of holomorphic functions (see [199] or [98, §VII.2.5, p. 148]) there is a uˆ1 ∈ M (Σ | E ∪ E0 ∪Θ ) satisfying the following conditions. (a1) uˆ1 − u∗1 ∈ OZ+D+2(θˆ )0 (K ∪ L) and uˆ1 − u∗1 C (K∪L) < ε1 . (a2) (uˆ1 |Σ \E ) = 2D1 + (u∗1 ) for some effective divisor D1 ∈ Div(Σ \ (K ∪ L ∪ E)); in particular, we have (uˆ1 |K∪L ) = (u∗1 ). Therefore, (4.48) and (a1) guarantee that uˆ1 θˆ ∈ M 1 (Σ | E0 ∪ E) and uˆ1 θˆ has no zeros on L. (a3) (uˆ1 θˆ )∞ ≥ ∑ p∈E p.   By (a1) and (a2) we have that uuˆ11 − 1 u2 ∈ O(K), and we may further assume that * u

* * * 1 − 1 u2 * < ε1 . * uˆ1 C (K)

(4.49)

For each q ∈ supp(D1 ) (see (a2)) we choose a closed disc Tq ⊂ Σ \ (K ∪ L ∪ E) with q ∈ T˚q such that the discs Tq are pairwise disjoint. Set T = q∈supp(D1 ) Tq . We have 2D1 = (uˆ1 |T ) = (uˆ1 |Σ \(K∪E) )0 .

(4.50)

Step 3. Next, we deal with the last n − 2 components of u. Pick functions v3 , . . . , vn in M (K ∪ T | E0 ) such that vi |K = ui for i = 3, . . . , n,

and

n

∑ v2i

i=3

  = uˆ1 |T . T

(4.51)

(Such functions exists by (4.50). This justifies the need of condition (ii) in Theorem 4.3.1.) Denote by v1 , v2 ∈ M (K ∪ T | E0 ) the functions defined by vi |K = ui for i = 1, 2,

v1 |T = uˆ1 |T ,

v2 |T ≡ 1.

(4.52)

4.5 Approximation and Interpolation

215

Fix another number ε2 , 0 < ε2 < ε1 , to be specified later. Applying Theorem 4.3.1 and Hurwitz’s theorem again, we obtain a map (uˆ3 , . . . , uˆn ) ∈ M (Σ | E ∪ E0 )n−2 satisfying the following conditions for i = 3, . . . , n. (b1) uˆi − vi ∈ OZ+D+4D1 (K ∪ T ) and uˆi − vi C (K∪T ) < ε2 . (b2) (uˆi |K ) = (ui ). Finally, we define the function uˆ2 by uˆ2 :=

1 n 2 ∑ uˆi . uˆ1 i=3

(4.53)

If ε2 > 0 is chosen small enough (in terms of the number ε1 > 0, which still needs to be specified) then the following conditions hold. (c1) uˆ2 ∈ M (Σ | E0 ∪ E). (c2) uˆ2 − v2 ∈ OD+(u2 )0 (K ∪ T ) and uˆ2 − v2 C (K∪T ) < ε1 . (c3) (uˆ2 |K∪T ) = (uˆ2 |K ) = (v2 ) = (u2 ). Indeed, by (4.44), (4.51), (b1), and (b2) we have for each i = 3, . . . , n that  2        uˆi K∪T − v2i = uˆi K∪T − vi + uˆi K∪T + vi ≥ Z + D + 4D1 − (ui )∞ ≥ D + 4D1 + 2(u1 )0 + (u2 )0 , and hence by (4.41), (4.44), (4.51), and (4.52) it follows that    (uˆ1 uˆ2 )K∪T − v1 v2 =

n

∑

 2  uˆi |K∪T − v2i ≥ D + 4D1 + 2(u1 )0 + (u2 )0 .

i=3

Thus, since (uˆ1 |K∪T )0 = (u1 )0 + 2D1 by (4.48), (a2), and (4.51), we have that uˆ2 |K∪T −

v1 v2 ∈ OD+2D1 +(u1 )0 +(u2 )0 (K ∪ T ). uˆ1 |K∪T

(4.54)

On the other hand, (4.48), (4.52), and (a2) ensure that v1 /uˆ1 |K∪T ∈ O(K ∪ T ). Since v2 ∈ M (K ∪ T | E0 ) (see (4.52)), we infer that v1 v2 /uˆ1 |K∪T ∈ M (K ∪ T | E0 ) and hence, by (4.54), uˆ2 |K∪T ∈ M (K ∪ T | E0 ) as well. Moreover, uˆi ∈ M (Σ | E ∪ E0 ) for all i ≥ 3 and uˆ1 vanishes nowhere off K ∪ T ∪ E (see (4.50)), and hence uˆ2 ∈ O(Σ \ (K ∪ T ∪ E)) (see (4.53)). This shows (c1). Let us now verify (c2). Conditions (4.44), (4.48), (a1), (a2), and (4.52) give that v

1 − 1 ≥ Z + D − (u1 )0 ≥ D + (u1 )0 + (u2 )∞ , uˆ1 |K∪T and hence (4.52) and (4.54) ensure that   v1  v1 v2 + v2 − 1 ≥ D + (u1 )0 + (u2 )0 . (uˆ2 |K∪T − v2 ) = uˆ2 K∪T − uˆ1 |K∪T uˆ1 |K∪T

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4 Complete Minimal Surfaces of Finite Total Curvature

This, (b1), (4.44), (4.49), (4.51), and (4.52) yield condition (c2) whenever that ε2 > 0 is chosen sufficiently small. Finally, condition (c3) follows from (c2) and Hurwitz’s theorem provided that ε2 > 0 is small enough. Set uˆ := (uˆ1 , . . . , uˆn ) ∈ M (Σ | E0 ∪ E ∪Θ ) × M (Σ | E0 ∪ E)n−1 . From (4.53) we see that uˆ ∈ M (Σ | E0 ∪ E ∪Θ , Sn−1 ∗ ). Conditions (a1)–(a3), (b1)– (b2), (c1)–(c3), (4.52), and (4.43) guarantee the following properties. (d1) uˆθˆ belongs to M 1 (Σ | E0 ∪ E)n and vanishes nowhere on Σ \ (E ∪ E0 ). √ (d2) uˆ − u ∈ OD (K)n and uˆ − uC (K) < nε1 . Thus, uˆ ∈ OD (K, u) ∩ M 1 (K | E0 )n ; see (4.31) for the notation. (d3) uˆ1 θˆ has an effective pole at each point of E. (d4) (uˆi |K ) = (ui ) for all i ∈ {1, . . . , n}. Step 4. It remains to deal with the periods of uˆθˆ on the curves C1 , . . . ,Cl . Let  l   h = (hi, j )i=1,...,n j=1,...,l ∈ OD0 (K)n be the map in (4.45) and ρ > 0 be the number in (4.47). By Theorem 4.3.1 there is a map    n l  hˆ = (hˆ i, j )i=1,...,n j=1,...,l ∈ OD0 (K) ∩ M (Σ | E) satisfying the following conditions. (e1) hˆ − hC (K) < ε1 .   n ˆ (e2) hˆ vanishes on the set i=1 supp((uˆi θ )0 ) \ K. (e3) (hˆ i, j )∞ = ∑ p∈E mi, j (p)p for all i = 1, . . . , n, j = 1, . . . , l, where mi, j (p) are pairwise distinct natural numbers for each p ∈ E. To obtain (e3), we fix an ordering of the set {1, . . . , n} × {1, . . . , l} and apply Theorem 4.3.1 recursively in such a way that the map {1, . . . , n} × {1, . . . , l}

(i, j) → mi, j (p) ∈ N is strictly increasing for each p ∈ E. For each ζ ∈ (Cn )l we consider the following map of the form (4.33):   s (ζ ) = (1 + ζ · h) ˆ 2 uˆ1 , (1 + ζ · h/2) ˆ 2 uˆ2 , (1 + ζ · h)(1 ˆ + ζ · h/2)( ˆ uˆi )i=3,...,n . Ψ s (ζ ) ∈ M (Σ | E ∪ E0 ∪Θ )n by (d1). If ε1 > 0 is small then, by (d2) and Note that Ψ nl s is  (e1), Ψs (ζ ) is C (K)-close to Ψs (ζ ) uniformly on ζ ∈ ρB , and hence P ◦ Ψ nl close to P ◦Ψs uniformly on ρB , where P is the period map 

P : C (C, u) → (Cn )l , P(h) = (h − u)θˆ (4.55) Cj

j=1,...,l

(cf. (4.32)). If ε1 > 0 is sufficiently small then, up to decreasing ρ > 0 if necessary, (d1)–(d4) and (e1)–(e3) ensure the following conditions. s (ζ ) ∈ OD (K, u) for all ζ ∈ ρBnl ; see (4.31). Recall that hˆ ∈ (OD (S)n )l , (f1) Ψ 0 ˆ C (K) , and take into account Remark 4.4.1. assume that ρ < 1/h

4.5 Approximation and Interpolation

217

s : ρBnl → (P ◦ Ψ s )(ρBnl ) is a biholomorphism with 0 ∈ (f2) The map P ◦ Ψ s )(ρBnl ); see (4.46) and use the Cauchy estimates. (P ◦ Ψ s (ζ ) ∈ M (Σ | E ∪ E0 ∪ Θ , Sn−1 ) and is full for all ζ ∈ ρBnl ; use Lemma (f3) Ψ ∗ ˆ vanish anywhere on 4.4.2-(i) and (ii). Recall that neither uˆ nor s(ζ · h) s (ζ ) does not vanish there either by (e2). Σ \ (E ∪ E0 ∪Θ ), and hence Ψ s (ζ ) − uC (K) < ε0 for all ζ ∈ ρBnl ; see (4.47). (f4) Ψ Moreover, (d1) and (e2) ensure that for all ζ ∈ (Cn )l we have that s (ζ )θˆ ∈ M 1 (Σ | E ∪ E0 )n Ψ

(4.56)

and it vanishes nowhere on Σ \ (K ∪ E). Thus, (d1) and (f1) give that s (ζ )θˆ has no zeros on Σ \ (E0 ∪ E) for all ζ ∈ ρBnl . Ψ

(4.57)

Furthermore, (d3) and (e3) imply that s (ζ )θˆ has an effective pole at each point p ∈ E for every ζ ∈ (Cn )l . Ψ

(4.58)

Finally, let ζ0 ∈ ρBnl be the unique point such that s (ζ0 )) = 0 P(Ψ

(4.59)

(see (f2)) and define s (ζ0 ) ∈ M (Σ | E ∪ E0 ∪Θ , An−1 ); fˆ = ( fˆ1 , . . . , fˆn ) = Ξ −1 ◦ Ψ ∗ see (4.19) and (f3). Then, the following conditions hold. (g1) fˆθˆ ∈ M 1 (Σ | E ∪E0 )n vanishes nowhere on Σ \(E0 ∪E); see (4.56) and (4.57). (g2) fˆθˆ has an effective pole at each point p ∈ E; see (4.58). (g3) (( fˆi − fi )θˆ |K ) ≥ D for all i ∈ {1, . . . , n}, where f = ( f1 , . . . , fn ) is given by (4.41); see (4.41) and (f1). In particular, for each p ∈ E0 there is an i ∈ {1, . . . , n} such that fˆi θˆ has an effective pole at p.  (g4) C j ( fˆ − f )θˆ = 0 for all j ∈ {1, . . . , l}, and hence ( fˆ − f )θˆ is exact on K; see (4.41), (4.55), and (4.59). (g5)  fˆ − f C (K) < 2ε0 ; see (4.19), (4.41), and (f4). Completion of the proof. Recall that p0 ∈ K˚ \ (Λ ∪ E0 ). Since K is a deformation retract of Σ \ E with the homology basis {C1 , . . . ,Cl1 }, properties (4.41), (4.55), and conditions (g1)–(g4) guarantee that the map xˆ : Σ \ (E0 ∪ E) → Rn given by p ˆ ˆ Σ \ (E0 ∪ E) p → x(p) ˆ = ℜ p0 f θ is a well defined complete conformal minimal immersion of finite total curvature with Fluxxˆ = Fluxx = p. Moreover, (4.55), (g3), (g4), (4.42), and the fact that the arcs Cl1 +1 , . . . ,Cl connect p0 to points in E0 ∪ Λ show that xˆ − x vanishes to order at least k at every point of E0 ∪ Λ . Finally, (g5)  ensures that xˆ − xC (K) < ε provided that ε0 > 0 is chosen small enough. 

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4 Complete Minimal Surfaces of Finite Total Curvature

4.6 Intersections with Affine Lines The following main result of this section, obtained in [13, Theorem 1.4], gives an explicit upper bound on the number of intersections of a complete conformal minimal surface x : M → R3 of finite total curvature with an affine line L ⊂ R3 not contained in x(M) in terms of the Euler number of M and the degree of the Gauss map of x or, equivalently, its total Gaussian curvature TC(x); see (2.96). Theorem 4.6.1. Let x : M → R3 be a complete conformal minimal immersion of finite total curvature with the complex Gauss map g : M → CP1 . If L ⊂ R3 is an affine line which is not contained in x(M), then   # x−1 (L) ≤ χ(M) + 6 deg(g) + 25 deg(g)3 . (4.60) As usual, # denotes the number of points in a given set, χ(M) is the Euler number of M (see (1.9), (1.10)), and deg(g) ∈ Z+ is the degree (1.132) of the extended Gauss map g : Σ → CP1 (see Theorem 4.1.3 (iii)). This seems to be the first known bound of this type in the literature. We expect that an analogous result holds for intersections of minimal surfaces of finite total curvature in Rn for n > 3 with affine subspaces of codimension 2. The bound in (4.60) is not sharp in general. For instance, any affine line in R3 intersects a catenoid in at most 4 points (see Example 2.8.1), while the upper bound provided by Theorem 4.6.1 in this case is 31. It seems a very challenging problem to find a precise upper bound. In the proof of Theorem 4.6.1 we shall use the following well known result. Proposition 4.6.2. If x : M → R3 is a complete conformal minimal immersion of finite total curvature and L ⊂ R3 is an affine line which is not contained in x(M), then x−1 (L) is a finite set. Proof. Up to a rigid motion we may assume that L = {(0, 0, z) : z ∈ R}. Write x = (x1 , x2 , x3 ) and recall that M = Σ \ {p1 , . . . , pk } for some compact Riemann surface Σ (see Theorem 4.1.1). Since L ⊂ x(M) and x is complete, x1 does not vanish identically on M. Denote by Z ⊂ Σ the topological closure of x1−1 (0). Since ∂ x is meromorphic on Σ by Theorem 4.1.1, every point p ∈ Z admits an open neighbourhood U p in Σ such that Z ∩ U p consists of a finite collection α1 , . . . , αm of regular analytic Jordan arcs that meet only at p and form an equiangular system there. The number m = m(p) is the zero order of ∂ x1 at p plus one if ∂ x1 is holomorphic at p, and is the pole order of ∂ x1 at p minus one if ∂ x1 has a pole at p; in the latter case, p ∈ Σ \ M = {p1 , . . . , pk }. Since ∂ x2 is meromorphic on Σ , the function h = x2 /(1 + x22 ) is analytic, bounded, and does not vanish identically, and hence the identity principle ensures that the zero set of h|α j has no accumulation points for any j. This implies that x−1 (L) ∩ U p has no accumulation points for any  p ∈ Σ . Since Σ is compact, we infer that x−1 (L) is finite.  We begin by recalling some elementary facts about planar curves.

4.6 Intersections with Affine Lines

219

Let γ : [0, 1] → R2 be a smooth path parameterizing a closed curve, i.e., γ(0) = γ(1). The winding number wγ (p) ∈ Z of γ with respect to a point p ∈ R2 \ γ is the γ(t)−p ∈ S1 , i.e., the number of times topological degree of the closed curve γ p (t) = |γ(t)−p| the curve winds around p in the positive (counterclockwise) direction. The winding number wγ (p) equals the oriented intersection number of γ with any oriented ray  = {p + sα : s ≥ 0} based at p, where α ∈ C∗ . By the transversality theorem (see Sect. 1.4), γ intersects most such rays transversely at finitely many points, and the intersection number equals the number of positive crossings minus the number of negative crossings of the ray with γ. (A crossing is called positive if the pair of vectors (α, γ  (t)) forms a positively oriented basis of R2 , and negative otherwise.) This number is independent of the choice of the ray; see Remark 1.5.5. We now restrict attention to nonconstant real analytic curves, i.e., curves γ(t) = (x(t), y(t)) ∈ R2 which are real analytic functions of the variable t on some interval in R. Such γ is piecewise regular in the sense that its velocity vector γ  (t) vanishes only at isolated points which are called singular points; γ is called regular if there are no singular points, i.e., if it is immersed. A real analytic curve has a well defined tangent line and normal line at any point γ(t0 ). Indeed, assuming as we may that t0 = 0 and γ(t0 ) = (0, 0), we have after a rotation of coordinates on R2 that   γ(t) = t m a + O(t), bt + O(t 2 ) (4.61) for some m ∈ N and a, b ∈ R, a > 0. The point γ(0) = (0, 0) is regular if m = 1 and is singular if m > 1. We have that   γ  (t) = t m−1 ma + O(t), (m + 1)bt + O(t 2 ) , and after dividing by t m−1 we see that the tangent line at t = 0 is R × {0} and the normal line is {0} × R. Furthermore, γ has locally a well defined unit normal vector field n(t) (i.e., γ  (t)· n(t) = 0 and |n(t)| = 1 for every t) which is real analytic, unique up to a sign, and is given locally near t = 0 by n(t) = ±N(t)/|N(t)|, where   N(t) = −(m + 1)bt + O(t 2 ), ma + O(t) . A simple calculation shows that n (0) = 0 if and only if b = 0 (see (4.61)); in this case γ is tangent to order > 1 (possibly fractional) to its tangent line at t = 0. The local geometric behaviour of the curve γ(t) given by (4.61) at t = 0 strongly depends on the parity of the integer m ∈ N. If m is odd, then locally the curve may be represented as a graph over an interval (−δ , +δ ) ⊂ R which is tangent to R × {0} at t = 0. If on the other hand m is even, then γ(t) turns back at t = 0, so it locally forms a two-sheeted graph over a one-sided interval [0, δ ) with δ > 0. (In general, δ will have the same sign as the coefficient a = 0 in the local expression (4.61).) In this case, γ(0) is said to be a cusp point of γ (see Figure 4.1). These definitions apply to any point γ(t0 ) by passing to suitable local coordinates. Assume now that γ(t) is a nonconstant closed real analytic curve in R2 . From what has been said, it follows that such a curve has at most finitely many singular

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4 Complete Minimal Surfaces of Finite Total Curvature

points, and it admits a globally defined real analytic unit normal vector field if and only if the number of its cusps is even; this is because at every cusp the curve turns but the normal vector field does not. If in addition the curve is nonflat, meaning that its image is not contained in an affine line of R2 , then its normal vector field n(t) is piecewise regular in the sense that its derivative n (t) has only finitely many zeros. Assuming that the normal field n(t) exists (i.e., the number of cusps is even), the turning number tγ ≥ 1 of γ is the number of full rotations made by n(t) during one traversal of the curve, which equals the absolute value of the degree of n as a map S1 → S1 . Clearly, tγ does not depend on the choice of the unit normal field n. We shall need the following lemma relating the winding number, the turning number, and the number of cusps of a closed real analytic curve. Lemma 4.6.3. Let γ = γ(t) be a nonconstant closed real analytic curve in R2 admitting a unit normal vector field n(t) satisfying n (t) = 0 for all t. Then for every point p ∈ R2 \ γ the winding number wγ (p) satisfies 1 |wγ (p)| ≤ 2tγ + mγ , 2 where mγ is the number of cusp points of γ (recall that mγ is even).

Fig. 4.1 A planar curve with eight cusp points, turning number 1, and winding number −3.

Proof. The previous discussion shows that, up to a rigid motion of R2 , we may assume that p = (0, 0) ∈ R2 , the curve γ intersects the ray  := {(0, y) : y ≥ 0} transversely (hence there are at most finitely many intersection points which are all regular points of γ), and the preimage n−1 ({(1, 0), (−1, 0)}) of the points (±1, 0) contains no singular points of γ. Recall that the winding number wγ of γ around the origin equals the intersection number of γ with . Consider the semicircles S1+ = {(x, y) ∈ S1 : y ≥ 0},

S1− = {(x, y) ∈ S1 : y ≤ 0}.

4.6 Intersections with Affine Lines

221

Denote by γ1 , . . . , γ2tγ the connected components (arcs) of n−1 (S1+ ) and n−1 (S1− ), ordered to lie end to end. Their total number equals twice the turning number of γ since n (t) = 0 for each t, so n(t) only moves in one direction along the circle S1 . The same condition implies that n does not assume the values (±1, 0) at any interior point of an arc γ j . We split each γ j into finitely many subarcs γ j,1 . . . , γ j,m j +1 lying end to end such that for any i = 1, . . . , m j + 1 the endpoints of γ j,i are either cusp points or points in n−1 ({(1, 0), (−1, 0)}) (the latter only happens at the endpoints of γ j ), and the interior of γ j,i contains no cusp points. Here, m j ∈ Z+ is the number of cusp points of γ contained in γ j . We obviously have that 2tγ

mγ =

∑ m j.

(4.62)

j=1

The arcs γ j,i may contain singular points of the form (4.61) with odd m (those are not cusp points), but this will not affect the subsequent arguments. Let π : R2 → R denote the projection π(x, y) = x. Since the interior of γ j,i has neither cusp points nor points where n = (±1, 0), the local analysis of the behaviour of a real analytic curve, made above, shows that the restriction of π to every arc γ j,i is injective. We say that γ j,i is positive if π ◦ γ j,i (t) is decreasing (the point is moving to the left), and is negative if π ◦ γ j,i (t) is increasing (the point is moving to the right). Since a common endpoint of any two adjacent subarcs γ j,i , γ j,i+1 of γ j is a cusp point where the curve turns back and the normal vector is not horizontal, such subarcs have opposite character, one positive and the other one negative. Recall that  = {(0, y) : y ≥ 0}. Since π|γ j,i is injective,  ∩ γ j,i is either empty or a single point; moreover, if  ∩ γ j,i = ∅ then the crossing of  with γ j,i is positive if γ j,i and every two adjacent arcs and only if γ j,i is positive. Since we have m j + 1 arcs m  have opposite character, there are at most 1 + 2j arcs of each character. (Here [·] denotes the integer part of a real number.) It follows that the absolute value of the m signed number of crossings of  with γ j is at most 1 + 2j . Taking into account     2tγ m 1 + 2j ≤ 2tγ + 12 mγ .  (4.62), we have that |wγ | ≤ ∑ j=1  Proof of Theorem 4.6.1. We may assume that x is primitive in the sense that there is no nontrivial finite holomorphic covering ρ : M → M0 onto an open Riemann surface M0 admitting a conformal minimal immersion x0 : M0 → R3 such that x = x0 ◦ ρ. Indeed, any such map  x0 is complete  andof finite total curvature, and if k ∈ N is the degree of ρ then # x−1 (L) = k # x0−1 (L) , χ(M) = kχ(M0 ), and deg(g) = k deg(g0 ) where g0 denotes the complex Gauss map of x0 ; hence, the theorem holds true for x provided that it holds for x0 . Furthermore, the conclusion of the theorem clearly holds when x is a plane or a catenoid; we assume in the sequel that it is not. By the Chern–Osserman Theorem 4.1.1 we have that M = Σ \ E for a compact Riemann surface Σ and a nonempty finite set E ⊂ Σ , and the complex Gauss map of x extends to a nonconstant holomorphic map g : Σ → CP1 . Under the stereographic projection S2 → CP1 , g corresponds to the extended classical Gauss map N : Σ → S2 of x compatible with the orientation on the Riemann surface Σ (see (2.82)).

222

4 Complete Minimal Surfaces of Finite Total Curvature

→ − For every affine line T ⊂ R3 we denote by T the directing vectorial line of T , i.e., the parallel translate of T passing through the origin. Denote by L the manifold of all affine lines in R3 , and let Lx be the subset of L consisting of lines T ∈ L satisfying the following conditions. → − (a) The set QT := {p ∈ M : N(p) is orthogonal to T } is compact. (b) The Gauss curvature function K : M → (−∞, 0] of x vanishes nowhere on QT . By (2.93) this is equivalent to g having no critical points on QT , so the Gauss map N : Σ → S2 of x is a local diffeomorphism at each point of QT . (c) x−1 (T ) ∩ QT = ∅; equivalently, x intersects the line T transversely. → − → − Denote by πT : R3 → T ⊥ ∼ = R2 the orthogonal projection with the kernel T . Then, p ∈ QT if and only if p is a critical point of the map πT ◦ x : M → R2 . Hence, condition (a) is equivalent to πT ◦ x having no critical points on a neighbourhood of the ends E of M in Σ . Since the Gauss map N extends holomorphically to Σ = M ∪E, an affine line T ⊂ R3 satisfies condition (a) if and only if → − (a ) N(p) is not orthogonal to T for any point p ∈ E. → − Indeed, the set {p ∈ Σ : N(p) is orthogonal to T } is the preimage by N of the → − unit circle in the 2-plane orthogonal to T , so it is a union of finitely many onedimensional real analytic subsets without isolated points. We claim that Lx is a dense open subset of L . Indeed, recall that condition (a) is equivalent to (a ), and hence it is satisfied by lines in an open dense subset of L . → − Moreover, given a vectorial line T satisfying (a ), conditions (b) and (c) hold for → − an open dense set of lines T with the direction T by the transversality theorem (for (c)) and the fact that the zero set of the real analytic curvature function K has no interior (for (b)). This proves the claim. We shall first prove the theorem under the assumption that the affine line L belongs to Lx . Write x = (x1 , x2 , x3 ) : M → R3 and N = (N1 , N2 , N3 ) : Σ → S2 ⊂ R3 . Up to a rigid motion, we may assume that → − L = L = {(0, 0, z) ∈ R3 : z ∈ R} and hence

QL = N −1 (S1 × {0}) = N3−1 (0).

Thus, conditions (a)⇔(a’), (b), and (c) guarantee that N3 (p) = 0 for all p ∈ x−1 (L) ∪ E and that 0 is a regular value of the real analytic function N3 : M → R. It follows that QL = N3−1 (0) is a compact subset of M = Σ \ E consisting of finitely many pairwise disjoint real analytic Jordan curves, and the open set Σ \ QL has finitely many connected components Ω1 , . . . , Ωa . Let Ω be one of these components. Clearly, bΩ = Ω ∩ QL and hence (x−1 (L) ∪ E) ∩ bΩ = ∅ and x−1 (L) ∩ Ω ⊂ Ω \ E.

4.6 Intersections with Affine Lines

223

Note that N maps the domain Ω properly onto one of the open hemispheres S˚ 2± = S2 ∩ {±N3 > 0}. Recall that the set x−1 (L) is finite (see Proposition 4.6.2) and x intersects L transversely by condition (c). Hence, writing x−1 (L) ∩ Ω = {p1 , . . . , ps } ⊂ Ω \ E, there are pairwise disjoint compact discs D1 , . . . , Ds in Ω \ E such that p j ∈ D˚ j and the map (x1 , x2 )|D j : D j → R2 is a diffeomorphism onto its image for each j ∈ {1, . . . , s}. Likewise, write E ∩ Ω = {q1 , . . . , qr } ⊂ Ω and choose pairwise disjoint compact discs U1 , . . . ,Ur in Ω \ each j ∈ {1, . . . , r} we have q j ∈ U˚ j and

s

j=1 D j

such that for

x(U j ) ⊂ R3 is an Iq j -sheeted multigraph over R2 \ B for some integer Iq j ≥ 1, where B is an open Euclidean disc in R2 not depending on j. Such discs U j exist by the asymptotic behaviour of the ends of a complete minimal surface of finite total curvature, described in Theorem 4.1.3 where the numbers Iq j are introduced. Denote by c1 , . . . , cm the connected components of bΩ = QL ∩ Ω . Also, write d j = bD j = D j \ D˚ j , j = 1, . . . , s, and u j = bU j = U j \ U˚ j , j = 1, . . . , r. This gives a collection of m + s + r pairwise disjoint Jordan curves. Set Ω0 := Ω \

s !

D˚ j ∪

j=1

r !

U˚ j

j=1

and note that Ω0 is a compact domain in M with boundary bΩ0 =

m ! j=1

The map

s r

!

!

cj ∪ dj ∪ uj . j=1

j=1

(x1 , x2 ) : Ω 0 → S1 β= 2 2 x1 + x2

is well defined and continuous since x(Ω0 ) is disjoint from the z-axis L. Let β∗ : H1 (Ω0 , Z) → H1 (S1 , Z) = Z denote the induced homomorphism between the first homology groups. Orienting the boundary curves c1 , . . . , cm , d1 , . . . , ds , and u1 , . . . , ur of Ω0 coherently, we have that ∑mj=1 c j + ∑sj=1 d j + ∑rj=1 u j = 0 in H1 (Ω0 , Z), and hence

224

4 Complete Minimal Surfaces of Finite Total Curvature s

r

m

j=1

j=1

j=1

∑ β∗ (d j ) = − ∑ β∗ (u j ) − ∑ β∗ (c j ) ∈ Z.

(4.63)

Since (x1 , x2 )|D j is injective and (x1 (p j ), x2 (p j )) = (0, 0), the winding number of β (d j ) around the origin is ±1. Further, since sj=1 D j ⊂ Ω and the Gauss map N|Ω assumes values in a hemisphere, the sign of this winding number depends on the orientation of Ω0 but not on j ∈ {1, . . . , s}. Thus, β∗ (d1 ) = . . . = β∗ (ds ) ∈ {1, −1}, and hence  s    (4.64)  ∑ β∗ (d j ) = s = #(x−1 (L) ∩ Ω ). j=1

Summing over all components Ω ∈ {Ω1 , . . . , Ωa } gives the number #(x−1 (L)) that we wish to estimate. The same argument shows that β∗ (u j ) = ±Iq j for j ∈ {1, . . . , r}, where Iq j ≥ 1 is the number in Theorem 4.1.3 and the sign of β∗ (u j ) does not depend on j, so   r    ∑ β∗ (u j ) = j=1

r

∑ Iq j .

(4.65)

j=1

With (4.63), (4.64), and (4.65) in hand, in order to conclude the proof it remains to give an upper bound on | ∑mj=1 β∗ (c j )| in terms of the degree deg(N) = deg(g) of the Gauss map and the Euler number χ(M) of M. Recall that c1 , . . . , cm are the connected components of bΩ . For each j ∈ {1, . . . , m} we consider the closed oriented planar curve α j := (x1 , x2 )(c j ) ⊂ R2 .

(4.66)

Since x(c j ) is a regular compact curve in R3 which is not contained in an affine line and is disjoint from the third coordinate axis L, its projection α j to R2 is a piecewise regular analytic curve in R2∗ with at most finitely many singular points. Since N(Ω ) is one of the hemispheres S2± = {±N3 ≥ 0} and N3 |bΩ = 0, we see that, up to the identification S1 ≡ S2 ∩ {z = 0}, the unit normal field to α j equals ±N|c j . Denote by w j , t j , and m j the winding number with respect to the origin, the turning number, and the number of cusp points of α j , respectively. Lemma 4.6.3 then gives that |w j | ≤ 2t j + 12 m j . Since α j and β (c j ) = α j /|α j | have the same winding number with respect to the origin, we infer that β∗ (c j ) = w j and hence   m    ∑ β∗ (c j ) ≤ j=1

m

m

j=1

j=1

∑ |w j | ≤ ∑



1  2t j + m j . 2

(4.67)

Since N(Ω ) is one of the hemispheres S2± = {±N3 ≥ 0}, N3 |bΩ = 0, and 0 is a regular value of N3 , it follows that the map N|Ω : Ω → N(Ω ) is a finite branched covering of degree deg(N|Ω ) ≤ deg(N) without branched points on bΩ . Since the normal vector field ±N|c j to α j is a regular curve (see condition (b) in the definition of Lx ), its turning number t j equals deg(N|c j ), and hence ∑mj=1 t j = deg(N|Ω ).

4.6 Intersections with Affine Lines

225

Setting mΩ = ∑mj=1 m j , this formula and (4.67) give  m  1    ∑ β∗ (c j ) ≤ 2 deg(N|Ω ) + mΩ . 2 j=1 In view of (4.63), (4.64), and (4.65) it follows that r     1 # x−1 (L) ∩ Ω = # x−1 (L) ∩ Ω ≤ 2 deg(N|Ω ) + ∑ Iq j + mΩ . 2 j=1

Using this bound for each of the connected components Ω1 , . . . , Ωa of Σ \ QL gives #(x−1 (L)) ≤ 4 deg(N) +

1

a

∑ Iq + 2 ∑ mΩi .

q∈E

(4.68)

i=1

The factor 2 in front of deg(N) has changed to 4 because deg(N) is the number of points in the fibre of N over a generic point in S2 ; choosing the point in S2+ we get 2 deg(N) by summing over those components Ωi which project onto S2+ , while the components projecting onto S2− contribute another such term. The Jorge–Meeks formula (4.5) gives ∑q∈E Iq = 2deg(N) + χ(M). Since N = g under the identification S2 ≡ CP1 , this and (4.68) will imply the estimate (4.60) in the statement of the theorem provided we can show that 1 a ∑ mΩi ≤ 25(deg(g))3 . 2 i=1

(4.69)

We shall now prove this last inequality. Recall that QL = N −1 (S1 × {0}) = |g|−1 (1) =

a !

bΩi .

(4.70)

i=1

Denote by C the finite set of points in QL whose image via (x1 , x2 ) is a singular point of the corresponding curve α j = (x1 , x2 )(c j ) (4.66). (This will of course include all cusp points.) Since every curve c j ⊂ QL lies in the boundary of exactly two components Ωi , to prove (4.69) it suffices to show that #C ≤ 25(deg(g))3 .

(4.71)

Let Φ = (φ1 , φ2 , φ3 ) = 2∂ x denote the Weierstrass data (2.83) of x. We have that (φ3 ) = (g)0 + (g)∞ −

∑ (Iq + 1)q.

(4.72)

q∈E

(See the explanation just below the formula (2.84) and note that φ3 = 2∂ x3 has a pole of order Iq + 1 at every point of q ∈ E.) Consider the meromorphic function

226

4 Complete Minimal Surfaces of Finite Total Curvature

F =g

φ3 φ3 = dg d log g

(4.73)

on the compact Riemann surface Σ = M ∪ E. In view of (4.72) we have that (F) = (g) + (φ3 ) − (dg) = 2(g)0 + (dg)∞ − (dg)0 −

∑ (Iq + 1)q.

q∈E

At any pole p ∈ Σ of g we have (dg)∞ (p) = (g)∞ (p) + 1 and hence (dg)∞ ≤ 2(g)∞ . It follows that deg(F) = (F)0 ≤ 2(g)0 + 2(g)∞ and therefore deg(F) ≤ 4 deg(g).

(4.74)

Fix a point p0 ∈ x−1 (L), so x1 (p0 ) = x2 (p0 ) = 0. By (2.83) we have that  p

i 1

 1 1 −g , + g F d log g, p ∈ M. (x1 , x2 )(p) = ℜ 2 g 2 g p0 Since |g|−1 (1) consists of regular analytic curves and g has no critical points on this set, we can choose a local parameter t on |g|−1 (1) such that g(t) = eit and hence

  1 (x˙1 (t), x˙2 (t)) = ℜ (e−it − eit , ie−it + ieit )F(t)i = ℜF(t) sin(t), − cos(t) . 2 Thus, the set C of singular points of (x1 , x2 ) restricted to |g|−1 (1) equals   C = p ∈ |g|−1 (1) : ℜF(p) = 0 .

(4.75)

Choose θ ∈ C with |θ | = 1 such that g−1 (θ ) is disjoint from C and set u := i A calculation shows that ℑu =

u+i g+θ ⇐⇒ g = θ . g−θ u−i 2(|g|2 −1) , |g−θ |2

(4.76)

which implies

|g|−1 (1) = u−1 (R ∪ {∞}) and C ⊂ u−1 (R).

(4.77)

Since we assumed at the very beginning of the proof that x is neither a plane nor a catenoid, u and F are nonconstant meromorphic functions on the compact Riemann surface Σ , and hence there exists a nonconstant irreducible complex polynomial P in two variables such that P(u, F) = 0 on Σ ; see [123, Proposition IV.11.6]. Denote by Σ the compact Riemann surface which normalizes the algebraic curve {P = 0}. We claim that (u, F) is a primitive pair on Σ , (4.78) in the sense that the holomorphic map ρ : Σ → Σ uniquely determined by (u, F) : Σ → {P = 0} is a biholomorphism. Indeed, let u˜ and F denote the unique meromorphic functions on Σ such that u = u˜ ◦ ρ and F = F ◦ ρ; likewise, define

4.6 Intersections with Affine Lines

g˜ = θ

u˜ + i , u˜ − i

F φ˜3 = d g˜ , g˜

227

1 φ˜1 = 2



 1 − g˜ φ˜3 , g˜

i φ˜2 = 2



 1 + g˜ φ˜3 , g˜

 = (φ˜1 , φ˜2 , φ˜3 ). It then follows from (4.76) and (4.73) that and Φ g = g˜ ◦ ρ

and

 Φ = ρ ∗ (Φ).

(4.79)

 integrates to a complete  = ρ(M). The key to prove (4.78) is to show that Φ Set M  → R3 of finite total curvature. Recall that Φ = 2∂ x conformal minimal immersion M is holomorphic and nonvanishing on M = Σ \ E, and it has an effective pole at every point of E (see Theorem 4.1.1). Taking into account (4.79) and that ρ  is is a holomorphic map between compact Riemann surfaces, it follows that Φ  and it has an effective pole at every point holomorphic and vanishes nowhere on M,  E = ρ −1 (E),  and that of E = ρ(E). From this it also follows that M = ρ −1 (M),  is an unbranched finitely sheeted holomorphic covering. ρ :M→M  has vanishing real periods on M,  we reason by contradiction In order to see that Φ  with γ(0) = γ(1) such that ν := and assume that there is a path γ : [0, 1] → M   = 0. In view of (4.79), this means that x(γ(1)) = x(γ(0)) + ν for any lifting ℜ γ Φ  The same formula holds (with γ : [0, 1] → M of γ via the covering ρ : M → M.  homotopic to γ, and hence, the same period ν ∈ R3 ) for each closed curve in M by analyticity, x(M) is invariant under the translation by the vector ν. This implies  is exact and hence, by that x has infinite total curvature, a contradiction. Thus, ℜΦ (4.79) and what has been said above, it integrates to a complete conformal minimal  = Σ \ E → R3 of finite total curvature such that x = x˜ ◦ ρ. Since x immersion x˜ : M was assumed to be primitive at the very beginning of the proof, this implies that ρ has degree one and hence is a biholomorphism. This proves (4.78). We continue proving (4.71). Since u is obtained from g by a M¨obius transformation (cf. (4.76)), we have that deg(u) = deg(g). By virtue of (4.78) and (4.74), it follows from [123, Proposition IV.11.9] that degF (P) = deg(u) = deg(g) and

degu (P) ≤ deg(F) ≤ 4 deg(g),

where degF (P) denotes the algebraic degree of the polynomial P in the variable F, and likewise for degu (P). In view of (4.77), writing f1 = ℜF and f2 = ℑF and identifying Σ with Σ via ρ, the set C (4.75) is given by   (4.80) C = (u, i f2 ) ∈ R × iR : P(u, i f2 ) = 0 . Write Q1 (u, f2 ) = ℜP(u, i f2 ) and Q2 (u, f2 ) = ℑP(u, i f2 ). Each Q j is a real polynomial in two real variables with the total algebraic degree deg(Q j ) ≤ deg(P) ≤ degF (P) + degu (P) ≤ 5deg(g). Hence, the Milnor–Thom inequality (see N. R. Wallach [336, Theorem 8.1]) guarantees that the set C0 = {(v, w) ∈ R2 : Q1 (v, w) = Q2 (v, w) = 0} satisfies

228

4 Complete Minimal Surfaces of Finite Total Curvature

#C0 ≤ (5deg(g))2 .

(4.81)

By (4.80), a point (v, w) lies in C0 if and only if (v, iw) lies in C ⊂ Σ = Σ, and hence C ⊂ u−1 (π1 (C0 )) where π1 (v, w) = v. Since deg(u) = deg(g), this inclusion and (4.81) imply (4.71), thereby proving (4.69). This shows that the estimate (4.60) holds for the affine lines in Lx . Assume now that the given affine line L ∈ L does not lie in Lx . We still take L to be the vertical z-axis. We may assume that x−1 (L) is nonempty, for otherwise there is nothing to prove. The set x−1 (L) is finite by Proposition 4.6.2; write x−1 (L) = {p1 , . . . , pm }. Our goal is to show that m ≤ χ(M) + 6 deg(g) + 25 deg(g)3 .  By what has already been proved, it suffices  to find an affine line L ∈ Lx which is −1  not contained in x(M) such that # x (L) ≥ m. Let us first see that there is a plane Π ⊂ R3 containing L and satisfying the following two conditions: (i) The intersection of Π and x(M) at x(p j ) is transverse for all j ∈ {1, . . . , m}. (ii) QL ∩ x−1 (Π ) is finite. (Recall that QL is given by 4.70.) Condition (i) clearly holds for all but finitely many planes containing L. Note that QL ∩ x−1 (Π ) is a real analytic subset of the real analytic set QL = |g|−1 (1) (see (4.70)). If condition (ii) fails for a plane Π ⊃ L satisfying (i), then QL ∩ x−1 (Π ) contains a proper analytic arc. If this holds for all planes containing L and satisfying (i), then since L ⊂ x(M) we see that QL is bidimensional and hence M = QL by the identity principle. But this implies that g is constant and thus x(M) is a plane containing L, a contradiction. Note that, in fact, the set of planes Π satisfying (i) and (ii) is open and dense in the set of all affine planes in R3 containing L. Fix a plane Π ⊃ L satisfying (i) and (ii) and choose pairwise disjoint neighbourhoods V j of p j , j ∈ {1, . . . , m}, such that the intersection of Π and x(V j ) is transverse, ϒj := V j ∩ x−1 (Π ) is a Jordan arc, and x(QL ∩ V j ) ∩ Π is either empty or equal to {x(p j )}. Denote by  the orthogonal projection of Π to the horizontal plane {z = 0} ≡ R2 ; note that  is an affine line since Π is vertical. Write x = (x1 , x2 , x3 ) = (x , x3 ) where x = (x1 , x2 ) : M → R2 . It is clear that x (ϒj ) contains a nontrivial compact segment δ j in  such that 0 ∈ δ j ⊂ R2 . We can choose δ j so that it either contains the origin 0 ∈ R2 as an interior point, or (x )−1 (t) ∩ϒj consists of at least two points for all t ∈ δ j \ {0}. We can then take an affine line L ⊂ Π \ {L} transverse to x(M) and such that x−1 (L ) ∩ mj=1 ϒj consists of at least m points; in fact, every L sufficiently close to L in one of the two components of Π \ L satisfies this condition. (Note that L need not intersect every ϒj , but it could intersect some ϒj ’s twice.) By condition (ii) it is clear that L may be chosen such that QL ∩ x−1 (L ) = ∅ (note that QL = QL ). Since Lx is an open dense subset of L ,  ∩ mj=1 V j ) ≥ m,  ∈ Lx so close to L that #(x−1 (L) this shows that there is a line L which concludes the proof.  

4.7 An Effective Obstruction to Hitting

229

4.7 An Effective Obstruction to Hitting In this section we approach the general question of determining the simplest complete minimal surface in R3 passing through a given finite set of points in R3 . Corollary 4.5.2 shows that for any parabolic open Riemann surface M of finite topology (these are precisely the finitely punctured compact Riemann surfaces) and any finite set A ⊂ Rn for n ≥ 3 there are complete conformally immersed minimal surfaces M → Rn of finite total curvature containing A. The following natural questions appear in this context. Question 4.7.1. Let A be a finite set in Rn for some n ≥ 3. (i) What is the minimum absolute value of the total curvature of complete minimal surfaces in Rn containing A? (ii) Assume that M is a smooth open surface of finite topology. What is the minimum absolute value of the total curvature of complete immersed minimal surfaces M → Rn containing A? (iii) Assume now that M is a finitely punctured compact Riemann surface. What is the minimum absolute value of the total curvature among all complete conformally immersed minimal surfaces M → Rn containing A? Equivalently (see Corollaries 2.6.6 and 2.6.8), what is the smallest degree of the Gauss map of such surfaces? Corollary 4.5.2 and the fact that the total curvature of any candidate surface is an integer multiple of −2π (see Corollary 2.6.8) imply that these minimum numbers are attained in all cases. It is reasonable to expect the existence of upper bounds for these minima in terms of the cardinal of A. However, nothing seems to be known in this direction. We now discuss a counterpart to questions (i) and (ii) in dimension n = 3. By Corollary 2.6.8 the total Gaussian curvature of a complete orientable minimal surface of finite total curvature in R3 is an integer multiple of −4π. For every integer r ≥ 1 we denote by ϒ [r] the space of open, orientable, complete, nonflat immersed minimal surfaces x : M → R3 with the absolute value of the total curvature |TC(x)| ≤ 4πr. Given another integer m ≤ 1, we denote by ϒm [r] the subset of ϒ [r] consisting of surfaces x : M → R3 with the Euler number χ(M) = m (see (1.9), (1.10)). The Jorge–Meeks formula (4.5) implies that ϒm [r] is nonempty only if 2r + m ≥ 2 (see Corollary 4.1.5), and hence ϒ [r] =

! m≤1

ϒm [r] =

!

ϒm [r].

(4.82)

m∈{2−2r,...,1}

Corollary 4.5.2 shows that for any nonempty finite set A ⊂ R3 and integer m ≤ 1 there exists a large enough integer r ≥ 1 such that A ⊂ x(M) for some surface x ∈ ϒm [r]. So, given integers r and m with 2 − 2r ≤ m ≤ 1 ≤ r, a natural question is whether there exist finite sets A ⊂ R3 that are against the family ϒm [r], in the sense

230

4 Complete Minimal Surfaces of Finite Total Curvature

that A lies in no surface x ∈ ϒm [r], and if such sets do exist, what is their smallest cardinal. The same questions may be asked for the bigger family ϒ [r] for any r ≥ 1. The following result by Alarc´on, Castro-Infantes, and L´opez [13, Theorem 1.3] shows that such sets A indeed exist, and it provides an effective upper bound for the number of points they must contain. Theorem 4.7.2. For any pair of integers r and m with 2 − 2r ≤ m ≤ 1 ≤ r there exists a set Am [r] ⊂ R2 × {0} ⊂ R3 consisting of 2m + 12r + 50r3 + 1 points which set A[r] = A1 [r] consisting of is against the family k≤m ϒk [r]. In particular, the 12r + 50r3 + 3 points is against the family ϒ [r] = k≤1 ϒk [r]. Thus, if x : M → R3 is an open, orientable, complete nonflat immersed minimal surface with χ(M) ≤ m and Am [r] ⊂ x(M), then |TC(x)| > 4πr. In particular, no open, orientable, complete nonflat immersed minimal surface with the absolute value of the total curvature smaller than or equal to 4πr contains A[r] = A1 [r]. In order to discard also planes, it suffices to add to Am [r] any point off R2 × {0}. m [r] ⊂ R3 Corollary 4.7.3. Let r and m be as in Theorem 4.7.2. There exists a set A 3 consisting of 2m + 12r + 50r + 2 points such that |TC(x)| > 4πr for any open, orientable, complete immersed minimal surface x : M → R3 with χ(M) ≤ m and m [r] ⊂ x(M). In particular, there exists a set, namely A[r]  =A 1 [r] ⊂ R3 , consisting A 3 of 12r + 50r + 4 points, which is not contained in any open, orientable, complete immersed minimal surface x : M → R3 with |TC(x)| ≤ 4πr. Remark 4.7.4. It is easily seen that the cardinals of the sets in Theorem 4.7.2 and Corollary 4.7.3 satisfy the respective estimates 63 ≤ 2m + 12r + 50r3 + 1 ≤ 12r + 50r3 + 3 and 64 ≤ 2m + 12r + 50r3 + 2 ≤ 12r + 50r3 + 4. These bounds are far from sharp. Determining the optimal bounds seems a very challenging task which we do not pursue here.   Before proving Theorem 4.7.2, we need some preparations. Let x : M = Σ \ E → R3 be a complete conformal minimal immersion of finite total curvature, where Σ is a compact Riemann surface and E ⊂ Σ is a finite set. (By Theorem 4.1.1, every complete orientable minimal surface of finite total curvature in R3 is of this kind.) A symmetry of x is a rigid motion R : R3 → R3 such that R(x(M)) = x(M). If L is an affine line in R3 and L ⊂ x(M), then the Schwarz reflection principle for minimal surfaces (see Corollary 2.3.6) implies that x(M) is invariant under the reflection on R3 about L. Proposition 4.7.5. Planes and catenoids are the only orientable complete minimal surfaces of finite total curvature in R3 with an infinite group of symmetries. In particular, no complete nonflat minimal surface of finite total curvature in R3 contains infinitely many affine lines.

4.7 An Effective Obstruction to Hitting

231

Proof. We give a simplified version of the proof from [13]. Assume that x : M = Σ \ E → R3 is a complete nonflat conformal minimal immersion of finite total curvature. Its Gaussian curvature K : M → (−∞, 0] does not vanish identically, and Corollary 4.1.4 shows that lim p→E K(p) = 0. It follows that   C = p ∈ M : K(p) = inf K < 0 M

is a nonempty compact set. The symmetry group G of x is a closed subgroup of the Lie group G of rigid motions in R3 and R(x(C)) = x(C) for all R ∈ G. Assume that G is infinite and let us see that x is a catenoid. Suppose first that G is a topologically discrete subset of G . Every rigid motion R : R3 → R3 is of the form R(z) = AR · z + bR where AR ∈ O(R3 ) and bR ∈ R3 . Hence, the set {(AR , bR ) : R ∈ G} ⊂ O(R3 ) × R3 is closed and discrete as well. Since G leaves the compact set x(C) ⊂ R3 invariant, the set {(AR , bR ) : R ∈ G} ⊂ R12 is bounded, hence finite. Thus, G is finite as well. This contradiction to the initial assumption shows that G is not discrete, so it contains a 1-parametric subgroup, say G0 , leaving invariant the compact set x(C). This implies that G0 contains a 1-parametric group of rotations about the same line, and hence, since x is nonflat and complete, x is a catenoid (see Example 2.8.1). This completes the proof of the first assertion in the proposition. For the second one, let x be as above. By the Schwarz reflection principle for minimal surfaces (see Corollary 2.3.6), every affine line contained in x(M) generates a symmetry of the surface given by the reflection about that line, and hence the existence of infinitely many lines in x(M) would imply (by the first assertion in the proposition) that the surface is a catenoid. It is however well known that catenoids do not contain any affine lines, which concludes the proof.   Proof of Theorem 4.7.2. Fix a pair of integers r and m with 2 − 2r ≤ m ≤ 1 ≤ r. Pick an irrational number a ∈ R \ Q and let L1 and L2 be a pair of vectorial lines in R2 × {0} ⊂ R3 making an angle of 2πa at {0} = L1 ∩ L2 . For each j ∈ {1, 2} take a set A j ⊂ L j such that 0 ∈ A j and #A j = m + 6r + 25r3 + 1.

(4.83)

The set Am [r] := A1 ∪ A2 then consists of 2m + 12r + 50r3 + 1 points. Assume that we have a surface x : M → R3 in k≤m ϒk [r] with Am [r] ⊂ x(M). By (4.83) and Theorem 4.6.1 we have that L1 ∪ L2 ⊂ x(M), and hence the Schwarz reflection principle (see Corollary 2.3.6) implies that x(M) is invariant under the reflection RL j : R3 → R3 about L j for j = 1, 2. Since the number a is irrational, the surface x(M) is invariant under an infinite group of symmetries generated by RL1 and RL2 . Proposition 4.7.5 implies that x is either a plane or a catenoid. The first possibility contradicts the assumption x ∈ ϒ [r] (see 4.82), while the second is impossible since a catenoid does not contain any affine lines. This shows that Am [r]  is against the family k≤m ϒk [r], thereby proving the theorem. 

232

4 Complete Minimal Surfaces of Finite Total Curvature

The following corollary to Theorem 4.6.1 and Proposition 4.7.5 provides many examples of closed discrete subsets of R3 which are against the family of all complete minimal surfaces of finite total curvature in R3 . Corollary 4.7.6. Let L be a family of affine lines in R3 such that the reflections about the lines in L generate an infinite group of rigid motions of R3 . For each line L ∈ L choose an infinite subset AL ⊂ L. Then, the set A = L∈L AL is against the family of all open, orientable, complete nonflat minimal surfaces of finite total curvature in R3 , i.e., no such surface contains A. Proof. Assume that there exists a complete immersed minimal surface x : M → R3 of finite total curvature with A ⊂ x(M). Theorem 4.6.1 implies that x(M) contains all affine lines in the family L , and hence, by the Schwarz reflection principle (see Corollary 2.3.6), the surface x(M) is invariant under the group of rigid motions of R3 generated by the reflections about these lines. Since catenoids do not contain any affine lines, Proposition 4.7.5 shows that x is flat.   Corollary 4.7.6 shows for instance that no complete nonflat minimal surface of finite total curvature in R3 contains the discrete set Z+ × {0, 1} × {0} ⊂ R3 . One can easily find many more examples of sets with this property by using the corollary. On the other hand, if the set A in the corollary is countable, then it is contained in a complete nonflat minimal surface in R3 by Theorems 3.8.2 and 3.9.1, and Corollary 4.7.6 implies that every such surface has infinite total curvature. The following corollary in the same direction is an immediate consequence of the fact that complete minimal surfaces of finite total curvature have finitely many ends (see Theorem 4.1.1) and of the Jorge–Meeks theorem (see Theorem 4.1.3). Corollary 4.7.7. Let F be an infinite family of vectorial planes in R3 , and let A ⊂ R3 be a discrete set such that for any plane Π ∈ F there is a sequence {a j } j∈N ⊂ A \ {0} with lim |a j | = ∞

j→∞

and

lim

j→∞

dist(a j , Π ) = 0. |a j |

Then A is against the family of all open, orientable, complete minimal surfaces of finite total curvature in R3 .

Chapter 5

The Gauss Map of a Minimal Surface

The Gauss map of a minimal surface is of major importance in the theory of minimal surfaces in Euclidean spaces. In this chapter we review some of the classical results of this theory and present recent developments on the subject. Recall from Section 2.5 that the Gauss map of a conformal minimal immersion X = (X1 , . . . , Xn ) : M → Rn (n ≥ 3) from an open Riemann surface is the holomorphic map G : M → Qn−2 ⊂ CPn−1 given by G = [∂ X1 : ∂ X2 : · · · : ∂ Xn ], where Qn−2 = {[z1 : · · · : zn ] ∈ CPn−1 : ∑nj=1 z2j = 0}. A recent discovery of the authors [32] is that, conversely, every holomorphic map M → Qn−2 is the Gauss map of a conformal minimal immersion M → Rn ; see Theorem 5.4.1. This is the main one among the new results presented in this chapter. Before proceeding with that, we develop in Sections 5.1 and 5.2 the technical tools required in the proof of this result. These are somewhat analogous to those in Section 3.2, but with the additional control of the Gauss map. Let M be a compact bordered Riemann surface, and let θ be a nowhere vanishing holomorphic 1-form on M. In Lemma 5.1.2 we show that for any given full holomorphic map f : M → Cn (n ≥ 2) there exists a period dominating spray of multipliers, i.e., a holomorphic spray of maps ht : M → C (t ∈ CN ), with h0 = 1, such that the period map, sending t ∈ CN to the periods of ht f θ on a given basis of the homology group H1 (M, Z), has maximal rank at t = 0. When n ≥ 3 and f has range in the null quadric A∗ ⊂ Cn (2.54), it follows that the space of conformal minimal immersions M → Rn of class C r (M) (r ∈ Z+ ) with the Gauss map π ◦ f : M → CPn−1 is a real analytic Banach manifold (see Theorem 5.1.3). Here, π : Cn∗ → CPn−1 is the canonical projection (2.74). In Section 5.3 we go further and prove that for any homotopy of full holomorphic maps ft : M → Cn (t ∈ [0, 1]) from an open Riemann surface M there is a homotopy of multipliers ht : M → C∗ which give prescribed periods of ht ft θ for any given nowhere vanishing holomorphic 1-form θ on M (see Theorem 5.3.1). If every map ft assumes values in the null quadric, then by choosing the functions ht such that h1 f1 θ has vanishing real periods we obtain by the Enneper–Weierstrass formula a conformal minimal immersion M → Rn with the Gauss map π ◦ f1 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_5

233

234

5 The Gauss Map of a Minimal Surface

Theorem 5.3.1 easily leads to the proof of Theorem 5.4.1 (see Section 5.4) to the effect that every holomorphic map M → Qn−2 is the Gauss map of a conformal minimal immersion M → Rn . In Section 5.5 we return to the classical global theory and study the value distribution properties of the Gauss map of complete minimal surfaces in Rn . In particular, we present H. Fujimoto’s theorem from [152] asserting that the Gauss map of a complete nonflat minimal surface in R3 can omit at most four points of the sphere (see Theorem 5.5.2). In higher dimensions, we recall M. Ru’s theorem [305] to the effect that the Gaussian image of a complete nonflat minimal surface in Rn (n ≥ 3) cannot fail to intersect a set of more than n(n + 1)/2 hyperplanes in CPn−1 in general position (see Theorem 5.5.4). In Section 5.6 we go in the opposite direction and prove that every open Riemann surface, M, admits a complete conformal minimal immersion M → Rn (n ≥ 3) whose Gauss map M → Qn−2 ⊂ CPn−1 is full and fails to intersect n hyperplanes of CPn−1 in general position (see Theorem 5.6.1). In particular, we show that there is a complete nonflat conformal minimal immersion M → R3 whose Gauss map M → CP1 omits any two given points of CP1 (see Corollary 5.6.4). These results, which are sharp by Picard’s theorem, are obtained as application of the approximation theorems for conformal minimal immersions, proved in Chapter 3. Period dominating multipliers constructed in Sections 5.1 and 5.3 have other applications. In particular, we obtain in Section 5.7 some general existence results for smooth 1-parameter families of conformal minimal immersions M → Rn with remarkable properties. One of the main results of that section is Corollary 5.7.7 which identifies the path components of the space of conformal minimal immersions M → Rn (n ≥ 3) from any open connected Riemann surface M.

5.1 Period Dominating Sprays of Multipliers In this section, M denotes a compact connected bordered Riemann surface. We prove a version of Lemma 3.2.1 in which a period dominating spray is obtained by multiplying a given holomorphic map M → Cn by a holomorphic function on M without zeros, called a multiplier; see Lemma 5.1.2. This will be used in Section 5.4 to construct minimal surfaces in Rn with a prescribed Gauss map. Recall (see Definition 2.5.2) that a path f : I = [0, 1] → Cn (n ∈ N) is said to be full if the C-linear span of its image equals Cn , and everywhere full if f |I  : I  → Cn is full for every nontrivial subinterval I  ⊂ I. Obviously, an everywhere full path is full; the converse holds for real analytic paths. We denote by P : C (I, Cn ) → Cn the period map P( f ) =

 1 0

f (s) ds ∈ Cn ,

The following result is [32, Lemma 2.1].

f ∈ C (I, Cn ).

(5.1)

5.1 Period Dominating Sprays of Multipliers

235

Lemma 5.1.1. Assume that I  is a nontrivial closed subinterval of I = [0, 1] and Q is a compact Hausdorff space. Given a continuous map f = ( f1 , . . . , fn ) : I × Q → Cn such that f (·, q) is full on I  for every q ∈ Q, there are finitely many continuous functions g1 , . . . , gN : I → C supported on I  such that the function h : I × CN → C given by N

h(s,t) = 1 + ∑ ti gi (s),

s ∈ I, t = (t1 , . . . ,tN ) ∈ CN

(5.2)

i=1

is a period dominating multiplier of f , in the sense that  ∂  P h(·,t) f (·, q)  : CN → Cn is surjective for every q ∈ Q. ∂t t=0

(5.3)

Proof. If h is given by (5.2) then   1  1  ∂ ∂ h(s,t)   P(h(·,t) f (·, q)) = f (s, q) ds = gi (s) f (s, q) ds ∂ti ∂ti t=0 t=0 0 0

(5.4)

for all q ∈ Q and i ∈ {1, . . . , N}. We now show how to find functions gi : I → C such that (5.3) holds. Since f (·, q) is full on I  for every q ∈ Q, compactness of Q and continuity of f ensure that there are points s1 , . . . , sN ∈ I˚ for a big N ∈ N such that   span f (s1 , q), . . . , f (sN , q) = Cn for all q ∈ Q. (5.5) Choose ε > 0 such that (si − ε, si + ε) ⊂ I  for all i ∈ {1, . . . , N}, and for each i ∈ {1, . . . , N} pick a continuous function gi : I → C supported on (si − ε, si + ε) such that   1

0

si +ε

gi (s) ds =

si −ε

gi (s) ds = 1.

(5.6)

If ε > 0 is chosen sufficiently small, then (5.4) and (5.6) ensure that  1 ∂ P(h(·,t) f (·.q))  gi (s) f (s, q) ds ≈ f (si , q)  = ∂ti t=0 0

for all q ∈ Q and i ∈ {1, . . . , N}. Assuming as we may that these approximations are close enough, (5.3) follows from (5.5).   Let M be a compact connected bordered Riemann surface and θ be a of holomorphic 1-form on M without zeros. Let C1 , . . . ,Cl (l ∈ N) be a collection ˚ Set C = li=1 Ci and smooth oriented embedded arcs and closed Jordan curves in M. consider the period map P = (P1 , . . . , Pl ) : C (C, Cn ) → (Cn )l given by (3.12) for this family of curves: C (C, Cn ) f −→ Pi ( f ) =

 Ci

f θ ∈ Cn ,

i = 1, . . . , l.

(5.7)

An application of Lemma 5.1.1 and of the Mergelyan approximation theorem gives the following result which is a minor generalization of [32, Lemma 3.2].

236

5 The Gauss Map of a Minimal Surface



Lemma 5.1.2. Assume that the compact set C = lj=1 C j is Runge in M and every curve C j contains a nontrivial arc I j disjoint from k= j Ck . Given a compact Hausdorff space Q, a continuous map f = ( f1 , . . . , fn ) : M × Q → Cn such that f (·, q) : M → Cn is a full map of class A (M) for every q ∈ Q, an integer r ∈ Z+ , and a finite set P ⊂ M, there are finitely many functions g1 , . . . , gN ∈ O(M) such that each gi vanishes to order r at every point p ∈ P

(5.8)

and the function h : M × CN → C, given by N

h(p,t) = 1 + ∑ ti gi (p),

p ∈ M, t = (t1 , . . . ,tN ) ∈ CN

i=1

is a period dominating multiplier of f , meaning that  ∂  P h(·,t) f (·, q)  : CN → (Cn )l is surjective for every q ∈ Q. ∂t t=0

(5.9)

The same result holds if M is a compact admissible subset of an open Riemann surface and the map f (·, q) is everywhere full (i.e., full on every relatively open subset of M; see Definitions 1.12.9 and 3.1.2 and Remark 3.2.2). By (5.8), for every t ∈ CN the function h(·,t) agrees with the constant 1 to order r at every point of P. Proof. For each j = 1, . . . , l we choose a nontrivial arc I j ⊂ C j which is disjoint from P ∪ ( k= j Ck ). Fix a parameterization γ j : [0, 1] → C j with I j ⊂ γ j ((0, 1)). The assumptions on f imply that the map f (·, q) ◦ γ j : [0, 1] → Cn is everywhere full for every q ∈ Q and j ∈ {1, . . . , l}. For each q ∈ Q we denote by P f ,q = (P1f ,q , . . . , Plf ,q ) : C (C) → (Cn )l the map whose j-th component is given by P jf ,q (g) =

 Cj

g f (·, q)θ ,

g ∈ C (C).

Lemma 5.1.1 furnishes for each j ∈ {1, . . . , l} an integer N j ≥ n and continuous functions g j,k : C → C (k = 1, . . . , N j ) with support on I j such that, setting Nj

h j (p,t j ) = 1 + ∑ t j,k g j,k (p),

p ∈ C j , t j = (t j,1 , . . . ,t j,N j ) ∈ CN j ,

k=1

we have that  ∂  P jf ,q (h j (·,t j )) : T0 CN j ∼ = CN j → Cn is surjective for every q ∈ Q. (5.10) ∂t j t j =0 Applying Mergelyan’s theorem with jet-interpolation (see Theorem 1.12.11 and Remark 1.12.12), we can approximate each g j,k uniformly on C by a holomorphic function gj,k ∈ O(M) vanishing to order r at every point p ∈ P. Set N = ∑lj=1 N j ≥ nl, identify CN = CN1 × · · · × CNl , and define

5.1 Period Dominating Sprays of Multipliers l

h(p,t) = 1 + ∑

237

Nj

∑ t j,k gj,k (p),

p ∈ M, t = (t1 , . . . ,tl ) ∈ CN1 × · · · × CNl .

j=1 k=1

Note that (5.8) is satisfied. If the approximations are close enough then (5.10) and the fact that each | g j,k | is small on C \ I j ensure that  ∂  P f ,q (h(·,t)) : T0 CN ∼ = CN → (Cn )l is surjective for every q ∈ Q, ∂t t=0 thereby completing the proof. (The details of the last argument are explained more carefully in the proof of Lemma 3.2.1.)   In light of the existence of period dominating sprays in Lemma 5.1.2 and following the spirit of Theorem 3.2.3, it is a natural problem to describe the space of conformal minimal immersions with the given Gauss map. In this direction, we have the following result which compiles [32, Corollaries 6.2 and 6.3]. Theorem 5.1.3. If M is a compact bordered Riemann surface and f : M → Cn∗ (n ∈ N) is a map of class A r−1 (M) for some r ∈ N, then (i) the space of holomorphic immersions F : M → Cn of class A r (M) having the Gauss map G = π ◦ f : M → CPn−1 is a complex Banach manifold. Assuming in addition that n ≥ 3 and f assumes values in the null quadric A∗ ⊂ Cn (2.54), the following assertions hold. (ii) The space of conformal minimal immersions M → Rn of class C r (M) having the Gauss map G = π ◦ f : M → CPn−1 is a real analytic Banach manifold. (iii) For any group homomorphism q : H1 (M, Z) → spanC ( f (M)) ∩ {z ∈ Cn : ℜ(z) = 0} ⊂ Cn the space of conformal minimal immersions X : M → Rn of class C r (M) having the Gauss map G = π ◦ f : M → CPn−1 and the flux map FluxX = iq : H1 (M, Z) → Rn is a real analytic Banach manifold. (iv) The space of holomorphic null immersions Z : M → Cn of class A r (M) having the Gauss map G = π ◦ f : M → CPn−1 is a complex Banach manifold. Remark 5.1.4. (A) Note that the Banach manifolds in Theorem 5.1.3 are nonempty in view of Theorem 5.4.1. (B) We remark that [32, Corollaries 6.2 and 6.3] were mistakenly stated with the assumption that f is of class A (M) (instead of A r−1 (M)). This weaker assumption suffices in Theorem 5.1.5 since the periods are computed on curves contained in the interior of M. However, in the proof of Theorem 5.1.3, the integral M p → Z(p) =   z0 + pp0 h f θ (see (5.11)) is a map of class A r (M) provided that f ∈ A r−1 (M) .  The proof of Theorem 5.1.3 depends on the following technical result.

238

5 The Gauss Map of a Minimal Surface

Theorem 5.1.5. Let M be a compact bordered Riemann surface, θ be a holomorphic 1-form on M without zeros, f : M → Cn∗ (n ∈ N) be a map of class A (M), and r ∈ Z+ be an integer. The following assertions hold true. (a) For any group homomorphism q : H1 (M, Z) → span( f (M)) ⊂ Cn the space of functions h ∈ A r (M, C∗ ) satisfying  γ

h f θ = q(γ) for every closed curve γ ⊂ M

is a complex Banach manifold with the natural C r (M) topology. (b) For any group homomorphism q : H1 (M, Z) → span( f (M)) ⊂ Cn , the space of functions h ∈ A r (M, C∗ ) satisfying  γ

ℜ(h f θ ) = ℜ(q(γ))

for every closed curve γ ⊂ M

is a real analytic Banach manifold with the natural C r (M) topology. Theorem 5.1.5 is obtained by following the proof of Theorem 3.2.3, but using Lemma 5.1.2 instead of Lemma 3.2.1. See [32, proof of Theorem 6.1] for the details. Proof of Theorem 5.1.3. Let θ be a holomorphic 1-form vanishing nowhere on M. We begin with (i). By Theorem 5.1.5 (a), applied with r − 1 ∈ Z+ and the group homomorphism q ≡ 0, the space Σ = {h ∈ A r−1 (M, C∗ ) : h f θ is exact on M} is a complex Banach manifold with the C r−1 (M) topology. The integration map M p −→ Z(p) = z0 +

 p p0

h fθ,

(5.11)

with any choice of the base point p0 ∈ M and the initial value z0 ∈ Cn , provides an isomorphism between the Banach manifold Σ × Cn and the space of holomorphic immersions Z : M → Cn of class A r (M) having the Gauss map π ◦ f ; hence the latter is also a complex Banach manifold. This shows (i), which in turn trivially implies (iv). We now prove (ii) and (iii). By Theorem 5.1.5 (b), applied with r − 1 and q ≡ 0, the space Σ  = {h ∈ A r−1 (M, C∗ ) : ℜ(h f θ ) is exact on M} is a real analytic Banach manifold with the Cr−1 (M) topology. Fix a point p0 ∈ M. The integration M p → X(p) = x0 + pp0 ℜ(h f θ ) (x0 ∈ Rn ) provides an isomorphism between the Banach manifold Σ  × Rn and the space of conformal minimal immersions X : M → Rn of class A r (M) with the Gauss map π ◦ f , so the latter is a Banach manifold as well. This proves (ii). Assertion (iii) follows from the same argument applied to the homomorphism −q and using Theorem 5.1.5 (a) instead of Theorem 5.1.5 (b).  

5.2 Paths With Given Integrals

239

5.2 Paths With Given Integrals In this section we construct paths in the punctured null quadric A∗ = A \ {0} ⊂ Cn (2.54) with prescribed integrals and also with prescribed Gauss map, an additional feature which will be crucial in Section 5.4. (A result of this type without paying attention to the Gauss map is given by Lemma 3.5.4.) We present the following 1-parametric version of [32, Lemma 2.3]. We shall write I = [0, 1]. Lemma 5.2.1. Let α : I → Cn and f : I × I → Cn be continuous maps such that the path ft := f (· ,t) : I → Cn is everywhere full for each t ∈ I (see p. 234). Then there is a continuous function h : I × I → C∗ such that h(s,t) = 1 for all (s,t) ∈ {0, 1} × I and  1

0

h(s,t) f (s,t) ds = α(t),

t ∈ [0, 1].

(5.12)



If in addition we have that 01 f (s, 0) ds = α(0), then the function h can be chosen such that h(s, 0) = 1 for all s ∈ [0, 1]. Proof. As in the proof of Lemma 3.5.4, it suffices to prove the lemma with the exact condition (5.12) replaced by the approximate condition  1     h(s,t) f (s,t) ds − α(t) < ε, t ∈ [0, 1] (5.13)   0

for an arbitrary ε > 0. Indeed, choose a pair of disjoint closed subintervals I1 , I2 of I˚ = (0, 1). It then suffices to obtain the approximate condition (5.13) for a small enough ε > 0 by a multiplier function h1 which equals 1 for all s ∈ I \ I1 , and then correct the error by applying Lemma 5.1.1 with a function h2 that equals 1 for all s ∈ I \ I2 . The product h = h1 h2 then satisfies the conclusion of the lemma. It remains to construct a function h satisfying (5.13). Since the map ft = f (· ,t) is full for each t ∈ I, there is a division 0 = s0 < s1 < · · · < sN = 1 of I such that for all t ∈ I we have span{ ft (s1 ), . . . , ft (sN )} = Cn . Set  sj

V j (t) =

s j−1

j = 1, . . . , N.

ft (s) ds,

Note that V j (t) is close to (s j − s j−1 ) ft (s j ) if the intervals [s j−1 , s j ] are sufficiently small.  Thus, passing to a finer division if necessary we may assume that span V1 (t), . . . ,VN (t) = Cn holds for all t ∈ I. For each t ∈ I we denote by Σt the affine complex hyperplane of CN given by  Σt = (g1 , . . . , gN ) ∈ CN :

N



∑ g jV j (t) = α(t)

.

j=1

Clearly, there is a continuous map g = (g1 , . . . , gN ) : I → CN such that g(t) ∈ Σt for every t ∈ I. (Note that g can be viewed as a section of the affine bundle over I whose fibre over the point t equals Σt .) It follows that

240

5 The Gauss Map of a Minimal Surface

 sj

N

∑

j=1 s j−1

N

g j (t) ft (s) ds =

∑ g j (t)V j (t) = α(t),

t ∈ I.

(5.14)

j=1





Since ∑Nj=1 V j (t) = 01 ft (s) ds, we infer that if 01 f (0, s) ds = α(0) then the map g can be chosen with g(0) = (1, . . . , 1) ∈ CN . We assume this to be the case, for the proof is even simpler otherwise. Further, by a small perturbation of each function g j ( j = 1, . . . , N) we may assume that it has no zeros on I, at the cost of replacing the exact condition (5.14) by the approximate condition    N  sj  ε   (5.15) g j (t) ft (s) ds − α(t) < , t ∈ I. ∑  j=1 s j−1  2 Here we have used that the parameter space I is real one-dimensional. For a fixed t ∈ I we consider the vector g(t) = (g j (t)) j ∈ (C∗ )N as a step function of s ∈ I which equals the constant g j (t) on s ∈ [s j−1 , s j ) for every j = 1, . . . , N. We now approximate this step function by a continuous function ht = h(· ,t) : I → C∗ which agrees with the step function except near the points s0 , s1 , . . . , sN , ensuring also that ht (0) = ht (1) = 1. Here are the details. Fix a constant C > 1 so large that max | f (s,t)| ≤ C,

max

t∈I, j=1,...,N

(s,t)∈I×I

|g j (t)| ≤ C.

(5.16)

For each j = 1, . . . , N there is a homotopy of continuous functions g j,τ : I → C∗ (0 ≤ τ ≤ 1) satisfying the following conditions. • g j,0 (t) = 1 for all t ∈ I. • g j,1 (t) = g j (t) for all t ∈ I. • g j,τ (0) = 1 for all τ ∈ [0, 1] (i.e., the homotopy is fixed at t = 0). • |g j,τ (t)| ≤ C for all t ∈ I and τ ∈ [0, 1]. Pick a number η satisfying 0 0 such that |K(p)| ≤

C , d(p)2

p ∈ M,

where K is the Gauss curvature of the metric g = X ∗ (ds2 ) and d(p) denotes the intrinsic distance in the metric g from p ∈ M to the boundary of M. If M is complete in the induced metric g, then d(p) = +∞ for every point p ∈ M and hence K ≡ 0, so X is flat and we get Theorem 5.5.2. Before recalling Fujimoto’s proof of Theorem 5.5.2 (see p. 253), we discuss the question of sharpness and the related results in higher dimensions. The Gauss map of Scherk’s minimal surfaces (see Subsection 2.8.3) omits four points of the sphere, and hence Fujimoto’s theorem is sharp. Moreover, Osserman gave in [279, Theorem 2] the following example of a complete, simply connected minimal surface in R3 whose normals omit precisely 4 directions. Consider the holomorphic function G on M := C \ {2kπi : k ∈ Z} given by G(z) =

1 , 1 − ez

z ∈ M.

(5.24)

The universal covering of M is the unit disc D = {z ∈ C : |z| < 1}. Denote by π : D → M the covering projection. Consider on D the following Weierstrass data:  φ3 (z) = dπ(z), g(z) = G(π(z)), z ∈ D.

5.5 The Gaussian Image of Complete Minimal Surfaces I

251

Since neither φ3 nor g vanish anywhere on D and g is holomorphic, the Weierstrass formula (2.84) for these data provides a conformal minimal immersion X : D → R3 which is easily seen to be complete. Moreover, the choice of G in (5.24) ensures that g omits precisely the values 0, 1, −1, and ∞ of CP1 . A bit later, K. Voss [335] proved that there are minimal surfaces such as those shown by Osserman with the complex Gauss map omitting any given four points of CP1 . For that, choose k ∈ N pairwise distinct complex numbers a1 , . . . , ak and consider on the open Riemann surface M = CP1 \ {a1 , . . . , ak } the Weierstrass data φ3 =

z dz ∏kj=1 (z − a j )

,

g = z.

Denote by X : D → R3 the associated conformal map given by (2.84) which is well defined on the universal covering D of M. It turns out that X is a minimal immersion if and only if k ≤ 4 (for k ≥ 5 there is a branch point at infinity); if this is the case then X is complete and its complex Gauss map omits the set {a1 , . . . , ak } ⊂ CP1 . Both Osserman’s and Voss’s examples are explicit and simply connected. F. J. L´opez obtained in [228] a general existence result for complete nonflat minimal surfaces in R3 of arbitrary genus and finite topology, and with the complex Gauss map omitting four points of the sphere. H. Fujimoto [150, 153] proved a Xavier type theorem in arbitrary dimension, showing that the Gaussian image of a complete full minimal surface in Rn (n ≥ 3) cannot omit a set of more than n(n + 1)/2 hyperplanes of CPn−1 in general position. The best result in this direction is due to M. Ru [305, Corollary, p. 412], who proved that the assumption in Fujimoto’s theorem from [153], that the complete minimal surface is full, may be relaxed to nonflat. Ru’s result is the following. Theorem 5.5.4 (Ru, [305]). The Gaussian image of a complete nonflat minimal surface in Rn for any n ≥ 3 cannot omit more than n(n + 1)/2 hyperplanes in CPn−1 in general position. The bound n(n + 1)/2 in Ru’s theorem is the best possible whenever n is odd or 3 ≤ n ≤ 17, as Fujimoto showed by examples in [151]. It remains an open problem whether this bound is sharp for even integers n ≥ 18. Let us illustrate that Theorem 5.5.4 is sharp in dimension n = 3. For any four distinct points a1 , . . . , a4 in CP1 there is a set of six projective lines H1 , . . . , H6 in CP2 in general position such that ( 6j=1 H j ) ∩ Q1 = τ({a1 , . . . , a4 }), where τ : CP1 → Q1 is the biholomorphic map (2.77). Indeed, up to a change of indices the lines Hi = τ(ai )τ(ai ), i = 1, . . . , 4, H5 = τ(a1 )τ(a2 ), and H6 = τ(a3 )τ(a4 ) satisfy this condition, where τ(ai )τ(a j ) denotes the tangent line to Q1 at τ(ai ) if i = j and the projective line passing through τ(ai ) and τ(a j ) if i = j (see [150]). Since the Gauss map G : M → Q1 ⊂ CP2 and the complex Gauss map g : M → CP1 of a conformal minimal immersion X : M → R3 are related by G = τ ◦ g (see (2.80)), this shows that the Gauss map of the aforementioned examples by R. Osserman [279], K. Voss [335], and F. J. L´opez [228] fail to intersect six hyperplanes of CP2 in general position, thereby showing that Theorem 5.5.4 is sharp for n = 3.

252

5 The Gauss Map of a Minimal Surface

We now give Fujimoto’s proof of Theorem 5.5.2 from [152]. It is based on the following Schwarz–Pick type lemma, see [152, Lemma 3.1]. Lemma 5.5.5. Take a pair of numbers ε > 0 and 0 < 4ε < 3 −

2 3

< p < 1 with

2 < 1. p

(5.25)

If h : DR = {w ∈ C : |w| < R} → C is a holomorphic function omitting four distinct values a1 , . . . , a4 ∈ C, then there is a number C > 0 such that 

1 + |h(w)|2

1/p

|h (w)|

∏4j=1 |h(w) − a j |1−ε

≤C

2R , R2 − |w|2

w ∈ DR .

Proof. We sketch the proof given by X. Mo and R. Osserman in [263]. Recall that the Poincar´e metric on the unit disc D = {|z| < 1} equals dσ02 = ρ0 (z)2 |dz|2 =

4|dz|2 2 , where ρ0 (z) = . 2 2 (1 − |z| ) 1 − |z|2

This metric is complete and has constant Gaussian curvature −1 (see Example 2.1.5). Let F : D → Ω denote the universal holomorphic covering of the domain Ω = C \ {a1 , . . . , a4 }. Since the deck transformations of F are holomorphic automorphism of D which are dσ02 -isometries, there is a unique complete metric dσ 2 = ρ(z)2 |dz|2 on Ω with constant curvature −1 (which is equivalent to Δ log ρ = ρ 2 by (2.32)) such that F ∗ (dσ 2 ) = dσ02 . This metric is called the Poincar´e metric of Ω . Furthermore, it is known that the limits lim ρ(z)|z − a j | log |z − a j |,

z→a j

lim ρ(z)|z| log |z|

z→∞

exist and are different from zero (see R. Nevanlinna [272, p. 250]). Therefore, in view of the choices of the constants p and ε in (5.25) there is a constant C > 0 such that (1 + |z|2 )1/p ≤ Cρ(z), z ∈ Ω . (5.26) ∏4j=1 |z − a j |1−ε Indeed, near each point a j such an estimate holds since |z − a j | · | log |z − a j || ≤ c|z − a j |1−ε for some c > 0. At infinity, the left-hand side of (5.26) is asymptotic to 1/|z|4−4ε−2/p while ρ(z) is asymptotic to 1/|z|| log |z||. Conditions (5.25) imply 4 − 4ε − 2/p > 1, which gives the desired inequality for a suitable value of C. (Note that the proof fails at this point with less than four exceptional points a j .) In the complement of

5.5 The Gaussian Image of Complete Minimal Surfaces I

253

any neighbourhood of the points {a1 , . . . , a4 , ∞} in CP1 the estimate (5.26) holds by compactness. Suppose now that h : DR → Ω is a holomorphic map. Since DR is simply connected and F : D → Ω is a covering, there is a holomorphic map h0 : DR → D such that h = F ◦ h0 . By the Schwarz–Pick lemma (see (2.34)) we have that ρ0 (h0 (w))|h0 (w)| =

2|h0 (w)| 2R ≤ 2 , 2 1 − |h0 (w)| R − |w|2

w ∈ DR .

Since F : (D, dσ02 ) → (Ω , dσ 2 ) is an isometry, it follows that ρ(h(w))|h (w)| ≤

2R R2 − |w|2

,

w ∈ DR .

(5.27)

Taking z = h(w) ∈ Ω in the estimate (5.26) gives (1 + |h(w)|2 )1/p ≤ Cρ(h(w)), ∏4j=1 |h(w) − a j |1−ε

w ∈ DR .

Together with (5.27) this completes the proof.   Proof of Theorem 5.5.2. We reason by contradiction. Assume that X : M → R3 is a complete nonflat conformal minimal immersion whose Gauss map g : M → CP1 (2.79) omits five points. Let (g, φ3 ) be the Weierstrass data (2.83) of X. Up to a rigid motion of R3 we may assume that g(M) ⊂ Ω = CP1 \ {a1 , a2 , a3 , a4 , ∞}, where {a1 , a2 , a3 , a4 } ⊂ C are pairwise distinct points. Set M  = {p ∈ M : dg(p) = 0}. The key step is to prove that M  carries a complete flat conformal metric ds21 . If so, then Huber’s theorem (see Theorem 2.6.4) implies that the Riemann surface M  is parabolic and of finite topology. Since the map g|M : M  → C is holomorphic, nonconstant, and it omits four values of C, the big Picard theorem implies that none of the ends of M  is an essential singularity of g. It follows that X : M → R3 is a complete minimal immersion of finite total curvature. Since its complex Gauss map omits five points of CP1 , Osserman’s three points theorem (see Theorem 4.1.8) shows that X is flat, a contradiction. The construction of the metric ds21 is explicit. Choose p and ε as in Lemma 5.5.5 and consider the conformal metric on M  defined by  ds21

=

4

∏ |g − a j | j=1

p(1−ε)/(1−p)

|φ3 |1/(1−p) 1/(1−p) |g| |dg| p/(1−p)

2 .

(5.28)

Note that the zeros of φ3 and g cancel as seen from (2.83). Since the difference of the p 1 − 1−p = 1, ds21 is indeed exponents of |φ3 | and |dg| in the above formula equals 1−p a conformal metric, and it is flat in view of (2.32). To see that ds21 is complete, we

254

5 The Gauss Map of a Minimal Surface

have to show that any divergent arc γ(t) : [0, 1) → M  has infinite length 1 (γ) in (M  , ds21 ). We distinguish two possibilities: (i) γ(t) diverges to a point of M \ M  = (dg)−1 (0) as t → 1. (ii) γ(t) diverges in M as t → 1. Suppose that γ(t) diverges to p0 ∈ M \ M  (case (i)). Pick a conformal parameter z around p0 with z(p0 ) = 0. Since p0 is a branch point of g, we have dg(z) = p > 2 and hence (czm + o(zm ))dz for some m ≥ 1 and c = 0. As p > 23 , we have 1−p ds1 (z) ∼

|dz| c |φ3 |1/(1−p) > c 2 . 1/(1−p) p/(1−p) |z| |g| |dg|

Taking into account that z(γ(t)) diverges to z(p0 ) = 0 at t → 1, we infer that 1 (γ) =

 γ

ds1 ≥ c

 γ

|dz| =∞ |z|2

and we are done. Assume now that γ(t) ⊂ M  is divergent in M (case (ii)). Reason by contradiction and suppose that 1 (γ) < +∞. Then, the biggest geodesic disc D in (M  , ds21 ) centred at γ(0) has finite radius R > 0, and there exists a geodesic ray Γ in (D, ds21 ) with the initial point γ(0) which is divergent in M  , hence also divergent in M by case (i). Consider the exponential map at γ(0) associated to ds21 ,  expγ(0) : DR , |dz|2 ) → (D, ds21 ), where we are identifying Tγ(0) M  with C, DR = {|z| < R} ⊂ C, and without loss of generality Γ = expγ(0) ([0, R)). Since ds21 is flat, the Cartan–Hadamard theorem (see M. P. do Carmo [110, p. 163]) implies that expγ(0) is a local isometry, hence a conformal map. Consider the conformal parameter (DR , expγ(0) ) on M  and write h = g ◦ expγ(0) : DR → C,

exp∗γ(0) (φ3 ) = f3 (z)dz

for the Weierstrass data of the conformal minimal immersion X ◦ expγ(0) : DR → R3 . Since expγ(0) : (DR , |dz|2 ) → (D, ds21 ) is a local isometry, we have exp∗γ(0) (ds21 )(z) = |dz|2 . Hence, from the definition of ds21 we see that |h (z)| p | f3 (z)| . = 4 |h(z)| ∏ j=1 |h(z) − α j | p(1−ε) This shows that the length of Γ in (M, X ∗ (ds20 )) equals

5.6 The Gaussian Image of Complete Minimal Surfaces II

0 (Γ ) =

1 2

255

 R | f3 (z)|  0

  1 R (1 + |h(z)|2 )|h (z)| p |dz|. 1 + |h(z)|2 |dz| = |h(z)| 2 0 ∏4j=1 |h(z) − α j | p(1−ε)

Applying Lemma 5.5.5 to h, this can be estimated by 0 (Γ ) ≤

Cp 2

 R 0

2R p (2C) p |dz| = 2 2 R − |z| 2R p−1

 1 0

dt < +∞, (1 − t 2 ) p

(5.29)

where the last inequality holds since p < 1. This contradicts the assumption that (M, X ∗ (ds20 )) is complete, hence 1 (γ) = +∞ and (M  , ds21 ) is complete.   Remark 5.5.6. The assumption p < 1 is crucial to ensure the estimate (5.29). On the other hand, the inequality (5.26) holds true for p < 1 if and only if the number of exceptional values a j ∈ C is at least four, so the number of exceptional values of g in CP1 is at least five. The proof of Fujimoto’s theorem breaks down without this condition, and it cannot be otherwise in view of the examples.  

5.6 The Gaussian Image of Complete Minimal Surfaces II By the Fujimoto–Ru Theorem 5.5.4, the maximum number of hyperplanes in CPn−1 in general position that can be omitted by the Gaussian image of a complete full minimal surface in Rn does not exceed n(n + 1)/2. However, the true maximum number depends on the underlying conformal structure of the surface and may be smaller than n(n + 1)/2. For instance, it was proved by L. V. Ahlfors [4] that a full holomorphic map G : C → CPn−1 (see Definition 2.5.2) can fail to intersect at most n hyperplanes of CPn−1 in general position. Hence, the Gauss map of a full conformal minimal immersion C → Rn cannot omit n + 1 hyperplanes in general position, even without asking that the surface be complete. Similarly, by the little Picard theorem, the complex Gauss map of a nonflat conformal minimal immersion C → R3 can omit at most two values. By the big Picard Theorem, the same holds true for any complete conformal minimal immersion M → R3 of infinite total curvature from a finitely punctured compact Riemann surface M, for in such case at least one of the punctures is an essential singularity of the Gauss map. In this section we prove that, in fact, every open Riemann surface admits a complete conformal minimal immersion in Rn for any n ≥ 3 whose Gauss map has n exceptional hyperplanes in general position, and whose complex Gauss map omits two values of CP1 if n = 3. This shows that the bounds mentioned in the previous paragraph can be achieved on any open Riemann surface. The following theorem is a compilation of results by A. Alarc´on, I. Fern´andez, and F. J. L´opez [17, 18] and by I. Castro-Infantes [78].

256

5 The Gauss Map of a Minimal Surface

Theorem 5.6.1. Let n ≥ 3. Every open Riemann surface, M, admits a complete conformal minimal immersion X : M → Rn whose Gauss map G : M → Qn−2 ⊂ CPn−1 is full and fails to intersect n hyperplanes of CPn−1 in general position. Furthermore, given a group homomorphism p : H1 (M, Z) → Rn and a map ϕ : P → Rn on a closed discrete subset P ⊂ M, we may choose X such that FluxX = p and X(p) = ϕ(p) for all p ∈ P. Proof. Let M, p = (p1 , . . . , pn ), and ϕ = (ϕ1 , . . . , ϕn ) be as in the statement. Case 1: n = 2k ≥ 4 is even. Let θ be an exact holomorphic 1-form vanishing nowhere on M (see Theorem 1.10.5). Let K ⊂ M \ P be a compact, smoothly bounded, simply connected domain. Fix a point p0 ∈ K˚ and numbers λ1 , . . . , λk ∈ C∗ satisfying k

∑ λ j2 = 0.

(5.30)

j=1

Choose holomorphic functions g j : K → C∗ = C \ {0} for j = 1, . . . , k such that the holomorphic map f = ( f1 , f2 , . . . , fn ) : K → Cn defined by     λ j2 λ j2 1 i gj − gj + , f2 j = , j ∈ {1, . . . , k} (5.31) f2 j−1 = 2 gj 2 gj is full. Note that f22j−1 + f22j = ( f2 j−1 − i f2 j )( f2 j−1 + i f2 j ) = −λ j2 = 0

for every j,

(5.32)

and hence (5.30) ensures that f takes values in the punctured null quadric An−1 ∗ . Moreover, for each j the map ( f2 j−1 , f2 j , λ j ) : K → C3 assumes values in A2∗ and, since λ j = 0 and in view of (5.32), the map [ f2 j−1 : f2 j : λ j ] : K → Q1 ⊂ CP2 fails to intersect the following two hyperplanes of CP2 in general position:   [z1 : z2 ] ∈ CP2 : z1 + (−1)ε iz2 = 0 , ε = 1, 2. (5.33) Since K is simply connected and θ is exact, the map Y j = (Y j,1 ,Y j,2 ,Y j,3 ) : K → R3 given by  p

Y j (p) = ℜ

p0

( f2 j−1 , f2 j , λ j )θ ,

p∈K

(5.34)

is a well defined conformal minimal immersion whose third component is harmonic on M. In this situation, Theorem 3.7.1 and Theorem 3.9.1 iii) enable us to approximate Y j uniformly on K by a complete conformal minimal immersion Yj = (Yj,1 , Yj,2 , Yj,3 ) : M → R3 satisfying the following conditions. (1 j ) Yj,3 = Y j,3 . (2 j ) FluxYj = (p2 j−1 , p2 j , 0). (3 j ) (Yj,1 , Yj,2 )(p) = (ϕ2 j−1 , ϕ2 j )(p) for all p ∈ P.

5.6 The Gaussian Image of Complete Minimal Surfaces II

257

Consider the map X = (Y1,1 , Y1,2 , . . . , Yk,1 , Yk,2 ) : M → Rn . Write 2∂ Yj,i = f˜j,i θ ( j = 1, . . . , k, i = 1, 2) and note that, by (1 j ), we have 2 2 + f˜j,2 = −λ j2 f˜j,1

for every j.

(5.35)

In view of (5.30) it follows that X is a conformal minimal immersion. By (5.35) and as above, the Gauss map of X omits the following n hyperplanes in general position:   [z1 : · · · : zn ] ∈ CPn−1 : z2 j−1 + (−1)ε iz2 j = 0 , j = 1, . . . , k, ε = 1, 2. (5.36)   Note that X ∗ (ds2 ) = ∑kj=1 | f˜j,1 |2 + | f˜j,2 |2 |θ |2 (see (2.55)). Since    1 1 X ∗ (ds2 ) ≥ | f˜1,1 |2 + | f˜1,2 |2 |θ |2 ≥ | f˜1,1 |2 + | f˜1,2 |2 + |λ1 |2 |θ |2 = Y1∗ (ds2 ) 2 2 and Y1 is complete, X is complete as well. Since f is full, so is the Gauss map of X provided the approximations are close enough. Finally, conditions (2 j ) and (3 j ) imply FluxX = p and X|P = ϕ. This completes the proof in case 1. Case 2: n = 2k + 1 ≥ 3 is odd. Fix K and p0 as in case 1. Let θ be a holomorphic 1-form vanishing nowhere on M such that  C

θ = ipn (C)  p

and ℜ

p0

for all loops C ⊂ M

θ = ϕ(p)

for all p ∈ P.

(5.37)

(5.38)

(See the generalizations of Theorem 1.10.5 by Y. Kusunoki and Y. Sainouchi in [212] and I. Castro-Infantes in [78, Lemma 2.3].) Choose positive real numbers λ1 , . . . , λk with k

∑ λ j2 = 1.

(5.39)

j=1

As in the previous case, pick holomorphic functions g j : K → C∗ ( j ∈ {1, . . . , k}) such that the holomorphic map f = ( f1 , f2 , . . . , f2k−1 , f2k , 1) : K → Cn is full, where the functions f j,i are defined by (5.31). By (5.32) and (5.39), f takes values in An−1 ∗ and the map ( f2 j−1 , f2 j , λ j ) : K → C3 assumes values in A2∗ for each j. Thus, the condition λ j = 0 ensures that the map [ f2 j−1 : f2 j : λ j ] : K → Q1 ⊂ CP2 omits the hyperplanes of CP2 in (5.33). Since K is simply connected, the 1-form ℜθ is exact, and λ j > 0, the map Y j = (Y j,1 ,Y j,2 ,Y j,3 ) : K → R3 defined by (5.34) is a conformal minimal immersion whose third component is harmonic on M. Reasoning as in the previous case and taking into account (5.37), we obtain a complete conformal minimal immersion

258

5 The Gauss Map of a Minimal Surface

       X = Y1,1 , Y1,2 , . . . , Yk,1 , Yk,2 , ℜ θ : M → Rn p0

satisfying conditions (1 j ), (3 j ), and also (2j ) FluxYj = (p2 j−1 , p2 j , λ j pn ). The Gauss map of X is full provided the approximations are close enough, and it omits the 2k = n − 1 hyperplanes of CPn−1 in (5.36) and also the following one:   [z1 : · · · : zn ] ∈ CPn−1 : zn = 0 . (5.40) Again, these n hyperplanes are in general position. In view of (5.37), (5.38), (1 j ),  (2j ), and (3 j ), this completes the proof.  Let us discuss implications of Theorem 5.6.1 in the case n = 3. Assume that X : M → R3 is a conformal minimal immersion and denote by G : M → Q1 ⊂ CP2 and g : M → CP1 the Gauss map and the complex Gauss map of X, respectively; see (2.73) and (2.79). Recall from (2.80) that G = τ ◦ g, where τ : CP1 → Q1 is the biholomorphic map in (2.77). Since Q1 is a quadratically embedded rational curve in CP2 , any projective line CP1 ∼ = H ⊂ CP2 intersects Q1 in one or two points, and 1 H ∩ Q is a singleton if and only if H is tangent to Q1 . Furthermore, G (M) ∩ H = ∅ if and only if g(M) ∩ τ −1 (H ∩ Q1 ) = ∅. Therefore, each projective line H ⊂ CP2 omitted by G determines two points of CP1 omitted by g if H is not tangent to Q1 , and one point if H is tangent to Q1 . This discussion justifies the following remark. Remark 5.6.2. The family of n hyperplanes of CPn−1 in general position in Theorem 5.6.1 cannot be chosen arbitrarily, for otherwise the theorem would provide complete nonflat minimal surfaces in R3 with the complex Gauss map omitting six points of CP1 , which is impossible by Fujimoto’s theorem (see Theorem 5.5.2). In fact, the mentioned result implies that if {Hλ : λ ∈ Λ } is a family of complex lines 2 by the Gauss map of a complete nonflat minimal surface in R3 , then in   CP omitted 1  λ ∈Λ Hλ ∩ Q consists of at most four points.  In the case n = 3, Theorem 5.6.1 provides a complete nonflat conformal minimal immersion X : M → R3 whose Gauss map G : M → CP2 omits the lines   H1 = [z1 : z2 : z3 ] ∈ CP2 : z1 − iz2 = 0 ,   H2 = [z1 : z2 : z3 ] ∈ CP2 : z1 + iz2 = 0 ,   H3 = [z1 : z2 : z3 ] ∈ CP2 : z3 = 0 ; see (5.36) and (5.40). It turns out that   (H1 ∪ H2 ∪ H3 ) ∩ Q1 = [1 : −i : 0], [1 : i : 0] .

5.6 The Gaussian Image of Complete Minimal Surfaces II

259

  From the formula τ(t) = 1 − t 2 : i(1 + t 2 ) : 2t ∈ Q1 (cf. (2.77)) we see that   τ −1 (H1 ∪ H2 ∪ H3 ) ∩ Q1 = {0, ∞} ⊂ CP1 . Thus, the Gauss map g : M → CP1 of X omits the pair of antipodal points 0 and ∞ of CP1 ; in other words, the unit normal vector field to X is nowhere vertical. This proves the following result from [79]; see also [17, Theorem II]. Corollary 5.6.3. For every open Riemann surface M there is a complete nonflat conformal minimal immersion X : M → R3 such that the image of its complex Gauss map M → CP1 omits two antipodal values. Furthermore, given a group homomorphism p : H1 (M, Z) → R3 and a map ϕ : P → R3 on a closed discrete subset P ⊂ M, we may choose X such that FluxX = p and X|P = ϕ. By Picard’s theorem there is no nonflat conformal minimal immersion C → R3 with the complex Gauss map C → CP1 omitting three points of CP1 , and hence Corollary 5.6.3 is sharp even without assuming completeness of X. We now show that any two values of CP1 can be omitted by the Gauss map of a complete nonflat minimal surface in R3 . We do not know whether the immersion X in the corollary can be chosen with arbitrary flux. Corollary 5.6.4. Every open Riemann surface M admits a complete nonflat conformal minimal immersion X : M → R3 with vanishing flux whose complex Gauss map M → CP1 omits any two given points of CP1 . Proof. Fix a point a ∈ C. It suffices to find a map X as in the corollary whose complex Gauss map g : M → CP1 omits the values a and ∞; the general case is then obtained by orthogonal rotations of R3 . Corollary 5.6.3 furnishes a complete nonflat conformal minimal immersion X = (X1 , X2 , X3 ) : M → R3 with vanishing flux whose complex Gauss map g˜ omits 0 and ∞; hence, ∂ X3 has no zeros. Consider the Weierstrass data (g, φ3 ) given by g = g˜ + a,

φ3 =

g˜ + a  g ∂ X3 = ∂ X3 . g˜ g˜

Note that g : M → C is holomorphic. Since ∂ X3 has no zeros, the zeros of φ3 are exactly the zeros of g of the same order. We claim that (g, φ3 ) determines by the Weierstrass formula (2.84) a conformal minimal immersion X : M → R3 satisfying the required conditions. Indeed, since the 1-form ∂ X has vanishing periods, the Weierstrass data determined by (g, φ3 ) according to the formula (2.83) also have vanishing periods, so X is well defined and has vanishing flux. Its complex Gauss map g = g˜ + a is nonconstant (so X is nonflat) and it omits the values a and ∞. Finally, the metrics g = X ∗ ds2 and g˜ = X∗ ds2 on M satisfy  g=

1 + |˜g + a|2 1 + |˜g|2

2 g˜ ≥ Cg˜

for some constant C > 0; see (2.85). Since g˜ is complete, so is g and hence X.  

260

5 The Gauss Map of a Minimal Surface

5.7 Isotopies of Conformal Minimal Immersions In this section we use Theorem 5.3.1 to obtain some general existence results for smooth 1-parameter isotopies of conformal minimal immersions with remarkable properties from any open Riemann surface M into Rn for any n ≥ 3. By Theorem 3.12.1 every conformal minimal immersion X : M → Rn is isotopic through conformal minimal immersions Xt : M → Rn (t ∈ [0, 1]) to a complete one, X1 , with vanishing flux, and with arbitrary flux if X is full. We now show that, if one does not insist on the completeness of X1 and only pays attention to the flux, there is such an isotopy preserving the Gauss map. The following is [32, Corollary 1.6]. Theorem 5.7.1. Let M be an open Riemann surface. For every conformal minimal immersion X : M → Rn (n ≥ 3) there is a smooth isotopy Xt : M → Rn (t ∈ [0, 1]) of conformal minimal immersions such that X0 = X, X1 has vanishing flux (i.e., it is the real part X1 = ℜZ of a holomorphic null curve Z : M → Cn ), and all immersions Xt in the family have the same Gauss map G : M → CPn−1 . Furthermore, if X is full (see Definition 2.5.2) then there is an isotopy Xt as above such that X1 has any given flux. Proof. We may assume that X is full since the proof is even simpler otherwise. Write p = FluxX and 2∂ X = f θ for a holomorphic 1-form θ vanishing nowhere on M. Let p : H1 (M, Z) → Rn be any group homomorphism. Consider the isotopy of homomorphisms qt = i(t p + (1 − t)p) : H1 (M, Z) → i Rn ,

t ∈ [0, 1].

Note that q0 = ip and q1 = ip . Theorem 5.3.1 applied to these data (with ft = f for all t ∈ [0, 1]) gives a continuous family of holomorphic functions ht : M → C∗ (t ∈ [0, 1]) such that h0 = 1 and  γ

ht f θ = qt (γ)

for all closed curves γ ⊂ M and all t ∈ [0, 1].

Integrating from a fixed base point p0 ∈ M we obtain an isotopy of conformal minimal immersions Xt : M → Rn (t ∈ [0, 1]),  p

Xt (p) = X(p0 ) +

p0

ℜ(ht f θ ),

p ∈ M,

which clearly satisfies the conclusion of the theorem.   The immersion X1 obtained in Theorem 5.7.1 cannot be complete in general since not every holomorphic map M → Qn−2 is the Gauss map of a complete conformal minimal immersion M → Rn (see Section 5.5). However, it is reasonable to ask whether X1 can be chosen to be complete if X is complete.

5.7 Isotopies of Conformal Minimal Immersions

261

Question 5.7.2. Assume that G : M → Qn−2 ⊂ CPn−1 (n ≥ 3) is the Gauss map of a complete conformal minimal immersion X : M → Rn . Does there exist a complete conformal minimal immersion X : M → Rn with the Gauss map G and vanishing flux? If X is full, is it possible to prescribe the flux of such an immersion? In general, it seems an interesting question to understand the geometry of the space of complete conformal minimal immersions M → Rn with a fixed Gauss map. The following second main result in this section shows that every conformal minimal immersion M → R3 is isotopic through conformal minimal immersions to one whose complex Gauss map is nonconstant and avoids two points of the Riemann sphere; this was proved in [32, Theorem 1.4]. Theorem 5.7.3. Let M be an open Riemann surface. For every conformal minimal immersion X : M → R3 and every pair of distinct points a, b ∈ CP1 there is a smooth isotopy Xt : M → R3 (t ∈ [0, 1]) of conformal minimal immersions such that X0 = X and the complex Gauss map g1 of X1 (2.79) is nonconstant and satisfies g1 (M) ⊂ CP1 \ {a, b}. There is also an isotopy Xt as above such that X1 is flat. This result is sharp by Picard’s theorem. Comparing with Corollary 5.6.4, the above theorem does not ensure completeness of X1 ; we anyway expect that this extra feature can be guaranteed by a more precise argument. The proof of Theorem 5.7.3 requires the following two lemmas. Lemma 5.7.4. For every holomorphic map g : M → CP1 and every pair of distinct points a, b ∈ CP1 there is a homotopy of holomorphic maps gt : M → CP1 (t ∈ [0, 1]) such that g0 = g, gt is nonconstant for every t ∈ (0, 1], and g1 (M) ⊂ CP1 \ {a, b}. Proof. Without loss of generality we may assume that g : M → CP1 is nonconstant. Take a 1-dimensional embedded CW-complex C ⊂ M such that g(C) ∩ {a, b} = ∅ and there is a strong deformation retraction ρt : M → M (t ∈ [0, 1]), i.e. ρ0 = IdM , ρt |C = IdC for all t ∈ [0, 1], and ρ1 (M) = C. (Such a CW-complex representing the topology of M can be obtained as the Morse complex of a Morse strongly subharmonic exhaustion function on M; see Section 1.4.) Consider the homotopy of continuous maps ht = g ◦ ρt : M → CP1 , t ∈ [0, 1]. We have that h0 = g and h1 = g ◦ ρ1 , and hence h1 (M) = g(C) ⊂ CP1 \ {a, b}. Since CP1 \ {a, b} is biholomorphic to C∗ and hence an Oka manifold, the basic Oka principle (see Theorem 1.13.3) furnishes a homotopy ht : M → CP1 \ {a, b} (t ∈ [1, 2]) connecting h1 to a nonconstant holomorphic map h2 : M → CP1 \ {a, b}. Pick a pair of points p, q ∈ M such that h2 (p) = h2 (q). By general position we may assume that ht (p) = ht (q) for all t ∈ (0, 2]. Since CP1 is an Oka manifold, the 1-parametric Oka property with interpolation (see Corollary 1.13.6) enables us to deform the homotopy (ht )t∈[0,2] with fixed ends h0 and h2 to a homotopy (gt )t∈[0,2] of holomorphic maps gt : M → CP1 such that gt (p) = ht (p) and gt (q) = ht (q) for all t ∈ [0, 2]. It follows that g0 = g, the map gt is nonconstant for each t ∈ (0, 2], and the map g2 = h2 assumes values in CP1 \ {a, b}. To complete the proof we reparameterize the interval [0, 2] of the homotopy back to [0, 1].  

262

5 The Gauss Map of a Minimal Surface

Lemma 5.7.5. Let M be an open Riemann surface and X : M → R3 be a conformal minimal immersion with vanishing flux whose complex Gauss map g : M → CP1 assumes values in C∗ . Then there is an isotopy of conformal minimal immersions Xt : M → Rn (t ∈ [0, 1]) with vanishing flux such that X1 = X and X0 is flat. Proof. Write X = (X1 , X2 , X3 ) and φ3 = 2∂ X3 . Since X has vanishing flux, ∂ X is exact, and hence φ3 , gφ3 , and φ3 /g are exact as well in view of the expression (2.83) for the Weierstrass data of X in terms of the pair (g, φ3 ). Consider the 1-parameter family of holomorphic 1-forms       1 1 2 i 1 2 Φt = −t g , + t g ,t φ3 , t ∈ C. 2 g 2 g It turns out Φt is exact and Φt /θ assumes values in A2∗ for every t ∈ C, whereas   φ3 Φ1 = ∂ X and Φ0 /θ = 12 , 2i , 0 gθ assumes values in a complex line. Fix a point p0 ∈ M and consider the isotopy of conformal minimal immersions Xt : M → R3 (t ∈ C) given by  p

Xt (p) = X(p0 ) + 2

p0

ℜΦt ,

p ∈ M.

 We have that X1 = X, Xt has vanishing flux for all t ∈ C, and X0 is flat.  Proof of Theorem 5.7.3. Let X : M → R3 be a conformal minimal immersion and denote by g : M → CP1 its complex Gauss map. Let a, b ∈ CP1 be distinct points. We identify CP1 with the quadric Q1 ⊂ CP2 as in (2.77), and hence g is related to the generalized Gauss map G of X by the formula (2.80). Let θ be a nowhere vanishing holomorphic 1-form on M and write f = ∂ X/θ : M → A2∗ ; thus, π ◦ f = g where π : C3∗ → CP2 denotes the canonical projection (2.74). Lemma 5.7.4 furnishes a homotopy of holomorphic maps gt : M → CP1 (t ∈ [0, 1]) such that g0 = g, gt is nonconstant for every t ∈ (0, 1], and g1 (M) ⊂ CP1 \ {a, b} (see [32, Lemma 5.2] for the details). Proposition 5.4.4 applied with Q = {0} ⊂ P = [0, 1] gives a homotopy of holomorphic maps ft : M → A2∗ such that f0 = f and π ◦ ft = gt for every t ∈ [0, 1]. Theorem 5.3.1 then provides a homotopy of holomorphic functions ht : M → C∗ (t ∈ [0, 1]) such that h0 = 1 and ht ft θ has vanishing real periods for every t ∈ [0, 1]; see Remark 5.3.2 (A). Fix a point p0 ∈ M. It follows that for every t ∈ [0, 1] the map Xt : M → Rn given by  p

Xt (p) = X(p0 ) + 2

p0

ℜ(ht ft θ ),

p∈M

is a conformal minimal immersion with the Gauss map π(ht ft ) = π( ft ) = gt . We have that X0 = X since h0 = 1 and f0 = f . As g1 is nonconstant and g1 (M) ⊂ CP1 \ {a, b}, this completes the proof of the first part of the theorem. For the second part we consider homotopies gt and Xt as above such that g1 (M) ⊂ C∗ and assume in view of Remark 5.3.2 that X1 has vanishing flux. By  Lemma 5.7.5 we can connect X1 to a flat conformal minimal immersion. 

5.7 Isotopies of Conformal Minimal Immersions

263

In conclusion, we show a kind of counterpart of Theorems 5.7.1 and 5.7.3. The following result is [32, Theorem 7.2]. Theorem 5.7.6. Let M be an open Riemann surface. For any flat conformal minimal immersion X : M → Rn (n ≥ 3) there is an isotopy Xt : M → Rn (t ∈ [0, 1]) of conformal minimal immersions such that X0 = X and X1 is nonflat. Proof. It suffices to consider the case n = 3. Thus, let X : M → R3 be a flat conformal minimal immersion. Up to a rigid motion we may assume that ∂ X = (1, i, 0)φ3 where φ3 is an exact nowhere vanishing holomorphic 1-form on M. Choose a nonconstant holomorphic function g : M → C∗ such that gφ3 and g2 φ3 are exact holomorphic 1forms on M; the existence of such a function is guaranteed by [32, Proposition 7.6]. (The mentioned result shows that for any nowhere vanishing holomorphic 1-form θ on M, group homomorphism q : H1 (M, Z) → C2 ,and Runge admissible compact set S ⊂ M, every function u ∈ A (S, C∗ ) such that γ (u, u2 )θ = q(γ) for all closed curves γ ⊂ S can be approximated uniformly on S by functions u˜ ∈ O(M, C∗ ) with  ˜ u˜2 )θ = q(γ) for all closed curves γ ⊂ M. The proof is very similar to that of γ (u, Theorem 5.3.1.) Obviously, the 1-forms gφ3 and g2 φ3 vanish nowhere on M. Set   Φt = 1 − t 2 g2 , i(1 + t 2 g2 ), 2tg φ3 , t ∈ C. It is clear that Φt is exact and the map Φt /φ3 : M → C3 assumes values in the punctured null quadric A2∗ for every t ∈ C. Thus, for any p0 ∈ M we obtain a conformal minimal immersion Xt : M → R3 by the Weierstrass–Enneper formula  p

Xt (p) = X(p0 ) + 2

p0

ℜ(Φt ),

p ∈ M.

Clearly, we have that X0 = X, whereas X1 is nonflat since g is nonconstant. Thus,  the isotopy Xt satisfies the conclusion of the theorem.  Corollary 5.7.7. Let M be an open connected Riemann surface and n ≥ 3. The natural inclusion CMInf (M, Rn ) → CMI(M, Rn ) of the space of nonflat conformal minimal immersions into the space of conformal minimal immersions M → Rn induces a bijection of path components of the two spaces. In particular: (i) The set of path components of CMI(M, R3 ) is in bijective correspondence with the elements of H 1 (M, Z2 ) = (Z2 )l , where H1 (M, Z) = Zl , l ∈ Z+ ∪ {∞}. (ii) The space CMI(M, Rn ) is path connected if n > 3. Proof. In view of Corollary 3.12.4 which identifies the path connected components of the space CMInf (M, Rn ) in terms of H 1 (M, Z2 ), the case n = 3 is implied by Corollary 5.3.3 whereas the case n > 3 follows from Theorem 5.7.6.  

Chapter 6

The Riemann–Hilbert Problem for Minimal Surfaces

In this chapter we adapt the classical Riemann–Hilbert boundary value problem for use in the theory of conformal minimal surfaces in Rn and holomorphic null curves in Cn . In the last section we also explain a method of exposing boundary points of Riemann surfaces. These techniques play a central role in subsequent chapters; in particular, in our treatment of the conformal Calabi–Yau problem in Chapter 7, in the construction of proper minimal surfaces in minimally convex domains in Chapter 8, and in the study of minimal and null hulls in Chapter 9. The Riemann–Hilbert problem has been used extensively in constructions of complex curves parameterized by compact bordered Riemann surfaces and satisfying various geometric conditions. It provides a deformation of such a curve in the direction given by a continuous family of holomorphic discs, attached to the curve along a boundary arc, while at the same time keeping a precise control on the geometric placement of the curve in the ambient space. Classical geometric applications of this technique which are most relevant to this book include the construction of proper holomorphic maps from bordered Riemann surfaces to a certain class of complex manifolds including all Stein manifolds of dimension > 1 (see the paper [114] whose introduction includes a historical survey of the subject), the Poletsky theory of hulls (see [116] and the references therein), and constructions of complete bounded complex curves (the Yang problem, see [21]). The Riemann–Hilbert problem was first introduced to the study of null curves and minimal surfaces in dimension n = 3 by the first two named authors in [23]; it was subsequently improved and extended to any dimension n ≥ 3 in [14]. Our exposition mainly follows the one in the latter paper, with improved and more complete explanations provided at several key steps. This technique is crucial in our construction of complete minimal surfaces and holomorphic null curves with Jordan boundaries (Section 7.4), in the construction of proper minimal surfaces in minimally convex domains (Section 8.3), and in the characterization of minimally convex hulls by sequences of minimal discs (Section 9.2). The Riemann–Hilbert method has also been adapted by the authors in [31, 25] for use in complex contact geometry, which has in turn been applied in [131] to establish the Calabi–Yau property of superminimal surfaces in certain Riemannian four-manifolds. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_6

265

266

6 The Riemann–Hilbert Problem for Minimal Surfaces

6.1 The Complex Analytic Case By way of motivation and to set the stage, we begin by recalling the Riemann– Hilbert problem in the most basic complex analytic case. Let D = {z ∈ C : |z| < 1} denote the unit disc in C and T = bD = {z ∈ C : |z| = 1} its boundary circle. A map Z : D → Cn of class A (D) is called a holomorphic disc in Cn . (One may consider holomorphic discs in an arbitrary complex space X instead of Cn ; see [116].) Let g : T × D → Cn be a continuous map such that for each z ∈ T we have g(z, · ) ∈ A (D, Cn ) and g(z, 0) = Z(z). In other words, {g(z, · ) : z ∈ T} is a continuous family of holomorphic discs in Cn centred at boundary points of the disc  = Z(0) Z. The problem is to find a holomorphic disc Z ∈ A (D, Cn ) with centre Z(0) whose image is close to the image of the given configuration Z(D) ∪ g(T × D) and whose boundary curve is close to the torus g(T2 ). Given a compact set K ⊂ Cn and a point p ∈ K, we set dist(p, K) = inf |p − q|. q∈K

(6.1)

Given a map g(z, ξ ) with ξ ∈ A for some set A, we write g(z, A) = {g(z, ξ ) : ξ ∈ A}.

(6.2)

The following result and the estimate (6.4) are the content of [116, Lemma 3.1]. Lemma 6.1.1. (Assumptions and notation as above.) Given numbers ε > 0 and 0 < ρ0 < 1, there are a number ρ  ∈ [ρ0 , 1) and a holomorphic disc Z ∈ A (D, Cn ) satisfying the following conditions.    (i) For every z ∈ T we have that dist Z(z), g(z, T) < ε. In particular, the circle 2 ).  Z(T) lies in the ε-neighbourhood of the torus g(T    (ii) For every z ∈ T and ρ ∈ [ρ  , 1] we have that dist Z(ρz), g(z, D) < ε. Hence, Z maps the annulus Aρ  = {ρ  ≤ |z| ≤ 1} into the ε-neighbourhood of the cylinder g(T × D).  − Z(z)| < ε. (iii) For every |z| ≤ ρ  we have that |Z(z) (iv) Z agrees with Z to a given finite order at a given finite set of points in D. (v) If I is a closed arc in T and g(z, · ) = Z(z) is the constant disc for all z ∈ T \ I, then for any given neighbourhood U ⊂ D of I the map Z can be chosen such that |Z − Z| < ε holds on D \U. We shall say that the holomorphic disc Z in the above lemma is obtained from the initial disc Z by a Riemann–Hilbert deformation. See Figures 6.1 and 6.2. Proof (sketch). We outline the idea of the proof and refer to [116, Lemma 3.1] for the details. It is easy to imagine the situation covered by this lemma in the simplest model case when n = 2, Z(z) = (z, 0) is the inclusion map D → D × {0} ⊂ C2 , and g(z, ξ ) = (z, ξ ) (ξ ∈ D) are the vertical discs centred at g(z, 0) = (z, 0) = Z(z). A

6.1 The Complex Analytic Case

267

g(p, D)

 Z(D)

g(p, D)

Z(p)

Z(p)

Z(D)

Z(D) g(p, T)

g(p, T)

Fig. 6.1 The Riemann–Hilbert Method: a 1-dimensional illustration  Z(D) g(p, D) Z(p)

g(p, D) Z(p)

Z(I) Z(I)

Z(D) Z(D)

Fig. 6.2 The Riemann–Hilbert Method: a 2-dimensional illustration

 = (z, h(z)N ) for large solution to the above problem is provided by the disc Z(z) enough N ∈ N, where h ∈ A (D) is any nonconstant function that vanishes at the given finite set of points in D and satisfies |h| = 1 on T. These are exact solutions  lies in the circle of the Riemann–Hilbert problem, in the sense that the point Z(z) g(z, T) for every z ∈ T, and not merely close to it as required by condition (iii). Although exact solutions exist only rarely, approximate solutions always exist and are found by the following elementary method. We approximate the map g(z, ξ ) − Z(z), which is continuous in the first variable z ∈ T but is holomorphic in the second variable ξ ∈ D and vanishes at ξ = 0 (and also for z ∈ T \ I in case (iv) of the lemma), by a map G(z, ξ ) defined on C∗ × C which is a polynomial in ξ without the constant term (i.e., G(z, 0) = 0) and whose coefficients are Laurent polynomials in z with the only pole at z = 0. Let a1 , . . . , ak ∈ D be a given finite set of points at which we wish

to interpolate to a given finite order s (see condition (iv)). Let h(z) = ∏kj=1

z−a j s . 1−a¯ j z

Then, for all sufficiently big N ∈ N the map  = Z(z) + G(z, h(z)zN ) Z(z)

(6.3)

satisfies conditions (i)–(v). For N chosen big enough, the term zN in the second component of G kills the poles of the Laurent coefficients of G at z = 0 and it ensures the approximation condition (iii).  

268

6 The Riemann–Hilbert Problem for Minimal Surfaces

A characteristic feature of these Riemann–Hilbert deformations is their fast spinning in the direction of the fibre circles g(z, T) as the point z traces the base circle. This has an averaging effect on integrals of continuous functions on Cn  In some applications, in particular those coming from along the boundary of Z. the Poletsky theory of hulls (see Chapter 9), the following additional property of the disc Z will be needed. Given an arc I ⊂ T and an upper semicontinuous function φ : Cn → R, we can arrange in addition to conditions (i)–(v) that 1 |I|

 I

 it   ) dt < φ Z(e

1 2π|I|

 2π 0

I

  φ g(eit , eiθ ) dt dθ + ε.

(6.4)

 is not much bigger than its average on That is, the average of φ on the arc Z(I) the cylinder g(I × T). This addition can be ensured by replacing Z in (6.3) with  = Z(z) + G(z, ch(z)zN ) for a suitable constant c ∈ C with |c| = 1. We shall Z(z) explain this in the proof of Lemma 6.2.1 in the following section. In applications of the Riemann–Hilbert technique, the central disc D (the domain of the map Z in Lemma 6.1.1) is often replaced by an arbitrary compact bordered Riemann surface M, but the boundary curves g(z, · ) for points z in an arc I ⊂ bM are always holomorphic discs. This method has been adapted to curves with values in an arbitrary complex manifold X in place of Cn (see [116] and the references therein). These extensions are obtained by using approximate solutions on discs furnished by Lemma 6.1.1, but with an additional holomorphic dependence on a complex parameter, together with the method of gluing holomorphic sprays (see Sect. 1.14). This will be explained more carefully in the sequel.

6.2 The Riemann–Hilbert Problem for Null Discs in C3 In this section we prove an analogue of Lemma 6.1.1 for holomorphic null discs in C3 ; see Lemma 6.2.1. The analogous result for conformal minimal discs in R3 is an immediate consequence by taking the real parts, but we shall not state it here since it is subsumed by the more general Theorem 6.4.1. The higher-dimensional case is treated separately in Sect. 6.5 since it requires additional analysis and the result is not as general as the one for n = 3. Proofs of both results are considerably more involved than that of Lemma 6.1.1 since we cannot work with null curves themselves, but with their derivative maps with values in the punctured null quadric A∗ = A \ {0} (see (2.54)). In Sect. 6.3 we also prove a parametric version of the same lemma which will be used in Sect. 6.4 for gluing purposes. We shall be using the notations (6.1) and (6.2). See also Figures 6.1 and 6.2 which illustrate the Riemann–Hilbert modifications provided by the lemma. Lemma 6.2.1. Let Z : D → C3 be a holomorphic null disc of class A 1 (D), i.e., Z ∈ NC1 (D, C3 ). Assume that I is a closed arc in T = bD (which may coincide with T), r : T → [0, 1] is a continuous function (the size function) supported on I,

6.2 The Riemann–Hilbert Problem for Null Discs in C3

269

and σ : I × D → C3 is a C 1 map such that for every z ∈ I, D ξ → σ (z, ξ ) is an immersed holomorphic null disc with σ (z, 0) = 0. If I = T, we also assume that the map T z −→

∂ σ (z, ξ )  ∈ A∗ is homotopically trivial in A∗ .  ∂ξ ξ =0

(6.5)

Let κ : T × D → C3 be given by   κ(z, ξ ) = Z(z) + σ z, r(z) ξ , z ∈ T, ξ ∈ D, (6.6)   where σ z, r(z) ξ = 0 for z ∈ T \ I. Given numbers ε > 0, 0 < ρ0 < 1 and an open neighbourhood U of I in D, there exist a number ρ  ∈ [ρ0 , 1) and a holomorphic  = Z(0) and the following conditions hold. null disc Z ∈ NC1 (D, C3 ) such that Z(0)  i) dist(Z(z), κ(z, T)) < ε for all z ∈ T.  ii) dist(Z(ρz), κ(z, D)) < ε for all z ∈ T and ρ ∈ [ρ  , 1). iii) Z is ε-close to Z in the C 1 topology on the set (D \U) ∪ (ρ  D). Moreover, given an upper semicontinuous function φ : C3 → R ∪ {−∞}, we can achieve in addition to the above that 1 |I|



 2π   it  dt  1  ) ≤ φ Z(e φ κ(eit , eis ) dtds + ε. 2π 2π|I| 0 I I

(6.7)

Remark 6.2.2. Condition (6.7), which is analogous to (6.4), will only be used in the description of minimally convex hulls given in Section 9.2. Lemma 6.2.1 essentially coincides with [14, Lemma 3.1] and it generalizes [23, Lemma 3.1], the latter pertaining to the special case when each σ (z, · ) is a linear null disc of the form D ξ → r(z)ξ u in a constant null direction u ∈ A∗ .   Proof. By Proposition 3.3.2 we may assume by approximation that Z is a nonflat holomorphic null immersion on the disc r0 D for some r0 > 1. Let ι : C2 → C3 denote the homogeneous quadratic map defined by   ι(u, v) = u2 − v2 , i(u2 + v2 ), 2uv , (u, v) ∈ C2 . (6.8) Note that ι is a two-sheeted parameterization of the null quadric A ⊂ C3 , commonly called the spinorial parameterization, which is branched only at the point (0, 0) ∈ C2 . In particular, ι : C2∗ → A∗ is a two-sheeted holomorphic covering projection. By introducing the coordinate g = v/u on CP1 , we see that 

i 1

 1 1 −g , + g , 1 2uv, ι(u, v) = 2 g 2 g which is reminiscent of the Weierstrass representation formula (2.83) in dimension 3, with g = v/u being the Gauss map. The conditions on the map σ in the lemma imply that the map

270

6 The Riemann–Hilbert Problem for Minimal Surfaces

σ2 (z, ξ ) :=

∂σ (z, ξ ) ∈ A∗ , ∂ξ

(z, ξ ) ∈ I × D

(6.9)

is continuous and σ2 (z, · ) ∈ A (D, A∗ ) for every z ∈ I. If I is a proper arc in T, then I × D is simply connected and hence σ2 lifts to a map 2 = σ2 . ι ◦σ

2 : I × D → C2∗ , σ

If I = T then the same holds by the assumption (6.5). Set    2 z, r(z) ξ , (z, ξ ) ∈ T × D, η(z, ξ ) := r(z) σ

(6.10)

where η(z, ξ ) = 0 if z ∈ T \ I. Then η(z, · ) ∈ A (D, C2 ) for every fixed z ∈ T and   ∂ σ (z, r(z)ξ ). ι η(z, ξ ) = r(z) σ2 (z, r(z) ξ ) = ∂ξ

(6.11)

2 and the function r by simpler ones, keeping We now approximate both the map σ 2 uniformly the same notation for the approximants. We begin by approximating σ on I × D by a rational map 2 (z, ξ ) = z−N1 σ

l

∑ B j (z) ξ j ,

(6.12)

j=0

where N1 ∈ Z+ and B j are C2 -valued holomorphic polynomials. (If I is a proper 2 (z, ξ ) by subarc of T, we may take N1 = 0.) This is done by first replacing σ σ2 (z, cξ ) for some 0 < c < 1 close to 1, approximating its Taylor series in the ξ variable by a Taylor polynomial whose coefficients are continuous functions of z ∈ I, and applying the Mergelyan theorem to each of these coefficients. The map σ (z, ξ ) =

 ξ  0

 ι σ2 (z,t) dt = z−N0

m

∑ Ak (z) ξ k ,

(6.13)

k=1

where m = 2l + 1, N0 = 2N1 and Ak are C3 -valued holomorphic polynomials, then approximates the original map σ in the lemma uniformly on I × D. Finally, we replace the size functions r : T → [0, 1] by a holomorphic function given by the following lemma. Lemma 6.2.3. Given δ ∈ (0, 1), there is a function r˜ ∈ A ∞ (D) such that δ ≤ |˜r| ≤ 1 on D, |˜r| = δ on T \ I, and ||˜r| − r| ≤ δ on T. Proof. We first δ -approximate r by a smooth function r  : T → [δ , 1] with r  = δ on T \ I. Let a be the harmonic function on D with boundary values a(eit ) = log r  (eit ), and let b be a harmonic conjugate of a. The function r˜ = ea+ib then satisfies the √ (a+ib)/2 lemma. Note that r˜ = e ∈ A ∞ (D).  

6.2 The Riemann–Hilbert Problem for Null Discs in C3

271

2 defined Remark 6.2.4. We now replace the map η (6.10) by using the new map σ by (6.12) and the new size function r given by Lemma 6.2.3 (and denoted r˜ in that lemma; the precise choice of r˜ and in particular of the constant δ will be determined later). We also replace the maps σ (6.13) and κ (6.6) accordingly. It clearly suffices to prove the theorem for these new functions.   We shall need the following key lemma. k 1 Lemma 6.2.5. Let σ (z, ξ ) = z−N0 ∑m k=1 Ak (z) ξ , where N0 ∈ Z+ and Ak ∈ A (D) for all k = 1, . . . , m. Write σ2 (z, ξ ) = ∂∂ σξ (z, ξ ) and let r ∈ A 1 (D). Then,

 z       lim sup  Nζ N−1 r(ζ )σ2 ζ , r(ζ )ζ N dζ − σ z, r(z)zN  = 0.

N→∞ |z|≤1

(6.14)

0

k−1 and hence Proof. We have that σ2 (ζ , ξ ) = ζ −N0 ∑m k=1 Ak (ζ ) kξ

  Nζ N−1 r(ζ ) σ2 ζ , r(ζ )ζ N =

m

∑ Ak (ζ )r(ζ )k kNζ kN−N0 −1 .

k=1

Assume that N > N0 , so kN − N0 > 0 for k ≥ 1. Let Ck,N =

kN − N0 − 1 N0 − 1 = 1− kN kN

and note that limN→∞ Ck,N = 1 for every k ∈ N. Integration by parts gives  z

Ck,N −

 z0 0

Ak (ζ )r(ζ )k kNζ kN−N0 −1 dζ = Ak (z)r(z)k zkN−N0

Ak (ζ )r(ζ )k ζ kN−N0 dζ −

 z 0

kAk (ζ )r(ζ )k−1 r (ζ )ζ kN−N0 dζ . (6.15)

Since r ∈ A 1 (D) and Ak ∈ A 1 (D), there is a constant C > 0 such that the following estimates hold for all |ζ | ≤ 1 and k = 1, . . . , m: |Ak (ζ )r(ζ )k | ≤ C,

|kAk (ζ )r(ζ )k−1 r (ζ )| ≤ C.

For N ≥ N0 we have that  z  1 |z|kN−N0 +1   ≤ ,  |ζ |kN−N0 |dζ | = kN − N0 + 1 kN − N0 + 1 0 where the integral is over the radial segment from 0 to z ∈ D. Hence, the absolute C value of each of the last two integrals in (6.15) is bounded by kN−N , so they 0 +1 converge to zero uniformly on D as N → ∞. The estimate (6.14) now follows by summing over k = 1, . . . , m and noting that the integrated terms in (6.15) give m

m

k=1

k=1

∑ Ak (z)r(z)k zkN−N0 = ∑ z−N0 Ak (z)(r(z)zN )k = σ (z, r(z)zN ).

272

6 The Riemann–Hilbert Problem for Minimal Surfaces

We continue with the proof of Lemma 6.2.1. Recall that the function η is given 2 by (6.10) and Remark 6.2.4. Note that ζ N η(ζ , ξ ) is defined and smooth on D for 2 every N ≥ N1 (cf. (6.12)). Consider the sequence of maps gN : D × T → C for N ≥ N1 defined by √ √ gN (ζ , c) = c 2N + 1 ζ N η(ζ , c ζ 2N+1 )    √ √ 2 ζ , cr(ζ )ζ 2N+1 . (6.16) = c 2N + 1 r(ζ ) ζ N σ Since the projection ι (6.8) is homogeneous quadratic, (6.11) implies ι(gN (ζ , c)) = c(2N + 1)ζ 2N r(ζ )σ2 (ζ , c r(ζ )ζ 2N+1 ). Integrating this equation with respect to ζ from 0 to z, Lemma 6.2.5 shows that  z       2N+1   (6.17) lim sup  ι gN (ζ , c) dζ − σ z, cr(z)z  = 0. N→∞ |z|≤1, c∈T

0

Let Z : D → C3 be the null disc in the lemma. Its derivative Z  : D → A2∗ lifts to a map h = (u, v) : D → C2∗ of class A (D) such that ι ◦ h = Z  . We may assume by approximation that h and Z are holomorphic on a neighbourhood of D. Consider the sequence of maps hN : D × T → C2 defined for N ≥ N1 by hN (ζ , c) = h(ζ ) + gN (ζ , c),

ζ ∈ D, c ∈ T.

(6.18)

We would like to arrange that the maps hN take values in C2∗ , i.e., they avoid the origin. Since D×T is three-dimensional while the point 0 ∈ C2 has real codimension four in C2 , this can be achieved by Whitney’s general position argument [344], moving Z (and hence h) slightly. Let us explain this. Lemma 3.2.1 shows that the derivatives Z  : D → A∗ of nonflat holomorphic null curves form a submersive family of maps into A∗ . More precisely, there is a family of deformations of any given nonflat map such that the differential with respect to the parameter is surjective at each point. The same is then true for the liftings h : D → C2∗ with ι ◦ h = Z  . Therefore, the transversality theorem (see Theorem 1.4.3) shows that for every integer N ≥ N1 there is an open dense set of Z ∈ NC2nf (D, C3 ) such that the map hN defined by (6.18) omits the origin. By Theorem 3.2.3 the space NC2nf (D, C3 ) is a complex Banach manifold, hence a Baire space. Since the intersection of a countable family of dense open sets in a Baire space is dense, we can find Z (and hence h) arbitrarily close to the original one for which the maps (6.18) assume values in C2∗ . Consider the sequence of families of immersed holomorphic null discs  z

ZN (z, c) = Z(0) +

0

ι(hN (ζ , c)) dζ ,

z ∈ D, N ≥ N1 ,

(6.19)

depending smoothly on c ∈ T. Since ι (6.8) is a homogeneous quadratic map, we have that

6.2 The Riemann–Hilbert Problem for Null Discs in C3

273

ι(hN (ζ , c)) = ι(h(ζ )) + ι(gN (ζ , c)) + RN (ζ , c),

(6.20)

Z  (ζ )

where ι(h(ζ )) = and each component of the remainder RN (ζ , c) is a linear combination with constant coefficients of terms gN, j (ζ , c)u(ζ ) and gN, j (ζ , c)v(ζ ) for j = 1, 2. (Here, gN = (gN,1 , gN,2 ) and h(ζ ) = (u(ζ ), v(ζ )).) We claim that  z    (6.21) lim sup  RN (ζ , c) dζ  = 0. N→∞ |z|≤1, c∈T

0

To see this, set C1 =

sup |ζ |≤1,|ξ |≤1

  N ζ 1 η(ζ , ξ ) ,

Then, sup |ζ |≤1, c∈T, N∈N

  C2 = max sup |u(ζ )|, sup |v(ζ )| . |ζ |≤1

|ζ |≤1

 N  ζ 1 η(ζ , cζ 2N+1 ) ≤ C1 .

Given z ∈ D, c ∈ T, j ∈ {1, 2}, and N ≥ N1 , we thus have the estimate   |z|  z   √  gN, j (ζ , c)u(ζ )dζ  ≤ 2N + 1 |ζ |N−N1 |ζ N1 η(ζ , c ζ 2N+1 )|· |u(ζ )| d|ζ |   0

0

≤ C1C2 ≤ C1C2

 |z| √ 0 √

2N + 1 |ζ |N−N1 d|ζ |

2N + 1 . N − N1 + 1

(6.22)

Clearly, the right-hand side converges to zero as N → +∞. The same estimate holds with u(ζ ) replaced by v(ζ ). Since RN (ζ , c) is a linear combination of finitely many such terms whose number is independent of N, (6.21) follows. Recall that ι(h(ζ )) = ι(u(ζ ), v(ζ )) = Z  (ζ ). Integrating the equation (6.20) and using the estimates (6.17) and (6.21), we obtain in view of (6.19) that     ZN (z, c) = Z(z) + σ z, cr(z)z2N+1 + EN (z, c) = κ z, cz2N+1 + EN (z, c) (6.23) for all z ∈ D and c ∈ T, where κ is given by (6.6) and Remark 6.2.4, and lim

sup

N→∞ |z|≤1, c∈T

|EN (z, c)| = 0.

(6.24)

Let us verify that the null disc Z = ZN (· , c) : D → C3 satisfies conditions i)–iii) in the lemma for every c ∈ T and every sufficiently big N ∈ N, provided that the constant δ > 0 in Lemma 6.2.3 is chosen small enough. By (6.24) there is an integer N2 ≥ N1 such that ε (6.25) sup |EN (z, c)| < , N ≥ N2 . 2 |z|≤1, c∈T By (6.23) this implies

274

6 The Riemann–Hilbert Problem for Minimal Surfaces

 ε  sup ZN (z, c) − κ(z, cz2N+1 ) < , 2 z∈D, c∈T

N ≥ N2 .

For points z ∈ T we thus obtain condition i) in the lemma. By Lemma 6.2.3 we have that |r| = δ on T \ I. Let J ⊂ U ∩ T be an arc containing I in its relative interior. (This argument is unnecessary if I = T.) Recall that σ (z, 0) = 0 for all z ∈ D \ {0}. Choosing δ > 0 small enough, there is a neighbourhood V = V (δ ) ⊂ D of T \ J in D such that   ε sup σ (z, cr(z)z2N+1 ) < , 2 z∈V, c∈T

N ∈ N.

Together with (6.25) it follows that sup |ZN (z, c) − Z(z)| < ε,

N ≥ N2 .

(6.26)

z∈V, c∈T

Let ρ0 ∈ (0, 1) and U ⊃ I be as in the lemma. Note that U ∪ V is a neighbourhood of T in D. Pick a number ρ  with ρ0 < ρ  < 1 and close enough to 1 such that A := {ρ  ≤ |z| ≤ 1} ⊂ U ∪V,

ε sup |κ(z, ξ ) − κ(z/|z|, ξ )| < . 2 z∈A, ξ ∈D

(6.27)

(By Remark 6.2.4 the function κ is defined and continuous on (D \ {0}) × D.) Inserting ξ = cz2N+1 into (6.27) and taking into account (6.23) and (6.25) we obtain   ZN (z, c) − κ(z/|z|, cz2N+1 ) < ε for all z ∈ A and N ≥ N2 , so condition ii) holds for all N ≥ N2 . We now hold r and ρ  fixed. Note that the sequence σ (z, cr(z)z2N+1 ) converges to zero uniformly on compacts in (z, c) ∈ D × T. Hence, increasing N2 ∈ N if necessary we have that sup

|z|≤ρ  , c∈T

ε |σ (z, cr(z)z2N+1 )| < , 2

N ≥ N2 .

Together with (6.23) and (6.25) it follows that sup|z|≤ρ  , c∈T |ZN (z, c) − Z(z)| < ε for all N ≥ N2 . From this, the estimate (6.26), and the inclusion (6.27) we infer that condition iii) in the lemma holds with the uniform estimate. The C 1 estimate in condition iii) is obtained by using Cauchy estimates as follows. Recall that Z is holomorphic on r0 D for some r0 > 1. Choose a C ∞ -small smooth perturbation D ⊂ C of D such that D ⊂ D ⊂ r0 D and bD ∩ bD = I. (Such a domain D is obtained by slightly bumping D out along the arc bD \ I.) Pick a  ⊂ D of the arc I such that the compact set D \ U is small closed neighbourhood U  Since D is conformally diffeomorphic to D, contained in the open domain D \ U. the already proven result applied on D gives a null curve Z : D → C3 approximating  and satisfying the other conditions. The Z as closely as desired uniformly on D \ U  by C 1 estimate on Z − Z on D \U then follows from the uniform estimate on D \ U

6.3 The Riemann–Hilbert Problem for Sprays of Null Discs in C3

275

 the Cauchy estimates, and it is easily seen that conditions i) and ii) still hold for Z| D (cf. [23, proof of Lemma 3.1]). It remains to show that the inequality (6.7), 1 |I|

 I

  φ ZN (eit , c) dt ≤

1 2π|I|

 2π 0

I

  φ κ(eit , eis ) dtds + ε,

can be achieved for all big N by a suitable choice of c = c(N) ∈ T. The change of variables formula shows that for any N ∈ N we have 1 2π|I|

 2π 0

       1 2π 1 φ κ(eit , eis ) dtds = φ κ eit , ei(s+(2N+1)t) dtds. 2π |I| 0 I I

Assume first that the function φ is continuous. The mean value theorem furnishes a number s0 = s0 (N) ∈ [0, 2π) such that the above expression equals 1 |I|

 I

   φ κ eit , ei(s0 +(2N+1)t) dt.

In view of (6.23),(6.24), the fact  and  that φ is continuous, this number differs by 1 it , eis0 ) dt if N is chosen big enough. If φ is only upper at most ε from |I| φ Z (e N I semicontinuous, it is a limit of a decreasing sequence of continuous functions, and hence the estimate (6.7) follows from the already established fact for continuous functions and Lebesgue’s monotone convergence theorem.  

6.3 The Riemann–Hilbert Problem for Sprays of Null Discs in C3 The following parametric version of Lemma 6.2.1 will be used in the proof of the main results on the Riemann–Hilbert problem for null curves and conformal minimal surfaces parameterized by an arbitrary bordered Riemann surface. Lemma 6.3.1. Let Zw ∈ NC1 (D, C3 ) be a family of holomorphic null disc of class A 1 (D) depending holomorphically on a parameter w in a compact ball B ⊂ Cm . Assume that r : T → [0, 1] is a continuous function supported on a proper closed arc I in the circle T = bD. Let σ : T × D → C3 be a map of class C 1 such that for every z ∈ T the map D ξ → σ (z, ξ ) is an immersed null disc of class A 1 (D, C3 ) with σ (z, 0) = 0. Assume in addition that condition (6.5) holds and ∂ σ (z, ξ )  are linearly independent for all (z, w) ∈ T × B.  ∂ξ ξ =0 (6.28) Let κ : T × B × D → C3 be given by   κ(z, w, ξ ) = κw (z, ξ ) := Zw (z) + σ z, r(z) ξ , z ∈ T, w ∈ B, ξ ∈ D, (6.29) the vectors Zw (z) and

276

6 The Riemann–Hilbert Problem for Minimal Surfaces

  where σ z, r(z) ξ = 0 for z ∈ T \ I. Given numbers ε > 0, 0 < ρ0 < 1, and an open neighbourhood U of I in D, there exist a number ρ  ∈ [ρ0 , 1) and a holomorphic family of holomorphic null discs Zw ∈ NC1 (D, C3 ) (w ∈ B) such that for all w ∈ B we have that Zw (0) = Zw (0) and the following conditions hold. i) dist(Zw (z), κw (z, T)) < ε for all z ∈ T. ii) dist(Zw (ρz), κw (z, D)) < ε for all z ∈ T and ρ ∈ [ρ  , 1). iii) Zw is ε-close to Zw in the C 1 topology on the set (D \U) ∪ (ρ  D). Note that we omitted condition (6.7), which will not be needed. Remark 6.3.2. Lemma 6.3.1 is not a straightforward extension of Lemma 6.2.1, the reason being that we must pay attention not to introduce branch points to our null curves when performing the Riemann–Hilbert deformation. This is where condition (6.28) will be used. The transversality argument in the proof of Lemma 6.2.1 no longer applies since the dimension of the domain is too big. This point was not explained in the use of this parametric version of Lemma 6.2.1 that was made in the paper [15]. Without this analysis, it cannot be guaranteed that the resulting null curves and conformal minimal surfaces, constructed in the cited paper, are nonbranched (immersions). We compensate this here. The special cases considered in the earlier two papers [14, 23] on the subject only used families of null discs lying in parallel planes; in this special case the situation is considerably simpler and the proof was adequately explained in the cited papers. Note that for any single map Zw0 in the family, condition (6.28) can be achieved by a small perturbation of either Zw0 or the null discs σ (z, · ) for z ∈ I. Furthermore,  this condition is open, so it holds for all w near w0 if it holds for w0 .  Proof. As in (6.9) we define the map σ2 (z, ξ ) =

∂σ (z, ξ ) ∈ A∗ , ∂ξ

(z, ξ ) ∈ T × D.

Note that σ2 (z, · ) ∈ A (D, A∗ ) for every z ∈ T. Let ι : C2 → A ⊂ C3 denote the projection (6.8); recall that ι : C2∗ → A∗ is a double-sheeted covering map. In view of condition (6.5), σ2 lifts to a map σ2 : T × D → C2∗ ,

2 = σ2 . ι ◦σ

Likewise, for every w ∈ B the derivative Zw : D → A∗ of Zw : D → C3 lifts to a map hw : D → C2∗ of class A (D, C2∗ ) depending holomorphically on w ∈ B such that ι(hw ) = Zw ,

w ∈ B.

(6.30)

By (6.28) the null vectors Zw (z) and σ2 (z, 0) are linearly independent for every z ∈ T and w ∈ B. Since ι maps complex lines through the origin in C2 onto rays of 2 (z, 0) in C2∗ are linearly the quadric A ⊂ C3 , it follows that the vectors hw (z) and σ independent for all (z, w) ∈ T × B.

6.3 The Riemann–Hilbert Problem for Sprays of Null Discs in C3

277

By approximation we may assume that σ2 (z, ξ ) =

l

∑ B j (z) ξ j ,

(6.31)

j=0

where the B j ’s are Laurent polynomials with the only pole at z = 0 (see (6.12) and 2 (z, 0), and hence (6.28) ensures that the explanation there). Note that B0 (z) = σ hw (z) and B0 (z) are linearly independent for all (z, w) ∈ T × B.

(6.32)

We replace r by a nowhere vanishing function in A ∞ (D) furnished by Lemma 6.2.3. By analogy with (6.16) we define the following sequences for z ∈ D and w ∈ B:  √   2 z, r(z)z2N+1 , (6.33) gN (z) = 2N + 1 r(z) zN σ hN (z, w) = h(z, w) + gN (z). (6.34) Here we set h(z, w) = hw (z), where hw is defined by (6.30). It is now possible to complete the proof of Lemma 6.3.1 by following the proof of Lemma 6.2.1 step by step, but taking into account also the following result which takes care of the potential problem pointed out in Remark 6.3.2. Lemma 6.3.3. (Notation and assumptions as above.) For all big enough N ∈ N we have that hN (z, w) ∈ C2∗ for all z ∈ D and w ∈ B. Proof. Let B(p, ε) ⊂ C2 denote the closed ball of radius ε centred at p ∈ C2 . Recall that the maps hw : D → C2 and σ2 (z, · ) : D → C2 have ranges in C2∗ for all z ∈ T and w ∈ B. Condition (6.32) implies that for every (z, w) ∈ T × B the affine complex line h(z, w) + C· B0 (z) ⊂ C2 is contained in C2∗ . Obviously, this condition is stable under small deformations of both vectors. Hence, there are numbers ρ1 ∈ [ρ0 , 1) and ε0 > 0 such that   Γ := h(z, w) + C· B(B0 (z), ε0 ) : ρ1 ≤ |z| ≤ 1, w ∈ B ⊂ C2∗ (6.35) and R1 :=

min

ρ1 ≤|z|≤1, ξ ∈D

2 (z, ξ )| > 0. |σ

(6.36)

We also define R0 := max |h(z, w)| > 0.

(6.37)

z∈D, w∈B

As N → ∞, the sequence gN (z) converges to 0 uniformly on |z| ≤ ρ1 , and for such z the point hN (z, w) (6.34) lies in C2∗ for all w ∈ B for all big enough N ∈ N. Let us now consider points in the annulus A1 = {z : ρ1 ≤ |z| ≤ 1}. Choose a number c ∈ (0, 1) such that l

sup

∑ |B j (z)|c j < ε0 ,

z∈A1 j=1

(6.38)

278

6 The Riemann–Hilbert Problem for Minimal Surfaces

where ε0 is as in (6.35). Let ρN > 0 for N ≥ 2 be determined by the equation √ √ 2N + 1 c (ρN )N R1 = 2R0 . (6.39) For every big enough N we have that 0 < ρN < 1 and limN→∞ ρN = 1. Consider points in the annulus {ρN ≤ |z| ≤ 1}. We have two cases to consider: either |r(z)| < c or |r(z)| ≥ c. If |r(z)| < c, (6.38) shows that l





l

∑ B j (z)(r(z)z2N+1 ) j  ≤ ∑ |B j (z)|c j < ε0 ,

j=1

j=1

and hence hN (z, w) ∈ Γ ⊂ C2∗ in view of (6.31) and (6.35). In the second case when |r(z)| ≥ c we have in view of (6.36), (6.37), and (6.39) that √ √ |hN (z, w)| ≥ |gN (z)| − |h(z, w)| ≥ 2N + 1 c (ρN )N R1 − R0 = R0 . It remains to consider points in the annulus ρ1 ≤ |z| ≤ ρN . Then, |r(z)z2N+1 | ≤ |z|2N+1 ≤ (ρN )2N =

4R20 3 are given in Sect. 6.6. We shall be using the notation (6.1) and (6.2). Theorem 6.4.1 (The Riemann–Hilbert Problem for Minimal Surfaces in R3 ). Let M be a compact bordered Riemann surface, I1 , . . . , Ik be pairwise disjoint compact arcs in the boundary bM which are not connected components of bM, and I = ki=1 Ii . Choose an annular neighbourhood A ⊂ M of bM and a smooth retraction ρ : A → bM. Assume that • X : M → R3 is a conformal minimal immersion of class C 1 (M), • r : bM → [0, 1] is a continuous function with support contained in the relative interior of I, • α : I × D → R3 is a map of class C 1 such that for every z ∈ I the map D ξ → α(z, ξ ) ∈ R3 is a conformal minimal immersion with α(z, 0) = 0, and • Λ ⊂ M˚ is a finite set.

6.4 The Riemann–Hilbert Problem for Minimal Surfaces in R3 and Null Curves in C3

279

Let the map κ : bM × D → R3 be given by   κ(z, ξ ) = X(z) + α z, r(z) ξ , z ∈ bM, ξ ∈ D, (6.40)   where we take α z, r(z) ξ = 0 for z ∈ bM \ I. Given ε > 0, d ∈ N, and a neighbourhood U ⊂ A of I in M, there exist a conformal minimal immersion X : M → R3 and a neighbourhood Ω ⊂ U of supp(r) in M such that the following conditions hold.    i) dist X(z), κ(z, T) < ε for all z ∈ bM.    ii) dist X(z), κ(ρ(z), D) < ε for all z ∈ Ω . iii) X − XC 1 (M\Ω ) < ε. iv) X − X vanishes to order d at each point in Λ . v) FluxX = FluxX . Proof. We may assume that M is a compact smoothly bounded domain in an open  Pick a holomorphic 1-form θ without zeros on M.  By Lemma Riemann surface M. 3.3.1 we can deform X slightly (without changing the flux and keeping it fixed at the points in Λ , which we assume nonempty) to ensure that X is a nonflat conformal  of M, so we have that minimal immersion on an open neighbourhood M0 ⊂ M the map f = 2∂ X/θ : M0 → A∗ is holomorphic and nonflat.

(6.41)

Write Λ = {a1 , . . . , am }. Choose a reference point p0 ∈ M˚ \ (A ∪ Λ ) and closed Jordan curves C1 , . . . ,Cl0 ⊂ M˚ \ (A ∪ Λ ) based at p0 which form a basis for the 0 homology group H1 (M, Z) such that C = li=1 Ci is Runge in M˚ (see Lemma 1.12.10). To this collection of curves we add embedded arcs Ci+l0 ⊂ M˚ connecting p0 to ai ∈ Λ for i = 1, . . . , m, chosen to be pairwise disjoint and contained in the complement of C except for the common endpoint p0 . Set l = l0 + m. The union ˚ C = li=1 Ci is then a compact Runge set in M. For each i = 1, . . . , k we choose a smaller arc Ii in the relative interior of Ii such that supp(r) ∩ Ii ⊂ Ii . Next, choose a smoothly bounded simply connected domain Di ⊂ M such that the arc Ii is contained in the relative interior of bDi ∩ bM ⊂ Ii . Set D = ki=1 Di . The domains Di are chosen pairwise disjoint and small enough such that they are contained in the domain A of the retraction ρ : A → bM, with ρ(Di ) ⊂ Ii for i = 1, . . . , k and C ∩ D = ∅. (See Figure 6.3.) Recall that α(z, · ) : D → R3 is a conformal minimal disc with α(z, 0) = 0 for every z ∈ I = m i=1 Ii . Replacing α(z, ξ ) by α(z,tξ ) for some t < 1 close to 1 we may assume that these discs are smooth up to the boundary of D. By taking their harmonic conjugates, we obtain a family of holomorphic null discs σ (z, · ) : D → C3 ,

σ (z, 0) = 0,

ℜσ (z, · ) = α(z, · ) (z ∈ I).

(6.42)

Recall that the map f = 2∂ X/θ : M0 → A∗ is defined by (6.41). By a small generic deformation of f we can ensure that

280

6 The Riemann–Hilbert Problem for Minimal Surfaces

Fig. 6.3 The domains Di and Ωi

the vectors f (z) and

∂ σ (z, ξ )  are linearly independent for all z ∈ I.  ∂ξ ξ =0

(6.43)

This uses a general position argument explained in the proof of Lemma 6.2.1. We now extend the families of discs in (6.42) to all points z ∈ D by setting σ (z, · ) = σ (ρ(z), · ),

z ∈ D,

(6.44)

and likewise for α(z, · ) = ℜσ (z, · ). By choosing the domains Di for i = 1, . . . , k small enough, we can ensure that the condition (6.43) holds for all points z ∈ D. By Lemma 3.2.1 there exists a spray of holomorphic maps fw : M0 → A∗ , depending holomorphically on a parameter w in a ball 0 ∈ B ⊂ CN for some N ∈ N, such that f0 = f and the following conditions hold. (a) The spray is dominating on D =

k

i=1 Di ,

i.e.,

 ∂ fw (z)w=0 : CN −→ T f (z) A is surjective for every z ∈ D. ∂w (b) The spray is period dominating, i.e.,  ∂  : CN −→ (C3 )l is surjective. P( fw ) ∂w w=0 Here, P is the period map (3.12) associated to the family C = {C1 , . . . ,Cl }. (c) The map fw agrees with f0 = f to order  d at all points in Λ for all w ∈ B. ∂ σ (z,ξ )  (d) The vectors fw (z) ∈ A∗ and ∂ ξ  ∈ A∗ are linearly independent for all ξ =0

z ∈ D and w ∈ B (see (6.43) and (6.44)).

Pick δ > 0 such that for every continuous map f˜ : M → C3 we have  f − f˜C (M,C3 ) < δ =⇒

 z p

| f − f˜|· |θ | < ε/2 for all p, z ∈ M,

(6.45)

where the integral is taken over the shortest path in M from p0 to z in some fixed metric on M. Shrinking the ball B if necessary, we may assume that (6.45) holds for all maps f˜ = fw with w ∈ B.

6.4 The Riemann–Hilbert Problem for Minimal Surfaces in R3 and Null Curves in C3

281

For every i ∈ {1, . . . , k} we extend the function r : bM → [0, 1] to bDi such that it vanishes on bDi \ Ii . Recall that U ⊂ A is a neighbourhood of I = ki=1 Ik . Pick a small relatively open neighbourhood Ωi ⊂ Di of the arc Ii such that Ω i ⊂ U. (See Fig. 6.3.) Fix a point pi ∈ D˚ i . For every i = 1, . . . , k the restriction fw |Di : Di → A∗ (w ∈ B) defines a holomorphic null disc Zi,w : Di → C3 by  z

Zi,w (z) := X(pi ) +

pi

fw θ ,

z ∈ Di , w ∈ B.

We now apply Lemma 6.3.1 on the disc Di (which is conformally diffeomorphic to the closed unit disc D) to the spray Zi,w : Di → C3 (w ∈ B) and the family of holomorphic null discs σ (z, · ) : D → C3 (z ∈ Di ) given by (6.42), (6.44). Note that the conditions in the lemma are satisfied; see in particular condition (d) above. Given ε  > 0, Lemma 6.3.1 furnishes a spray gi,w : Di → A∗ of class A (D), depending holomorphically on w ∈ B, such that the family of conformal minimal immersions Xi,w : Di → R3 (w ∈ B) defined by  z

Xi,w (z) := X(pi ) +

pi

2ℜ(gi,w θ ),

z ∈ Di ,

satisfies the following conditions for all w ∈ B.   (i) dist Xi,w (z), κ(z, T) < ε/2 for all z ∈ bM ∩ bDi .   (ii) dist Xi,w (z), κ(ρ(z), D) < ε/2 for all z ∈ Ωi . (iii) gi,w − fw C (Di \Ωi ) < ε  . The number ε in the first two conditions is as in the theorem. (These two conditions depend on the estimate (6.45) which holds for all f˜ = fw , w ∈ B, so we need not redefine the map κ (6.40) which uses X as the central surface.) On the other hand, the number ε  > 0 in condition (iii) may be chosen arbitrarily small. Let Ω = ki=1 Ωi ; note that Ω ⊂ M ∩U. The compact sets D0 = M \ Ω ,

D=

k !

Di

(6.46)

i=1

form a Cartan pair, i.e., we have D \ D0 ∩ D0 \ D = ∅. Note also that M = D0 ∪ D and D0 ∩D = D\Ω = ki=1 Di \Ωi . Since the spray fw is dominating on D = ki=1 Di by (a), the gluing lemma for holomorphic sprays (see [140, Proposition 5.9.2]) shows that there are a smaller ball B ⊂ B around the origin in CN (depending on the spray { fw }w∈B ) and for every ε  > 0 an ε  > 0 such that for any collection of holomorphic sprays gi,w : Di → A∗ (w ∈ B, i = 1, . . . , k) satisfying the estimate (iii) (with this ε  ) there is a holomorphic spray f˜w : M → A∗ , w ∈ B , of the form  f (z), if z ∈ D0 ; f˜w (z) = a(z,w) (6.47) gi,b(z,w) (z), if z ∈ Di for i = 1, . . . , k,

282

6 The Riemann–Hilbert Problem for Minimal Surfaces

where a : D0 × B → B and b : D × B → B are continuous maps holomorphic on the interior of the respective set and satisfying the estimates sup

z∈D0 , w∈B

|a(z, w) − w| < ε  ,

sup

z∈D, w∈B

|b(z, w) − w| < ε  .

(6.48)

Assuming that ε  > 0 is small enough, the first estimate above clearly implies  f − f˜w C (D0 ,C3 ) ≤  f − fw C (D0 ,C3 ) +  fw − f˜w C (D0 ,C3 ) < δ /2 + δ /2 = δ for all w ∈ B , and hence (6.45) holds for any map f˜ = f˜w on D0 . Furthermore, the period domination property of the spray fw (see condition (b)) gives a point w0 ∈ B such that the map f˜w0 : M → A∗ satisfies the period conditions P( f˜w0 ) = P( f ).

(6.49)

It follows that the map X : M → R3 given by  = X(p0 ) + X(z)

 z p0

2ℜ( f˜w0 θ ),

z ∈ M,

(6.50)

is a well defined conformal minimal immersion. We claim that X satisfies the theorem for suitable choices of sprays gi,w . Conditions iii)–v) are obvious from the construction. Note in particular that the basic interpolation condition iv) at the points in Λ = {a1 , . . . , am } follows from (6.49) since the collection C includes arcs connecting p0 to these points, and the order d interpolation at the same points then follows from condition (c). From what has been said, conditions i) and ii) hold provided we have the following estimates:  z   z    gi,b(·,w ) θ − gi,w θ  < ε , z ∈ Di , i = 1, . . . , k. 0   p 0 2 pi i We see from (6.20) that every map gi,w : Di → A∗ is of the form gi,w (z) = ι(hi,w (z)) = fw (z) + ι(g˜i,N (z)) + RN (z, w),

z ∈ Di

(6.51)

for some integer N ∈ N, where ι : C2 → A∗ is the projection (6.8), the maps 2 is obtained from the null discs σ (z, · ) g˜i,N : Di → C2 are of the form (6.16) where σ given by (6.42) as explained there (in particular, the maps g˜i,N do not depend on the parameter w ∈ B), and the integral of the remainder RN is as small as desired uniformly on Di × B for big N (see the proof of the estimate (6.21), especially (6.22)). Replacing w0 by the function b(z, w0 ) appearing in (6.47), the integral of the first term on the right-hand side of (6.51) changes by at most ε/2 in view of (6.45), the integral of the second term does not change, and the integral of RN (z, b(z, w)) is arbitrarily small for N big enough (independently of b(z, w0 )). Hence, the conformal  minimal immersion X : M → R3 given by (6.50) also satisfies i) and ii).  The same proof yields the following analogue of Theorem 6.4.1.

6.5 The Riemann–Hilbert Problem for Null Discs in Cn for n > 3

283

Theorem 6.4.2 (The Riemann–Hilbert Problem for Null Curves in C3 ). Let M be a compact bordered Riemann surface with nonempty boundary bM, let I1 , . . . , Ik be pairwise disjoint compact arcs in bM which are not connected components of bM, and set I = ki=1 Ii . Choose an annular neighbourhood A ⊂ M of bM and a smooth retraction ρ : A → bM. Assume that • Z : M → C3 is a holomorphic null immersion of class C 1 (M), • r : bM → [0, 1] is a continuous function with support contained in the relative interior of I, • σ : I × D → C3 is a C 1 map such that σ (z, · ) : D → C3 is a holomorphic null immersion with σ (z, 0) = 0 for every z ∈ I, and • Λ ⊂ M˚ is a finite set. Let the map κ : bM × D → C3 be given by   κ(z, ξ ) = Z(z) + σ z, r(z) ξ , z ∈ bM, ξ ∈ D,   where we take σ z, r(z) ξ = 0 for z ∈ bM \ I. Given ε > 0, d ∈ N, and a neighbourhood U ⊂ M of I, there exist a holomorphic null immersion Z : M → C3 and an open neighbourhood Ω ⊂ M ∩ A ∩U of I such that the following conditions hold.    i) dist Z(z), κ(z, T) < ε for all z ∈ bM.    ii) dist Z(z), κ(ρ(z), D) < ε for all z ∈ Ω . iii) Z − ZC 1 (M\Ω ) < ε. iv) Z − Z vanishes to order d at all points in Λ .

6.5 The Riemann–Hilbert Problem for Null Discs in Cn for n > 3 We now proceed to the treatment of the Riemann–Hilbert problem in dimensions n > 3. This requires some additional preparations. The main new difficulty is that the null quadric A = An−1 for n > 3 does not have a simple parameterization comparable to the one given for n = 3 by (6.8). Denote by Θ the holomorphic bilinear form on Cn given by Θ (z, w) =

n

∑ z jw j.

(6.52)

j=1

Let u, v, w ∈ A∗ be linearly independent null vectors such that a := Θ (v, w) = 0,

b := Θ (u, w) = 0,

c := Θ (u, v) = 0.

(6.53)

Denote by A(u,v,w) the intersection of A with the complex 3-dimensional subspace

284

6 The Riemann–Hilbert Problem for Minimal Surfaces

L (u, v, w) = span{u, v, w} ⊂ Cn . Condition (6.53) ensures that A(u,v,w) is biholomorphic (indeed, linearly equivalent) to the 2-dimensional null quadric A2 ⊂ C3 . In fact, a calculation shows that αu + β v + γw ∈ A for some (α, β , γ) ∈ C3 if and only if αβ Θ (u, v) + αγ Θ (u, w) + β γ Θ (v, w) = 0. Using the notation (6.53), the above equation is equivalent to 

α β −i a b

2



β γ + −i b c

2 +

γ c

−i

α 2 = 0. a

This is the equation of the null quadric A2 ⊂ C3 (2.54) in the coordinates z1 =

β α −i , a b

z2 =

γ β −i , b c

z3 =

α γ −i . c a

Note that z = (z1 , z2 , z3 ) = (α, β , γ)· A(a, b, c) where

⎛

⎞ 1/a 0 −i/a A(a, b, c) = ⎝−i/b 1/b 0 ⎠ 0 −i/c 1/c

(6.54)

is a nonsingular 3 × 3 matrix with holomorphic coefficients. (We are using row vectors and matrix product on the right for the convenience of notation.) Let ι : C2 → C3 be the homogeneous quadratic map given by (6.8):   ι(u, v) = u2 − v2 , i(u2 + v2 ), 2uv , (u, v) ∈ C2 . Recall that ι : C2∗ → A2∗ is a doubly sheeted holomorphic covering projection. We have that     1 1 −i i = , 0, − ι √ ,√ = (1, 0, 0)· A(a, b, c), a a 2a 2a     1 i 1 ι √ , 0 = − , , 0 = (0, 1, 0)· A(a, b, c). b b ib √ The choice of is fine on any simply connected subset in the domain space. Pick a holomorphically varying family of linear automorphisms φ(a,b) of C2 such that  φ(a,b) (0, 0) = (0, 0),

φ(a,b) (1, 0) =

 1 −i √ ,√ , 2a 2a

 φ(a,b) (0, 1) =

 1 √ ,0 . ib

This is achieved by taking φ(a,b) (s,t) = (s,t)· B(a, b) where B is the 2 × 2 matrix

6.5 The Riemann–Hilbert Problem for Null Discs in Cn for n > 3



B(a, b) =

√1 √−i 2a 2a √1 0 ib

285



.

(6.55)

The map     C2 (s,t) → α(s,t), β (s,t), γ(s,t) = ι (s,t)· B(a, b) · A(a, b, c)−1 ∈ C3 is homogeneous quadratic in (s,t) and depends locally holomorphically on (a, b, c), and hence on the triple (u, v, w) of null vectors satisfying the general position condition (6.53). By the construction, the associated map C2 (s,t) → ψ(u,v,w) (s,t) = α(s,t)u + β (s,t)v + γ(s,t)w

(6.56)

is a holomorphically varying parameterization of the quadric A(u,v,w) satisfying ψ(u,v,w) (e1 ) = u,

ψ(u,v,w) (e2 ) = v,

(6.57)

where e1 = (1, 0) and e2 = (0, 1). Note that ψ(u,v,w) is well defined on the set of triples (u, v, w) ∈ (An−1 )3 satisfying condition (6.53), except for the indeterminacies caused by the square roots in the entries of the matrix B (6.55). These reflect the fact that we have four different choices providing (6.57), √ √ the normalization corresponding to the possible choices of the roots ± a and ± b. In the sequel we shall hold fixed a pair of null vectors u, v ∈ A∗ with c = Θ (u, v) = 0 and will assume that w = f (z), where f : D → A∗ is a holomorphic map such that the triple of null vectors (u, v, f (z)) satisfies (6.53) for every z ∈ D. The following lemma provides an approximate solution of the Riemann–Hilbert problem for holomorphic null discs in Cn for any n ≥ 3 under the condition that the null discs attached at boundary points of the given central null disc Z have a constant direction vector u ∈ A∗ . (The original source is [14, Lemma 3.3]. For n = 3 this result is subsumed by Lemma 6.2.1. We have been unable to prove the exact analogue of Lemma 6.2.1 in dimensions n > 3.) Lemma 6.5.1. Fix an integer n ≥ 3 and let A = An−1 be the null quadric (2.54). Assume that u, v ∈ A∗ are null vectors such that Θ (u, v) = 0. Let Z : D → Cn be a holomorphic null immersion of class A 1 (D) whose derivative f = Z  : D → A∗ satisfies the following nondegeneracy condition: Θ (u, f (z)) = 0 and Θ (v, f (z)) = 0 for all z ∈ D.

(6.58)

Let r : T = bD → [0, 1] be a continuous function, and let σ : T×D → C be a function of class C 1 such that for every z ∈ T we have σ (z, · ) ∈ A 1 (D), σ (z, 0) = 0, the partial derivative σ2 := ∂∂ σξ is nonvanishing on T × D, and the winding number of the function T z → σ2 (z, 0) ∈ C∗ equals zero. Let κ : T × D → Cn be defined by κ(z, ξ ) = Z(z) + σ (z, r(z)ξ ) u,

z ∈ T, ξ ∈ D.

286

6 The Riemann–Hilbert Problem for Minimal Surfaces

Given numbers ε > 0 and 0 < ρ0 < 1, there exist a number ρ  ∈ [ρ0 , 1) and a  = Z(0) and the holomorphic null disc Z : D → Cn of class A 1 (D) such that Z(0) following conditions hold.  i) dist(Z(z), κ(z, T)) < ε for all z ∈ T.  ii) dist(Z(ρz), κ(z, D)) < ε for all z ∈ T and all ρ ∈ [ρ  , 1]. iii) Z is ε-close to Z in the C 1 topology on {z ∈ C : |z| ≤ ρ  }. Furthermore, if the function r is supported on a closed arc I ⊂ T and U is an open neighbourhood of I in D, then in addition to the above iv) we can choose Z to be ε-close to Z in the C 1 topology on D \U. Moreover, given an upper semicontinuous function φ : Cn → R∪{−∞} and a closed arc I ⊂ T, we can ensure in addition that the following condition holds: 1 |I|



 2π   it  dt  1  ) φ Z(e φ κ(eit , eis ) dtds + ε. ≤ 2π 2π|I| 0 I I

(6.59)

Remark 6.5.2. For every z ∈ T the map D ξ → σ (z, ξ )u is an immersed holomorphic disc directed by the null vector u ∈ A∗ ; the nonnegative function r(z) ≥ 0 is used to rescale them. If r is supported in a proper subarc I of T then it suffices to assume that σ (z, ξ ) is defined for z ∈ I, and in this case the winding number condition on the function ∂∂ σξ (z, ξ ) = 0 is irrelevant. (Compare with Lemma 6.2.1.) Condition (6.59) will be used in the analysis of minimal hulls in Sect. 9.2. Proof. Write A = An−1 . Fix a pair of null vectors u, v ∈ A∗ as in the lemma. Given a vector w ∈ A∗ such that the triple (u, v, w) satisfies condition (6.53), we write ψw := ψ(u,v,w) : C2 −→ A

(6.60)

where the map ψ(u,v,w) is defined by (6.56). By (6.57) we have that ψw (e1 ) = u and ψw (e2 ) = v. The conditions on σ ensure that the partial derivative σ2 (z, ξ ) :=

∂σ (z, ξ ), ∂ξ

(z, ξ ) ∈ T × D

(6.61)

(z, ξ ) ∈ T × D

(6.62)

admits a continuous square root 2 (z, ξ ) = σ



σ2 (z, ξ ),

such that σ2 (z, · ) ∈ A (D) for every z ∈ T. From this point on, the proof follows rather closely that of Lemma 6.2.1. We 2 as closely as desired uniformly on T × D by a rational function approximate σ 2 (z, ξ ) = σ

l

∑ B j (z) ξ j ,

j=0

(6.63)

6.5 The Riemann–Hilbert Problem for Null Discs in Cn for n > 3

287

where each B j is a C2 -valued Laurent polynomial. We adjust first the function  2 )2 and then σ (z, ξ ) = 0ξ σ2 (z,t)dt accordingly so that (6.61) and (6.62) still σ2 = ( σ hold. We also approximate (and replace) r by a nonvanishing function in A ∞ (D), with |r| ≤ 1, furnished by Lemma 6.2.3. Consider the sequence of functions √   √  2 z, cr(z)z2N+1 gN (z, c) = 2N + 1 c r(z) zN σ for z ∈ D, c ∈ T and N ∈ N. For big enough N, gN has no pole at z = 0 and hence  z 0

gN (ζ , c)2 dζ =

 z 0

  (2N + 1)cr(ζ )ζ 2N σ2 ζ , c r(ζ )ζ 2N+1 dζ .

By Lemma 6.2.5 we have that  z     2 2N+1   lim sup  gN (ζ , c) dζ − σ z, cr(z)z  = 0. N→∞ |z|≤1, c∈T

(6.64)

0

Note that the map f = Z  : D → A∗ is of class A (D), and (6.58) implies that the triple of null vectors u, v, f (z) ∈ A∗ satisfies condition (6.53) for every z ∈ D. Recall that the map ψw is defined by (6.60). Due to simple connectivity of the disc, the coefficients of the matrix function B (6.55) are well defined functions of class A (D), and there is a holomorphic map h = (u, v) : D → C2∗ satisfying ψ f (z) (h(z)) = ψ f (z) (u(z), v(z)) = f (z),

z ∈ D.

For big N ∈ N consider the map hN = (uN , vN ) : D × T → C2 ,   hN (z, c) = h(z) + gN (z, c) e1 = u(z) + gN (z, c), v(z) .

(6.65)

(6.66)

By general position, moving f and hence h slightly if necessary, we can ensure that the function v (the second component of h) has no zeros on T. Then, hN (z, c) = (0, 0) for all (z, c) ∈ D × T and all sufficiently big N ∈ N.

(6.67)

Indeed, we have v(z) = 0 in an annulus ρ1 ≤ |z| ≤ 1 for some ρ1 < 1, and hence (6.67) holds for such z. Since gN → 0 uniformly on {|z| ≤ ρ1 } × T as N → +∞ and h does not assume the value (0, 0), we see that (6.67) holds for N ≥ N1 for some N1 ∈ N. For such N the map   fN (z, c) := ψ f (z) hN (z, c) ∈ Cn , z ∈ D, c ∈ T has range in the punctured null quadric A∗ , and hence the map D z −→ ZN (z, c) = Z(0) +

 z 0

fN (ζ , c) dζ ∈ Cn

is an immersed holomorphic null disc for every c ∈ T.

288

6 The Riemann–Hilbert Problem for Minimal Surfaces

We claim that if N is big enough then Z = ZN (·, c) satisfies Lemma 6.5.1 for a suitable choice of c = c(N) ∈ T. Indeed, since all maps in the definition of ψ f (z) are either linear (given by a product with the matrices A−1 and B) or homogeneous quadratic (the projection ι given by (6.8)), we infer that ψ f (z) (gN (z, c)e1 ) = gN (z, c)2 u,

(z, c) ∈ D × T.

(6.68)

Since we also have ψ f (z) (h(z)) = f (z) by (6.65), we get that fN (z, c) = ψ f (z) (hN (z, c)) = f (z) + gN (z, c)2 u + RN (z, c)

(6.69)

for all (z, c) ∈ D × T. In order to estimate the remainder RN (z, c), note that the terms in fN (z, c) are of the following three kinds: (a) Terms which contain u2 , v2 or uv (where h = (u, v)); the sum of all such terms equals f (z) in view of (6.65). (b) Terms which do not contain any component u, v of h; the sum of all such terms equals gN (z, c)2 u in view of (6.68). (c) Terms which contain exactly one component u, v of h and exactly one copy of the function gN (z, c). All such terms are placed in the remainder RN . The terms described above are multiplied by various elements of the matrices A−1 (6.54) and B (6.55); those are functions in A (D) depending on f and h, but not depending on N. Integrating the equation (6.69) with respect to z as in (6.19), we see that the map ZN (z, c) is of the form (6.23) and the remainder EN (z, c) satisfies (6.24) (i.e., it goes to zero uniformly on D × T) as seen from (6.22) and (6.64). The proof is then concluded exactly as in the proof of Lemma 6.2.1.   Remark 6.5.3. An inspection of the proof of Lemma 6.5.1 gives the analogous conclusion for any family Zw : D → Cn (a spray) of null discs of class A 1 (D) depending holomorphically on a parameter w = (w1 , . . . , wm ) in a ball B ⊂ Cm centred at the origin, provided a couple of additional conditions hold. (Compare with Lemma 6.3.1 for the case n = 3.) First, we ask that the family of derivatives fw := Zw : D → A∗ for w ∈ B all satisfy condition (6.58) with respect to a given pair of null vectors u, v ∈ A∗ with Θ (u, v) = 0: Θ (u, fw (z)) = 0 and Θ (v, fw (z)) = 0 for all z ∈ D and w ∈ B.

(6.70)

With σ and r as in Lemma 6.5.1, we define the w-dependent family of maps κw (z, ξ ) = Zw (z) + σ (z, r(z)ξ ) u,

z ∈ T, ξ ∈ D, w ∈ B.

(6.71)

As before, there is a family of maps hw = (uw , vw ) : D → C2∗ of class A (D) depending holomorphically on w ∈ B and satisfying the analogue of (6.65): ψ fw (z) (hw (z)) = ψ fw (z) (uw (z), vw (z)) = fw (z),

z ∈ D, w ∈ B.

(6.72)

6.6 The Riemann–Hilbert Problem for Null Curves in Cn and Minimal Surfaces in Rn

289

The parameter w also enters in an obvious way in the definition (6.66) of the maps hw,N : D → C2 . We also assume that vw (z) = 0 for all z ∈ T and w ∈ B.

(6.73)

It follows as before that the map hw,N has range in C2∗ for all w ∈ B and all big enough N ∈ N (see (6.67)). The proof of Lemma 6.5.1 then furnishes a family of null curves Zw : D → Cn of class A 1 (D), depending holomorphically on w ∈ B, such that Zw satisfies conditions i)–iv) with respect to κw (6.71) for every w ∈ B. (Condition (6.59) will not be needed in this parametric case.)  

6.6 The Riemann–Hilbert Problem for Null Curves in Cn and Minimal Surfaces in Rn We now apply Lemma 6.5.1 and Remark 6.5.3 to obtain approximate solutions of the Riemann–Hilbert problem for null curves in Cn and minimal surfaces in Rn (n ≥ 3) parameterized by an arbitrary bordered Riemann surface. Theorem 6.6.1 (The Riemann–Hilbert problem for null curves in Cn ). Fix an integer n ≥ 3 and let A = An−1 ⊂ Cn denote the null quadric (2.54). Let M be a compact bordered Riemann surface, and let I1 , . . . , Ik be pairwise disjoint closed arcs in bM which are not connected components of bM. Choose an annular neighbourhood A ⊂ M of bM and a retraction ρ : A → bM. Assume that Z : M → Cn is a holomorphic null immersion of class A 1 (M), u1 , . . . , uk ∈ A∗ = A \ {0} are null vectors (the direction vectors), r : bM → [0, 1] is a continuous function supported on I := ki=1 Ii , and σ : I × D → C is a C 1 function such that for every z ∈ I, σ (z, · ) ∈ A 1 (D), σ (z, 0) = 0, ∂∂ σξ is nowhere vanishing on I × D, and • Λ ⊂ M˚ is a finite set.

• • • •

Consider the continuous map κ : bM × D → Cn given by  Z(z), if z ∈ bM \ I; κ(z, ξ ) = Z(z) + σ (z, r(z)ξ )ui , if z ∈ Ii , i ∈ {1, . . . , k}. Given ε > 0, d ∈ N, and a neighbourhood U ⊂ M of I, there exist a neighbourhood Ω ⊂ U ∩ A of I and a holomorphic null immersion Z : M → Cn of class A 1 (M) satisfying the following conditions.  i) dist(Z(z), κ(z, T)) < ε for all z ∈ bM.  ii) dist(Z(z), κ(ρ(z), D)) < ε for all z ∈ Ω . iii) Z is ε-close to Z in the C 1 topology on M \ Ω . iv) Z − Z vanishes to order d at all points in Λ .

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6 The Riemann–Hilbert Problem for Minimal Surfaces

Except for iv), the case n = 3 of this result corresponds to [23, Theorem 3.4] (this is subsumed by Theorem 6.4.2 above), and the general case is [14, Theorem 3.5]. Proof. Recall that Θ is the bilinear form (6.52) on Cn . For each i ∈ {1, . . . , k} we choose a null vector vi ∈ A∗ such that Θ (ui , vi ) = 0. Pick a holomorphic 1-form θ without zeros on M. Then dZ = f θ where f : M → A∗ is a map of class A (M). By Lemma 3.3.1 we can deform Z slightly and assume that f = dZ/θ is full. By a small deformation of the pairs of null vectors (ui , vi ) we may also assume that the following conditions hold on each of the arcs Ii (cf. (6.58)): Θ (ui , f (z)) = 0 and Θ (vi , f (z)) = 0 for all z ∈ Ii . By continuity there is a neighbourhood Ui ⊂ A ∩ U ⊂ M of the arc Ii such that the same conditions hold for all z ∈ Ui and i = 1, . . . , k. We now proceed as in the proof of Theorem 6.4.1. Write Λ = {a1 , . . . , am }. Choose a point p0 ∈ M˚ \ Λ and a family of embedded curves C = {C1 , . . . ,Cl } ˚ based at p0 , which contains a basis of the homology group H1 (M, Z) as well in M, as arcs from p0 to the points a1 , . . . , am , such that C = li=1 Cl is Runge in M. Let P denote the period map (3.12) associated to the family C . For each i = 1, . . . , k we choose a compact smoothly bounded simply connected domain Di ⊂ Ui abutting the boundary bM along the arc Ii such that Ii is contained in the relative interior of bDi ∩ bM. Set D = ki=1 Di . The domains Di are chosen pairwise disjoint and small enough such that C ∩ D = ∅. By Lemma 3.2.1 there exists a spray of maps fw : M → A∗ of class A (M), depending holomorphically on a parameter w in a ball B ⊂ CN for some N ∈ N and satisfying the following conditions. (a) The spray is dominating on D, i.e.,  ∂  fw (z) : CN −→ T f (z) A is surjective for every z ∈ D. ∂w w=0 (b) The spray is period dominating, i.e.,  ∂  P( fw ) : CN −→ (Cn )l is surjective. ∂w w=0 (c) The spray is fixed to order d at all points in Λ . For every i = 1, . . . , k the function r extends to bDi such that it vanishes on bDi \Ii . We can now complete the proof exactly as in Theorem 6.4.1. First, after shrinking the parameter ball B around 0, we can apply the parametric version of Lemma 6.5.1 (see Remark 6.5.3) on each disc Di in order to approximate the restricted spray fw |Di as closely as desired, uniformly on Di \ Ωi for a small neighbourhood Ωi ⊂ Di of the arc Ii , by a holomorphic spray gi,w : Di → A∗ of class A (Di ) such that the  integrals Zi,w (z) = z gi,w θ (z ∈ Di , w ∈ B) with suitably chosen initial values at some point pi ∈ Di satisfy conditions i)–iv) in Lemma 6.5.1. Assuming as we may

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291

that the approximation of fw |Di ⊂Ωi by gi,w is close enough for every i = 1, . . . , k, the domination property (a) of the spray fw allows us to glue the sprays fw and gi,w for i = 1, . . . , k on the Cartan pair (D0 , D) (see (6.46)) with M = D0 ∪ D into a new holomorphic spray f˜w : M → A∗ for w in a somewhat smaller ball B ⊂ B. (See (6.47) for the details of this gluing.) The period domination property of fw (condition (b)) implies that there exists a point w0 ∈ B close to 0 such that the map f˜w0 : M → A∗ satisfies P( f˜w0 ) = P( f ) (see (6.49)). We conclude as in the proof of Theorem   = Z(p0 ) + z f˜w θ 6.4.1 that the holomorphic map Z : M → Cn , defined by Z(z) p0 0 (z ∈ M), is a holomorphic null immersion satisfying the theorem provided that the approximations were sufficiently close.   We now adapt Theorem 6.6.1 to conformal minimal immersions. The original source for the next result is [14, Theorem 3.6]. Theorem 6.6.2 (Riemann–Hilbert method for minimal surfaces in Rn ). Assume that n ≥ 3 and the data M, I1 , . . . , Ik ⊂ bM, I = ki=1 Ii , r : bM → [0, 1], σ : I ×D → C, A ⊂ M, ρ : A → bM, and Λ are as in Theorem 6.6.1. Let X ∈ CMI1 (M, Rn ). For each i = 1, . . . , k let ui , vi ∈ Rn be a pair of orthogonal vectors satisfying |ui | = |vi | > 0. Consider the continuous map κ : bM × D → Rn given by  X(z), if z ∈ bM \ I; κ(z, ξ ) = X(z) + ℜσ (z, r(z)ξ )ui + ℑσ (z, r(z)ξ )vi , if z ∈ Ii , i ∈ {1, . . . , k}. Given ε > 0, d ∈ N, and a neighbourhood U ⊂ M of I, there exist a neighbourhood Ω ⊂ U ∩ A of I = ki=1 Ii and a conformal minimal immersion X ∈ CMI1 (M, Rn ) satisfying the following conditions.  i) dist(X(z), κ(z, T)) < ε for all z ∈ bM.  ii) dist(X(z), κ(ρ(z), D)) < ε for all z ∈ Ω . iii) X is ε-close to X in the C 1 norm on M \ Ω . iv) X − X vanishes to order d at all points in Λ . v) FluxX = FluxX . i = ui − ivi ∈ A∗ is a null vector Proof. The conditions on ui , vi ∈ Rn imply that u for every i = 1, . . . , k. Pick a holomorphic 1-form θ without zeros on M. Then, 2∂ X = f θ where f : M → A∗ is a holomorphic map of class A (M). We apply the i , but with the following proof of Theorem 6.6.1 to the map f and the null vectors u difference. At the last step of the proof we obtain a holomorphic map f˜w0 : M → A∗ in the new spray such that P( f˜w0 ) = P( f ). It follows that the real periods of f˜w0 over the closed curves in the family C vanish and the imaginary periods equal those of f . Hence, the map X : M → Rn given by  = X(p0 ) + X(z)

 z p0

ℜ( f˜w0 θ ),

z ∈ M,

is a conformal minimal immersion satisfying condition v). Conditions i)–iv) of X are verified exactly as in the proof of Theorem 6.6.1.  

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6.7 Exposing Boundary Points of Bordered Riemann Surfaces We shall need the following result on exposing boundary points of bordered Riemann surfaces from [147, Theorem 2.3] (see also [140, Theorem 9.9.1]). This result has been used in the construction of proper holomorphic embeddings of bordered Riemann surfaces into C2 (see [147, 148]), and it was generalized to strongly pseudoconvex domains of dimension > 1 in a series of papers where it was applied to the study of boundary behaviour of intrinsic metrics. Theorem 6.7.1. Assume that M is a compact connected smoothly bounded domain in a Riemann surface R. Choose finitely many pairwise distinct points a1 , . . . , ak ∈ M˚ and p1 , . . . , pm ∈ bM, and let E1 , . . . , Em be smooth embedded pairwise disjoint arcs in R such that Ei ∩ M = {pi } for i = 1, . . . , m. Let qi ∈ Ei be the other endpoint of Ei , and let bi be a point in the relative interior of Ei . Suppose that for every i = 1, . . . , m we are given neighbourhoods Ui  Ui of the point pi and a neighbourhood Vi ⊂ R of the arc Ei . Given s ∈ N there exists a smooth conformal diffeomorphism φ : M → φ (M) ⊂ R satisfying the following conditions. (a) φ is as close as desired to the identity in the smooth topology on M \ (b) φ is tangent to the identity map to order s at each point a1 , . . . , ak . (c) For every i = 1, . . . , m we have that φ (M ∩Ui ) ⊂ Ui ∪Vi ,

φ (pi ) = qi ,

˚ bi ∈ φ (M).

m

 i=1 Ui .

(6.74)

The last condition in (6.74) is new; it will be used in Chapters 7 and 8 to obtain hitting results for conformal minimal immersions of bordered Riemann surfaces. These conditions mean that the image M  = φ (M) of M is contained in a small neighbourhood of M ∪ ( m i=1 Ei ), it is close to M away from a small neighbourhood of each point pi ∈ bM, at pi it includes a spike reaching out to qi = φ (pi ) ∈ bM  , and every point bi is an interior point of M  ; see Figure 6.4.

Fig. 6.4 Effect of Theorem 6.7.1.

6.7 Exposing Boundary Points of Bordered Riemann Surfaces

293

In the proof of Theorem 6.7.1 we shall need the following lemma concerning small movements of finite sets of points in an open Riemann surface by injective holomorphic maps close to the identity on a smaller domain. The analogous result holds, with essentially the same proof, on any Stein manifold. Lemma 6.7.2. Let dist be a Riemannian distance function on an open Riemann surface R, and let Ω be an open relatively compact domain in R. Given a pair of disjoint finite sets A = {a1 , . . . , ak } ⊂ Ω , B = {b1 , . . . , bm } ⊂ Ω , and numbers s ∈ N and ε > 0, there is a δ > 0 such that the following holds. Given any set of points B = {b1 , . . . , bm } ⊂ R with dist(bi , bi ) < δ for i = 1, . . . , m, there exists an injective holomorphic map θ : Ω → R satisfying the following conditions. (i) θ (bi ) = bi for i = 1, . . . , m. (ii) θ agrees with the identity map on R to order s at every point of A. (iii) supz∈Ω dist(θ (z), z) < ε. Proof. For each i = 1, . . . , m we choose a holomorphic vector field ξi on R which vanishes to order s at all points in A ∪ B \ {bi } but does not vanish at bi . For example, we may take ξi = gi ξ where ξ is a nowhere vanishing holomorphic vector field on R and gi ∈ O(R) is such that gi (bi ) = 1 and gi vanishes to order s at the points in A ∪ B \ {bi }. Let θti denote the flow of ξi for complex time t. Choose a relatively compact domain Ω0 in R with Ω  Ω0 . There is an r0 > 0 such that for any z ∈ Ω the flow θti (z) with the initial point θ0i (z) = z exists for all |t| < r0 . Furthermore, there is an r ∈ (0, r0 ) such that the composition Θ (z,t1 , . . . ,tm ) = θt11 ◦ · · · ◦ θtmm (z) ∈ R,

z ∈ Ω,

is a well defined holomorphic spray of maps Ω → R defined for all t = (t1 , . . . ,tm ) ∈ rDm = (rD)m . Clearly we have Θ (· , 0) = Id. Decreasing r > 0 if necessary we may assume that Θ (· ,t) is ε-close to the identity map on Ω for all t ∈ rDm . Note also that Θ (z, · ) agrees with the identity to order s at all points z ∈ A, and  ∂  Θ (b j ,t) = ξi (b j ), ∂ti t=0

i, j = 1, . . . , m.

Our choice of the vector fields ξi implies that the m × m matrix (ξi (b j )) is diagonal with nonzero diagonal entries. The inverse function theorem then gives a δ > 0 such that for every n-tuple of points B = {b1 , . . . , bm } ⊂ R with dist(bi , bi ) < δ for i = 1, . . . , m there is a t ∈ rDm such that the map θ = Θ (· ,t) : Ω → R satisfies condition (i). By the construction, θ also satisfies conditions (ii) and (iii).   Proof of Theorem 6.7.1. We shall use [147, Theorem 2.3] and will only explain how ˚ for i = 1, . . . , m. (The to ensure the additional condition in (6.74) that bi ∈ φ (M) mentioned theorem and its proof can also be found in [140, Theorem 9.9.1].) Let Vi ⊂ Ω be a thin tubular neighbourhood of Ei as in the theorem, and let Ei ⊂ Ω be a smooth embedded arc intersecting Ei transversely at the point bi for i = 1, . . . , m. If Vi is chosen thin enough then Vi \ Ei is disconnected. By

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[147, Theorem 2.3] there is a conformal diffeomorphism φ : M → φ(M) satisfying all conditions in the theorem except perhaps the very last one in (6.74). These conditions imply that φ(M) intersects each of the arcs Ei (since M is connected and φ(M) contains the point qi = φ(pi ) as well as points close to pi , and these lie in different connected components of Vi \ Ei ). ˚ ∩ Ei for i = 1, . . . , m. If the neighbourhoods Vi are Choose a point bi ∈ φ(M) chosen thin enough, then bi is so close to bi that Lemma 6.7.2 applies with a given ε > 0, the pair of sets A = {a1 , . . . , ak , q1 , . . . , qm } and B = {b1 , . . . , bm }, and a relatively compact domain Ω  R containing φ(M). Let θ : Ω → R be the injective holomorphic map furnished by Lemma 6.7.2, fixing the points in A to order s and moving bi to bi for each i = 1, . . . , m. The conformal diffeomorphism φ = θ −1 ◦ φ : M → φ (M) then satisfies all conditions provided the number ε > 0 in the lemma is small enough. In particular, each point bi is an interior point of φ (M). This completes the proof.  

Chapter 7

The Calabi-Yau Problem for Minimal Surfaces

The famous Calabi-Yau problem for minimal surfaces concerns the existence and topological, geometric, and conformal properties of complete bounded (relatively compact) minimal surfaces in Rn and null curves in Cn for any n ≥ 3. The emphasis in our exposition is on the conformal Calabi-Yau problem, asking which open Riemann surfaces are the complex structures of complete bounded minimal surfaces in R3 ? Until recently, only the disc D = {z ∈ C : |z| < 1} was known to be such, because the complex structure of other previously known examples discussed in Section 7.1 could not be identified. Note that every Riemann surface satisfying this condition is hyperbolic in the sense of Definition 1.10.11. There are no restrictions on the topology of such surfaces according to L. Ferrer, F. Mart´ın, and W.H. Meeks [126]. Our main results on this subject are presented in Section 7.4. They were obtained by using new construction methods introduced in the papers [23, 14, 20] during 2015–2020. We focus on the orientable case; their analogues in the nonorientable framework can be found in our separate publication [29]. We begin with a brief history of the Calabi-Yau problem in Section 7.1. We recall Calabi’s conjectures from 1965 (see Conjectures 7.1.1 and 7.1.2) and explain the groundbreaking examples by L. P. Jorge and F. Xavier (of a complete nonflat minimal disc contained between two parallel planes in R3 , see Theorem 7.1.3) and N. Nadirashvili (of a complete bounded minimal disc in R3 , see Theorem 7.1.5). We also recall S.-T. Yau’s questions about this kind of minimal surfaces and review the literature which has been motivated by them. In Section 7.2 we show that the intrinsic diameter of an immersed compact manifold with boundary does not decrease much under a C 0 small perturbation; see Lemmas 7.2.1 and 7.2.2. This is used to guarantee completeness of limit maps in the proofs of our theorems on the Calabi-Yau problem. In Section 7.3 we prove the main new technical result, Lemma 7.3.1, which is used in our constructions on the Calabi-Yau problem. This lemma was obtained in [14]; an earlier version in a special case can be found in [23]. It shows that every conformal minimal immersion x : M → Rn of class CMI1 (M, Rn ) from a compact © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_7

295

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bordered Riemann surface can be uniformly approximated by immersions of the same type for which the boundary distance from a given interior point is as big as desired. The chief ingredient in the proof is the Riemann-Hilbert deformation method developed in Chapter 6. For the nonorientable case, see [29, Lemma 6.7]. Building upon Lemma 7.3.1, we develop the construction of complete bounded minimal surfaces in Euclidean spaces with the conformal structure of any given bordered Riemann surface, M, and with many additional features. In Theorem 7.4.1 we show the existence of a continuous map X : M = M ∪ bM → Rn for any n ≥ 3 whose restriction to the interior, M, is a complete conformal minimal immersion and whose restriction to the boundary, bM, is a topological embedding. Thus, X(M) is a complete minimal surface in Rn bounded by finitely many pairwise disjoint Jordan curves constituting X(bM). If n ≥ 5 then X can be chosen a topological embedding. The analogous result is proved for compact bordered surfaces of finite genus and countably many ends; see Theorem 7.4.3. Finally, we obtain complete bounded minimal surfaces whose complex structure is any proper domain in a compact Riemann surface in which the distance to the compact set of non-disc ends is infinite; see Theorem 7.4.9, Example 7.4.10, and Proposition 7.4.11. Analogous results hold in the nonorientable case; see Remark 7.4.5 and [29, Theorem 6.6]. Lemma 7.3.1 is also used in the construction of complete proper minimal surfaces in minimally convex domains in Rn for any n ≥ 3, given in Chapter 8. These results provide approximate solutions of the classical Plateau problem by complete minimal surfaces. (For a survey of the Plateau problem we refer for instance to H. B. Lawson [216, Chap. II] and R. Osserman [282].) Apart from the case of the disc D, they also furnish the first fairly general answer to the aforementioned conformal Calabi-Yau problem, showing in particular that every bordered Riemann surface is the complex structure of a complete bounded minimal surface. Combining our results with classical uniformization theorems, it follows that the same holds for every open Riemann surface of finite topological type without point ends and, more generally, for every open Riemann surface of finite genus and at most countably ends, none of which are point ends (see Corollary 7.4.7). In another direction, we furnish in Theorem 7.6.2 and Corollary 7.6.3 a complete conformal minimal immersion X : M → Rn from any bordered Riemann surface whose image X(M) contains any given countable subset of an arbitrary domain D ⊂ Rn , thereby yielding a hitting theorem. In particular, X may be chosen such that its image is a dense subset of the given domain D ⊂ Rn . Analogues of these results also hold for holomorphic (null) curves, and they are presented or mentioned throughout the chapter. In contrast to the results in Chapter 3, we do not know whether the results in this chapter generalize to immersed holomorphic curves in Cn which are directed by more general conical complex subvarieties A ⊂ Cn such that A\{0} is an Oka manifold, other than the null quadric. The reason is that the Riemann-Hilbert boundary value problem (see Chapter 6) has not been treated in such generality yet; every geometry requires a specific approach to this problem. So, this remains an open problem.

7.1 Examples by Jorge and Xavier and by Nadirashvili

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7.1 Examples by Jorge and Xavier and by Nadirashvili The Calabi-Yau problem for minimal hypersurfaces asks whether there are complete bounded (i.e., relatively compact) minimal hypersurfaces in Rn for n ≥ 3. Nothing seems known in this respect for n ≥ 4. However, the Calabi-Yau problem for surfaces, studying complete bounded minimal surfaces in R3 and, more generally, in Rn for arbitrary dimension n ≥ 3, has been an active focus of research in the past four decades, and a plethora of existence results are available in the literature. In this section we briefly discuss the amazing history of this subject. The story began in 1965 when E. Calabi posed the following conjecture. Conjecture 7.1.1. [209, p. 170] Every complete minimal hypersurface in Rn for any n ≥ 3 is unbounded. Calabi’s conjecture was also promoted by S. S. Chern [86, p. 212]. It is classical that there are no compact minimal submanifolds of Rn without boundary; this motivated Calabi to pose also the following more ambitious conjecture. Conjecture 7.1.2. [209, p. 170] Every complete nonflat minimal hypersurface in Rn has an unbounded projection to every (n − 2)-dimensional affine subspace. The following groundbreaking counterexample to Conjecture 7.1.2 was given by L. P. de M. Jorge and F. Xavier in 1980. Theorem 7.1.3 (Jorge and Xavier [203]). There is a complete nonflat conformal minimal immersion D → R3 with the range contained between two parallel planes. In fact, Jorge and Xavier constructed a complete conformal minimal immersion x = (x1 , x2 , x3 ) : D = {z ∈ C : |z| < 1} → R3 with x3 (z) = ℜ(z) for all z ∈ D, so the image of x is contained in the slab {x ∈ R3 : −1 < x3 < 1}. This was the first known application of the Runge approximation theorem for holomorphic functions (see Theorem 1.12.1) in the theory of minimal surfaces. We recall their proof. Proof. Choose an increasing sequence 0 < s0 < s1 < s2 < · · · with lim j→∞ s j = 1. For each j ∈ N set s j − s j−1 >0 (7.1) δj = 3 and   (7.2) K j = z ∈ C : s j−1 + δ j ≤ |z| ≤ s j − δ j , | arg((−1) j z)| ≥ δ j , where arg(·) is the principal branch of the argument with values in (−π, π]. So, K j is a compact domain in D obtained by deleting from the annulus of radii s j−1 + δ j and s j − δ j a small neighbourhood of a radial segment; if j is odd then the deleted radius lies in (−∞, 0), while if j is even then it lies in (0, +∞). (See Figure 7.1.) It turns out that the closed set K=

! j∈N

Kj ⊂ D

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Fig. 7.1 Jorge-Xavier’s labyrinth of compact sets in D.

forms a labyrinth of compact sets in D having the following two main properties. (i) D \ K is connected. (ii) Every divergent path in D of finite Euclidean length crosses infinitely many components of K. A path γ : [0, 1) → D is said to be divergent if the point γ(t) leaves any compact subset of D as t → 1, and that it crosses K j if there is a subpath in K j intersecting both the inner and the outer arcs of the annulus from which K j is obtained. Given a sequence of positive numbers {c j } j∈N , an inductive application of Runge’s Theorem 1.12.1 furnishes a nowhere vanishing holomorphic function g on D with (7.3) |g − c j | < 1 on K j for all j ∈ N. (Since D is simply connected, we can apply Runge’s theorem to the logarithms and exponentiate back. Alternatively, one may refer to the more general Theorem 1.13.3 with the target Oka manifold X = C∗ ; cf. Corollary 1.13.9.) Set φ3 = dz and denote by x = (x1 , x2 , x3 ) : D → R3 the conformal minimal immersion with the Gauss map g, given by the Weierstrass formula (2.84) for these data:  z

i 1

 1 1 −g , + g , 1 dζ , z ∈ D. x(z) = ℜ 2 g 2 g 0 Clearly, x3 (z) = ℜ(z) and hence the range of x lies in the slab {−1 < x3 < +1}. We claim that x is complete if the sequence {c j } j∈N is chosen to diverge fast enough. To this end, we have to show that any divergent curve in D has infinite length in (D, g), where g = x∗ ds2 is the metric induced on D via x by the Euclidean metric ds2 in R3 . In view of (2.85) we have that g=

1 4



2

1 + |g| |g|

|dz|2 ≥ max{1, |g|2 /4}|dz|2 .

(7.4)

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Let α be a divergent path in D. If α has infinite Euclidean length, then it also has infinite g-length by (7.4). If on the contrary α has finite Euclidean length then, in view of (ii), there is a strictly increasing map σ : N → N such that α crosses Kσ ( j) for all j ∈ N. By (7.1) and (7.2) the Euclidean length of this crossing is at least δσ ( j) . By (7.3) we have that |g| > cσ ( j) − 1 on Kσ ( j) , so in view of (7.4) the g-length of the crossing is > 12 (cσ ( j) − 1)δσ ( j) . Assuming as we may that c j ≥ 1 + 2/δ j for all j ∈ N, this number is ≥ 1. Since there are infinitely many crossings, the total g-length of α is +∞. This completes the proof.   The method by Jorge and Xavier gave rise to examples of complete nonflat minimal surfaces in R3 contained in a slab and having more complicated topologies. For example, H. Rosenberg and E. Toubiana [303] constructed annuli of this kind, while F. J. L´opez found a M¨obius strip [223] and orientable surfaces with arbitrary genus and finite topology [228]. A different method for constructing complete nonflat minimal surfaces between two parallel planes of R3 , relying on the use of power series with Hadamard gaps, was developed by F. F. de Brito [103] in 1992. His technique was further used by C. J. Costa and P. A. Q. Sim˜oes [102] in the construction of examples with any finite genus and finite number of ends. Combining the ideas of Jorge and Xavier with the approximation methods explained in Chapter 3 leads to the existence of complete conformal minimal immersions with one or more prescribed component functions. Indeed, as shown in Theorems 3.7.1 and 3.9.1, every nonconstant harmonic function on an open Riemann surface, M, is a component function of a complete conformal nonflat minimal immersion M → R3 . This extension of Jorge-Xavier’s Theorem 7.1.3 gives the following classification result which was obtained by A. Alarc´on, I. Fern´andez, and F. J. L´opez in [17, Corollary 4.3-(a)]; see also [16, 18]. Theorem 7.1.4. The following are equivalent for an open Riemann surface M. (i) M admits a nonconstant bounded harmonic function M → R. (ii) There is a complete nonflat conformal minimal immersion M → R3 with range contained between two parallel planes. This settles in an optimal way the conformal Calabi-Yau problem for surfaces in R3 with a bounded component function, asking when is an open Riemann surface the complex structure of a complete nonflat minimal surface in R3 with bounded projection into an affine line; see Conjecture 7.1.2. Coming back to the chronological history of Calabi’s conjecture, S.-T. Yau pointed out in his 1982 problem list [353, Problem 91] that, in spite of JorgeXavier’s Theorem 7.1.3, the question whether there are complete bounded minimal surfaces in R3 (i.e., Conjecture 7.1.1 for n = 3) remained open. This question became known in the literature as the Calabi-Yau problem. The following affirmative answer was given by N. Nadirashvili [268] in 1996. Theorem 7.1.5 (Nadirashvili [268]). There is a complete conformal minimal immersion D → R3 with the range contained in a ball.

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The example in Theorem 7.1.5 has negative Gaussian curvature, and hence Nadirashvili’s result also settled the question of J. Hadamard [177] from 1898 whether there is a complete bounded immersed surface in R3 with negative curvature. Like the construction by Jorge and Xavier in the proof of Theorem 7.1.3, Nadirashvili’s construction uses the Weierstrass representation of minimal surfaces and the Runge approximation theorem on a labyrinth of compact sets in the disc. The groundbreaking new idea in his method is to begin with a compact minimal disc in R3 and modify it near its boundary by applying a sequence of deformations which are approximately orthogonal to the position vector at every point. In view of Pythagoras’ theorem, this allows one to increase the intrinsic diameter of the surface while keeping the extrinsic diameter suitably bounded. Here is a brief explanation; details can be found in Nadirashvili’s original paper [268] and in the note by P. Collin and H. Rosenberg [97]. Proof of Theorem 7.1.5. Let X = (X1 , X2 , X3 ) : D = {z ∈ C : |z| ≤ 1} → R3 be a conformal minimal immersion of class CMI1 (D, R3 ); see (3.3). Assume that X(0) = 0 and choose r > 0 such that X(D) ⊂ B3R (0, r) = {x ∈ R3 : |x| < r}. Consider a labyrinth of finitely many compact sets in D as shown in Figure 7.2. Each connected component of the labyrinth consists of finitely many annular sectors joined together by a neighbourhood of a radial segment terminating on the circle T.

Fig. 7.2 Nadirashvili’s labyrinth of compact sets in D.

The following observations are in order. • Every path from the origin 0 ∈ D to the unit circle T = bD which avoids the labyrinth is fairly long both in the Euclidean metric and in the metric X ∗ ds2 , provided that the number of annular sectors in each connected component of the labyrinth is large and each component of the labyrinth is well intertwined with the two adjacent ones. • If the labyrinth is placed close enough to the boundary of the disc and the number of components of the labyrinth is large, then the diameter of the image X(K) ⊂ R3 of each component K is close to 0.

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301

Take a point p in a component K of the labyrinth. Up to composing by a linear isometry of R3 , we may assume that X(p) lies in the x3 -axis, and hence X(p) = (0, 0, X3 (p)),

|X3 (p)| < r.

(7.5)

Denote by (g, φ3 ) the Weierstrass data (2.83) of X. Fix a number s > 0. Given c > 0, Runge’s Theorem 1.12.1 provides a nowhere vanishing function h ∈ A (D, C∗ ) which is close to c on K and is close to 1 outside a small connected compact neighbourhood K  ⊂ D of K. Denote by X = (X1 , X2 , X3 ) : D → R3 the conformal minimal immersion of class CMI1 (D, R3 ) given by the Weierstrass data (hg, φ3 ) and  the initial condition X(0) = 0. The effect of this deformation is the following. • X3 = X3 everywhere on D (see the Enneper-Weierstrass formula (2.83)). • X is close to X in the C 1 topology on D \ K  (since h is close to 1 there). In particular, every path from the origin to the unit circle which avoids the union of K  and the labyrinth remains fairly long in the metric X∗ ds2 . Even more, if we have chosen the labyrinth with the components having enough annular sectors, then every path from the origin to the unit circle which avoids the labyrinth is fairly long, say of length > distX (0, T) + s, in the (possibly singular) metric X3∗ ds2 = X3∗ ds2 ; note that the singularities of this metric, if any, are isolated. This condition does not depend on the number of components of the labyrinth, but only on the number of annular sectors in them. • If c > 0 is big and α is a path in D crossing K, then the path X ◦ α is long, say of length > distX (0, T) + s, in view of (2.85); cf. (7.4). Here we say that a path α crosses K if it crosses one of the annular sectors of K as in the proof of Jorge-Xavier’s Theorem 7.1.3; see page 298. Thus, this deformation has enlarged the intrinsic diameter of the surface near K. We repeat this procedure in each component K of the labyrinth. Choosing the number c = cK > 0 sufficiently big at each step, we obtain a conformal minimal immersion X ∈ CMI1 (D, R3 ) such that distX (0, T) ≥ distX (0, T) + s, where T = bD. However, the problem is that the above procedure may also have√enlarged the extrinsic diameter and some points now lie outside of the ball B3R (0, r2 + s2 ). To amend this and keep the surface bounded, we need to cut away some pieces of the surface. We now explain this. Denote by distR3 the Euclidean distance on R3 and by distX the distance function on D induced by the metric X∗√ds2 . The above observations imply that for any point  ∈ R3 \ B3 (0, r2 + s2 ) the following holds, where Γ = bK  \ T: z ∈ K with X(z) R

302

7 The Calabi-Yau Problem for Minimal Surfaces (a)

  )) distX (z,Γ ) ≥ distR3 (X(z), X(Γ (b)

 ≈ distR3 (X(z), X(Γ ))

(c)

 − X(z)| ≈ |X(z)  (d) = (X1 (z) − X1 (z))2 + (X2 (z) − X2 (z))2 (e)

(f)

≈ |(X1 (z), X2 (z))| ≥ s.

The inequality (a) is obvious; (b) holds since X is close to X on D \ K˚  ; (c) follows from the fact that the Euclidean diameter of X(K  ) is close to 0 (here, we need to choose K  close enough to K at the beginning of the construction); (d) is implied by X3 = X3 on D; (e) follows from the fact that |(X1 , X2 )(z)| ≈ |(X1 , X2 )(p)| = 0 finally, the inequality (f) (see (7.5) and recall that X|K  is approximately constant); √  is ensured by Pythagoras’ theorem since |X(z)| ≥ r2 + s2 and |X3 (z)| = |X3 (z)| ≈ |X3 (p)| < r (again, see (7.5) and recall that X|K  is approximately constant). By the estimates obtained above, we may find a Jordan curve ϒ in D \ {0} bounding a simply connected domain D with 0 ∈ D ⊂ D ⊂ D such that   ⊂ B3R (0, r2 + s2 ) X(D) and distX (0, bD = ϒ ) ≥ distX (0,ϒ ) + s ≈ distX (0, T) + s, where the last estimate is ensured by choosing the labyrinth close enough to T. Roughly speaking, we obtain D by √ removing from D an annulus containing T and the preimage by X of R3 \ B3R (0, r2 + s2 ). These ideas, suitably elaborated, lead to the proof of the following key lemma of Nadirashvili. Lemma 7.1.6 ([268, p. 459]). Let X : D → R3 be a conformal minimal immersion of class CMI1 (D, R3 ) with X(0) = 0. Assume that X(D) ⊂ B3R (0, r) for some r > 0 and distX (0, T) > ρ for some ρ > 0, where distX denotes the distance function associated to the metric X ∗ ds2 on D. For any pair of positive numbers ε > 0, s > 0 there are a smoothly bounded simply connected domain D ⊂ D and a conformal minimal immersion X : D → R3 of class CMI1 (D, R3 ) satisfying the following conditions. • • • • •

(1 − ε)D = {z ∈ C : |z| ≤ 1 − ε} ⊂ D.  X(0) = 0.  − X(z)| < ε for all z ∈ (1 − ε)D. |X(z) √  ⊂ B3R (0, r2 + s2 ). X(D) distX (0, bD) > ρ + s.

Moreover, by Painlev´e’s theorem [283, 284] (see also Goluzin [160, p. 55]) we may precompose X by a smooth conformal diffeomorphism D → D and assume that D = D in the lemma.

7.1 Examples by Jorge and Xavier and by Nadirashvili

303

Theorem 7.1.5 is now proved as follows. Let X 1 : D → R3 be a conformal minimal immersion of class CMI1 (D, R3 ) with X 1 (0) = 0. Choose positive numbers r1 and ρ1 such that X 1 (D) ⊂ B3R (0, r1 ) and distX 1 (0, T) > ρ1 . Let ε1 > ε2 > ε3 > · · · be a sequence of positive numbers decreasing to 0 which will be specified later. Set s j = 1/ j for all j ∈ N and define, in a recursive way,  r j = r2j−1 + s2j , ρ j = ρ j−1 + s j , j ≥ 2. Note that the sequence r j is increasing and bounded, and hence it converges to some r > 0, whereas lim j→∞ ρ j = +∞. A standard inductive application of Lemma 7.1.6 furnishes a sequence of conformal minimal immersions X j : D → R3 ( j ≥ 2) of class CMI1 (D, R3 ) satisfying the following conditions. • • • •

X j (0) = 0. |X j (p) − X j−1 (p)| < ε j for all p ∈ D(0, 1 − ε j ). X j (D) ⊂ B3R (0, r j ). distX j (0, T) > ρ j .

If the sequence ε j decreases to 0 fast enough, then it is clear from these properties that the limit map X = lim j→∞ X j : D → R3 exists and is a complete conformal  minimal immersion whose image X(D) is contained in the ball B3R (0, r).  After Nadirashvili’s paper [268], Yau revisited the Calabi-Yau conjectures in his 2000 millenium lecture and proposed various new questions (see [354, p. 360] or [355, p. 241]). He asked in particular: What is the geometry of complete bounded minimal surfaces in R3 ? Can they be embedded? Are their spectra discrete? He further mentioned that, since their curvature tend to minus infinity, it is important to provide information on the asymptotic behaviour of these surfaces near their ends. Considerable progress has been made about these questions during the last two decades. Concerning the embedded Calabi-Yau problem, i.e., to determine whether there are complete bounded minimal surfaces in R3 without self-intersections, T. H. Colding and W. P. Minicozzi [93] proved that there is no such surface of finite topology. In fact, they showed that every complete embedded finitely connected minimal surface in R3 is proper in R3 , hence unbounded. Their result was extended by W. H. Meeks, J. P´erez, and A. Ros [251] to complete embedded minimal surfaces with finite genus and at most countably many ends. Thus, Calabi’s original conjecture holds in dimension 3 under these additional assumptions. The topological Calabi-Yau problem and the asymptotic Calabi-Yau problem deal, respectively, with the possible topological types and the asymptotic behaviour of complete bounded minimal surfaces in R3 . In this direction, Nadirashvili’s method was the seed of several construction techniques which gave rise to a large number of examples. Pioneering results are due to F. J. L´opez, F. Mart´ın, and S. Morales [232, 233] who provided examples with nontrivial topology, and to F. Mart´ın and S. Morales [241, 242, 243] who constructed complete minimal surfaces properly immersed in any given domain in R3 which is either convex, or bounded with smooth boundary. Following some partial results (see [240, 332, 19] and the

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7 The Calabi-Yau Problem for Minimal Surfaces

references in the following sections), L. Ferrer, F. Mart´ın, and W. H. Meeks [126] constructed complete properly immersed minimal surfaces with arbitrary topology, both orientable and nonorientable, in any domain in R3 as above. Concerning the spectrum of the Laplacian of complete bounded minimal surfaces in R3 , G. P. Bessa, L. P. Jorge, and J. P. Montenegro [64] answered Yau’s question affirmatively by showing, more generally, that the spectrum of a complete properly immersed surface in a ball of R3 is discrete whenever the norm of the mean curvature is sufficiently small. Later, G. P. Bessa, L. P. Jorge, and L. Mari [63] proved that the spectrum of a complete bounded minimal surface in R3 is discrete under some restrictions on the Hausdorff measure of its limit set. This result applies to many of the aforementioned examples. With such a rich collection of results and examples in hand, a natural question was whether there are complete bounded minimal surfaces in R3 whose conjugate surfaces are well defined and also bounded; equivalently, whether there are complete bounded holomorphic null curves in C3 . This is known as the Calabi-Yau problem for holomorphic null curves. In particular, F. Mart´ın, M. Umehara, and K. Yamada asked whether there exist complete properly immersed null curves in a ball, and complete bounded embedded null curves in C3 (see [245, Problems 1 and 2]). These questions were answered affirmatively in [40, 22] where complete properly embedded holomorphic null curves with arbitrary topology in any given convex domain of C3 were constructed; see [22, Corollary 6.2]. Despite an impressive body of literature on the subject, the following two questions remained open. Question 7.1.7. Let M be a compact bordered Riemann surface. • Is it possible to find a complete conformal minimal immersion X : M˚ → R3 , and a complete holomorphic null immersion Z : M˚ → C3 , without having to change the complex structure on M? • In addition, can one arrange that X is continuous on M and the image surface X(M) is bounded by finitely many Jordan curves constituting X(bM)? When M is the disc D, a partial positive answer to the second problem was given in 2007 by Mart´ın and Nadirashvili [244]. By a clever modification of Nadirashvili’s method in [268], they improved Theorem 7.1.5 as follows. Recall that T = bD denotes the unit circle. Theorem 7.1.8 (Mart´ın and Nadirashvili [244]). There exists a continuous map X : D → R3 whose restriction to the open disc D is a complete conformal minimal immersion and such that the curve X(T) ⊂ R3 is nonrectifiable and has Hausdorff dimension 1. Moreover, given a Jordan curve Γ in R3 and a number ε > 0, there is an X as above with δ H (Γ , X(T)) < ε, where δ H denotes the Hausdorff distance. Remark 7.1.9. In [244, Theorem 1] the authors claim that the continuous map X|T is an embedding, and hence X(T) is a Jordan curve. We have not been able to see how this follows from their exposition since the estimate (12) in [244, p. 297], which is crucial to ensure injectivity of the map X|T , does not seem to hold.  

7.1 Examples by Jorge and Xavier and by Nadirashvili

305

By using the same technique, Alarc´on [9] extended Theorem 7.1.8 to show the existence of compact complete minimal immersions in R3 with arbitrary finite topology, i.e., continuous maps X : Ω → R3 such that X|Ω : Ω → R3 is a complete conformal minimal immersion, where Ω is a relatively compact domain in an open Riemann surface. However, neither the conformal structure of Ω nor the topology of its frontier Ω \ Ω can be controlled by this method. Indeed, the use of Runge’s theorem in Nadirashvili’s technique does not enable one to control the placement in R3 of the entire surface, and hence one is forced to cut away small pieces of the surface in order to keep the image suitably bounded, thereby causing the aforementioned problems. By a different method, which relies also on RungeMergelyan type theorems from [38, 44] (see Sect. 3.6), Alarc´on and L´opez obtained analogous results for nonorientable minimal surfaces in R3 [42], holomorphic null curves in C3 , and complex curves in C2 [39]. Their technique still does not suffice to control the conformal structure of the surface or the topology of its limit set. An entirely new approach to the Calabi-Yau problem was presented by Alarc´on and Forstneriˇc in 2015 [23]. (Their paper was available on the arXiv in the Summer of 2013.) In this paper, the authors introduced the Riemann-Hilbert deformation method and the exposing of boundary points technique into this subject. These new techniques, presented in Chapter 6, enabled major breakthroughs in the theory of the Calabi-Yau problem. Their main advantage, when compared to those originating in Nadirashvili’s paper [268], is that they provide a precise control on the placement of the whole surface in the ambient space when performing modifications which increase the intrinsic diameter; in particular, no change of the complex structure on the surface is required in order to keep the extrinsic diameter bounded. The key technical ingredient is given by Lemma 7.3.1. By using this new technique, Alarc´on and Forstneriˇc proved in [23] that every bordered Riemann surface, M, embeds properly into the ball of C3 as a complete holomorphic null curve [23, Theorem 1]; its projection to R3 is then a bounded complete conformal minimal immersion M → R3 . Soon thereafter, the authors together with B. Drinovec Drnovˇsek improved the construction from [23] and extended it to any dimension n ≥ 3, thereby obtaining an affirmative answer to both the conformal and the asymptotic Calabi-Yau problem for an arbitrary bordered Riemann surface (see [14, Theorem 1.1]). In 2019, Alarc´on and Forstneriˇc [20] further extended this technique to bordered surfaces of finite genus and countably many non-point ends, showing that any such surface admits a complete conformal minimal immersion into R3 bounded by Jordan curves. These are currently the strongest known results on the Calabi-Yau problem, and they seem fairly close to the optimal ones. They are presented in Section 7.4 and proved in Section 7.5; see Theorems 7.4.1, 7.4.3, 7.4.9, and Corollary 7.4.7. A similar proof gives the Calabi-Yau property in several other geometries; see Theorems 7.4.12 and 7.4.14 where the latter result pertains to superminimal surfaces in any self-dual Einstein four-manifold. Reader, follow us to the continuation of the chapter where this new technique, and the results that it gives, are explained in detail.

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7.2 A Lower Bound on the Intrinsic Diameter In this section we show that the intrinsic diameter of an immersed compact manifold with boundary does not decrease much under a C 0 small perturbation of the immersion; see Lemmas 7.2.1 and 7.2.2. This will be used to guarantee completeness of limit maps in the proofs of theorems on the Calabi-Yau problem. Given a continuous path λ : [0, 1] → Rn , we denote by length(λ ) ∈ [0, +∞] its Euclidean length, i.e., the supremum of the sums ∑m i=1 |λ (ti ) − λ (ti−1 )| over all subdivisions 0 = t0 < t1 < · · · < tm = 1 of the interval [0, 1]. If λ is piecewise C 1  then length(λ ) = 01 |λ˙ (t)| dt, where |· | denotes the Euclidean length of a vector. For convenience of the reader we begin by recalling the relevant notions, some of which had been introduced earlier. Let M be an open (noncompact) connected manifold. A path γ : [0, 1) → M is said to be divergent if γ(t) leaves every compact subset of M as t → 1. More precisely, given a compact set K ⊂ M, there is a t0 ∈ (0, 1) such that γ(t) ∈ M \ K for all t0 ≤ t < 1. The path γ is said to be quasidivergent if for every compact K ⊂ M there is a t0 ∈ [0, 1) such that γ(t0 ) ∈ M \ K. Given a point p0 ∈ M we denote by Γ (M, p0 ) the set of divergent piecewise C 1 paths γ in M with γ(0) = p0 , and by Γqd (M, p0 ) the bigger set of quasidivergent piecewise C 1 paths in M with γ(0) = p0 . If g is a Riemannian metric on M then the intrinsic diameter of (M, g) with respect to a base point p0 ∈ M is the number   diamg (M, p0 ) = inf lengthg (γ) : γ ∈ Γ (M, p0 ) ∈ (0, +∞]. (7.6) 

˙ g dt. Changing the base point changes diamg (M, p0 ) by Here, lengthg (γ) = 01 |γ(t)| an additive constant. The Riemannian manifold (M, g) is complete, in the sense that the induced distance function distg : M × M → R+ is a complete metric on M, if and only if diamg (M, p0 ) = +∞ for some (and hence for all) p0 ∈ M. It is easily seen that the same minimum in (7.6) is obtained by using the bigger class Γqd (M, p0 ) of quasidivergent paths (see [20, Lemma 2.1]). More generally, given a continuous map X : M → Rn we define   diamX (M, p0 ) = inf length(X ◦ γ) : γ ∈ Γ (M, p0 ) ∈ [0, +∞]. (7.7) Here, length(X ◦ γ) is the Euclidean length of the path X ◦ γ : [0, 1) → Rn . If X : M → Rn is a smooth immersion and g = X ∗ ds2 is the induced Riemannian metric on M, then clearly diamX (M, p0 ) = diamg (M, p0 ). If M is the interior of a compact manifold M with boundary bM = M \ M, we also write distg (p0 , bM) = diamg (M, p0 ),

distX (p0 , bM) = diamX (M, p0 ),

(7.8)

and call these quantities the intrinsic diameter of (M, g) or (M, X), respectively. According to a theorem of J. Nash [269] from 1956 (see also Gromov [170]), every smooth Riemannian metric g on M is induced by a smooth embedding X : M → Rn for a sufficiently big n; however, (7.7) also applies to continuous maps.

7.2 A Lower Bound on the Intrinsic Diameter

307

Lemma 7.2.1. Let M be a compact connected C 1 manifold with boundary bM, and let Y : M → Rn be a C 1 immersion. Given a point p0 ∈ M˚ and a number η > 0, there exists a number ε > 0 such that for every continuous map X : M → Rn with X −Y C 0 (M) < ε we have that   ˚ p0 ) ≥ distY (p0 , bM) − η. inf length(X ◦ γ) : γ ∈ Γqd (M, Proof. This obviously holds if X is uniformly C 1 -close to Y on M˚ since a small C 1 perturbation changes lengths of curves by a small amount. Furthermore, any C 1 structure on a manifold is equivalent to a smooth structure by a theorem of H. Whitney [344, Lemma 24]. Hence, we may assume that M is a compact domain with  and Y is a smooth immersion Y : M  → Rn . C 1 boundary in a smooth manifold M (The latter statement again depends on a theorem of Whitney [343].)  denote the smooth normal bundle of the immersion Y : M  → Rn , so Let N → M  dim N = n. We identify M with the zero section of N. By the tubular neighbourhood theorem, Y extends to a smooth immersion F : N → Rn which agrees with Y on   Then, g = F ∗ (ds2 ) is a smooth Riemannian metric on N whose restriction to M M. ∗ 2 n 2 equals Y (ds ), and the map F : (N, g) → (R , ds ) is a local isometry. Let distg denote the distance function on N induced by the Riemannian metric g. We claim that there is a neighbourhood U ⊂ N of M such that distg,U (p0 , bM) > distg,M (p0 , bM) − η/2 = distY (p0 , bM) − η/2,

(7.9)

where distg,U (p0 , bM) is the distance from p0 to bM over all paths in U and η > 0 is as in the lemma. Here is an elementary proof. After shrinking N around M if  such that at any point x ∈ M,  the necessary, there is a smooth retraction ρ : N → M  in kernel ker(dρx ) of the differential dρx is the g-orthogonal complement of Tx M  Tx N. Since dρx equals the identity on Tx M, dρx has norm 1 in the metric g. Hence for any r > 1 there is a neighbourhood U ⊂ N of M such that dρx : Tx N → Tρ(x) N has g-norm less than r for every point x ∈ U. It follows that for every piecewise C 1 path γ : [0, 1] → U we have lengthg (ρ ◦ γ) ≤ r · lengthg (γ). Choosing γ to be a path in U connecting γ(0) = p0 to a point γ(1) ∈ bM, we obtain distg,M (p0 , bM) ≤ lengthg (ρ ◦ γ) ≤ r · lengthg (γ). The first inequality obviously holds even if ρ ◦ γ is not contained in M since it then crosses bM at some time t0 ∈ (0, 1), and the length of this shorter path is still ≥ distg,M (p0 , bM). Taking the infimum over all such paths γ gives distg,M (p0 , bM) ≤ r · distg,U (p0 , bM). If r is chosen close enough to 1 then (7.9) holds, thereby proving the claim. Since F : N → F(N) ⊂ Rn is a local isometry and M is compact, there is a number ε0 > 0 such that for every point p ∈ M the closed ball Bg (p, ε0 ) := {q ∈ N : distg (p, q) ≤ ε0 }

(7.10)

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7 The Calabi-Yau Problem for Minimal Surfaces

is contained in U, and F maps Bg (p, ε0 ) isometrically onto the closed Euclidean ball B(Y (p), ε0 ) ⊂ Rn . We may assume that 0 < ε0 < η/2. Fix a number ε with 0 < ε < ε0 /2. Assume that X : M → Rn is a continuous map satisfying X − Y C 0 (M) < ε. The above implies that there is a unique continuous map X : M → U such that X = F ◦ X

and

 distg (p, X(p)) = |X(p) −Y (p)| < ε for all p ∈ M.

(7.11)

Let γ ∈ Γqd (M, p0 ). There are a boundary point p ∈ bM and t0 ∈ (0, 1) such that distg (γ(t0 ), p) < ε.

(7.12)

By (7.11) we have that X ◦ γ = F ◦ γ, where the path γ = X ◦ γ : [0, 1] → U satisfies distg (γ(t), γ(t)) < ε for all t ∈ [0, 1).

(7.13)

Since F : (N, g) → (Rn , ds2 ) is a local isometry, we also have that lengthg (γ) = length(F ◦ γ) = length(X ◦ γ).

(7.14)

 By (7.12), (7.13), and the triangle inequality, the point γ(t0 ) = X(γ(t 0 )) satisfies distg (γ(t0 ), p) ≤ distg (γ(t0 ), γ(t0 )) + distg (γ(t0 ), p) < ε + ε ≤ ε0 < η/2. By adding to the path γ : [0,t0 ] → N an arc in the ball Bg (p, ε) (7.10) of g-length < η/2 connecting the point γ(t0 ) to p ∈ bM, we obtain a path λ : [0, 1] → U connecting λ (0) = p0 to λ (1) = p ∈ bM such that lengthg (λ ) < lengthg (γ) + η/2. We obviously have lengthg (λ ) ≥ distg,U (p0 , bM). Together with (7.9) we obtain lengthg (γ) > lengthg (λ ) − η/2 ≥ distg,U (p0 , bM) − η/2 > distY (p0 , bM) − η. In view of (7.14) it follows that length(X ◦ γ) > distY (p0 , bM) − η. Since γ ∈  Γqd (M, p0 ) was arbitrary, the lemma is proved.  The proof of Lemma 7.2.1 also provides the following lower estimate of the distance from an interior point p0 ∈ M˚ to a given compact subset E ⊂ M. Lemma 7.2.2. Let M be a compact connected C 1 manifold (either closed or with boundary), and let Y : M → Rn be a C 1 immersion. Given a nonempty compact subset E ⊂ M, a point p0 ∈ M \ E, and a number η > 0, there is a number ε > 0 such that for every continuous map X : M → Rn with X − Y C 0 (M) < ε and for every path γ : [0, 1) → M \ E with γ(0) = p0 such that γ(t) has a limit point in E as t → 1 we have that length(X ◦ γ) ≥ distY (p0 , E) − η.

7.3 C 0 Small Perturbations Enlarging the Intrinsic Diameter

309

7.3 C 0 Small Perturbations Enlarging the Intrinsic Diameter In this section we prove the following main technical lemma of this chapter. Lemma 7.3.1. Assume that M is a compact bordered Riemann surface with smooth boundary, n ≥ 3 is an integer, and X : M → Rn is a conformal minimal immersion of ˚ a point p0 ∈ M˚ \Λ , a positive integer class CMI1 (M, Rn ). Given a finite set Λ ⊂ M, d ∈ N, and numbers ε > 0 (small) and τ > 0 (big), there is a conformal minimal immersion X ∈ CMI1 (M, Rn ) satisfying the following conditions.  (i) |X(p) − X(p)| < ε for all p ∈ M. (ii) distX (p0 , bM) > τ. (iii) X − X vanishes to order d at every point in Λ . (iv) FluxX = FluxX .  bM : bM → Rn is an embedding. (v) X| Intuitively speaking, the lemma says that the intrinsic diameter of a conformal minimal immersion from a compact bordered Riemann surface can be increased by any given amount while keeping the map arbitrarily uniformly close to the original one. This lemma coincides with [14, Lemma 4.1], except for the interpolation condition (iii) which has been prepared in the previous chapter, and condition (v) which is achieved by the general position theorem; see Theorem 3.4.1 (d). A first partial result in the direction of Lemma 7.3.1 in the case M = D was given by F. Mart´ın and N. Nadirashvili in [244], and our proof uses some of their ideas. Lemma 7.3.1 provides an extremely useful tool for constructing complete conformal minimal immersions from a given open Riemann surface M to Rn . In particular, it is the principal ingredient in the proof of our main results on the Calabi-Yau problem, given by Theorems 7.4.1, 7.4.3, and 7.4.9. It is also used in the construction of proper complete conformal minimal surfaces in minimally convex domains in the following chapter. Basically, the lemma shows that, when inductively constructing a conformal minimal immersion M → Rn satisfying some desired conditions, guaranteeing completeness comes for free. For instance, Theorem 3.9.1 is trivially obtained by applying Lemma 7.3.1 in the proof of Theorem 3.6.1 (cf. condition (ii) in the lemma and property (gi ) in the proof of Theorem 3.9.1). Remark 7.3.2. Since conformal minimal immersions are harmonic maps, uniform approximation in C 0 (M) implies C r approximation on compact subsets of M˚ for all r ∈ N. However, if the number τ in Lemma 7.3.1 is chosen bigger than distX (p0 , bM), then the approximation in the C 1 (M) topology is clearly impossible. The true increase of the boundary distance takes place near the boundary bM.   Our proof Lemma 7.3.1 relies on the complex analytic methods developed in Chapters 3 and 6. In particular, we shall use the Mergelyan theorem for conformal minimal immersions (see Theorem 1.12.7), the Riemann-Hilbert problem for minimal surfaces (see Theorem 6.6.2), and the method of exposing boundary points

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of bordered Riemann surfaces (see Theorem 6.7.1). As in the proof of Nadirashvili’s Theorem 7.1.5, Pythagoras’ theorem plays a fundamental role. We shall obtain Lemma 7.3.1 by a finite inductive application of the following result. Lemma 7.3.3. Let M, n, X, Λ , p0 , and d be as in Lemma 7.3.1. Assume that Y : bM → Rn is a smooth map and choose positive numbers δ and μ such that |X(p) −Y (p)| < δ

for all p ∈ bM

(7.15)

and 0 < μ < distX (p0 , bM).

(7.16)

Given a compact set K ⊂ M˚ and numbers η > 0 and ε > 0, there is a conformal minimal immersion X ∈ CMI1 (M, Rn ) satisfying the following conditions.   (i) |X(p) −Y (p)| < δ 2 + η 2 for all p ∈ bM. (ii) distX (p0 , bM) > μ + η.  (iii) |X(p) − X(p)| < ε for all p ∈ K.  (iv) X − X vanishes to order d at every point in Λ . (v) FluxX = FluxX . Assuming for the moment that Lemma 7.3.3 holds, we now give Proof of Lemma 7.3.1. Choose numbers η0 and δ0 such that 0 < η0 < distX (p0 , bM) and 0 < δ0 < ε. Set  6(ε 2 − δ02 )

> 0. π Consider the following recursively defined sequences: 0 2 c 2 + c > 0, η j = η j−1 + > 0, δ j = δ j−1 j j2 c :=

j ∈ N.

It is easily seen that {η j } j∈Z+

+∞,

{δ j } j∈Z+

ε.

(7.17)

By using Lemma 7.3.3 we shall construct a sequence of conformal minimal immersions X j ∈ CMI1 (M, Rn ) ( j ∈ Z+ ) satisfying the following conditions: (i j ) (ii j ) (iii j ) (iv j )

|X j (p) − X(p)| < δ j for all p ∈ bM, distX j (p0 , bM) > η j , X j − X vanishes to order d at every point in Λ , and FluxX j = FluxX .

We begin by X0 := X. Assume inductively that we have a suitable X j for some j ∈ Z+ . To obtain a conformal minimal immersion X j+1 ∈ CMI1 (M, Rn ) satisfying conditions (i j+1 )–(iv j+1 ), it suffices to apply Lemma 7.3.3 to the data

7.3 C 0 Small Perturbations Enlarging the Intrinsic Diameter

X = Xj,

Λ,

p0 ,

d,

Y = X|bM ,

δ = δ j,

311

η=

c , j+1

μ = η j.

By the maximum principle and conditions (i j ) and (7.17), X j is ε-close to X in the C 0 (M) topology for all j ∈ Z+ . On the other hand, (ii j ) and (7.17) ensure that distX j (p0 , bM) > η j > τ for any large enough j ∈ Z+ . Therefore, in view of (iii j ) and (iv j ), X := X j satisfies Lemma 7.3.1 for any big enough j ∈ N.   Proof of Lemma 7.3.3. As in the proof of Nadirashvili’s Lemma 7.1.6, the key idea is to push the X-image of each boundary point of M for a distance approximately η in a direction approximately orthogonal to the vector X(p) − Y (p) ∈ Rn (cf. [244, proof of Lemma 3]). However, the realization of this idea is fairly different from Nadirashvili’s proof. In our case, this is done by combining two newly developed techniques presented in Chapter 6; here is the outline. In the first step, given by Lemma 7.3.4, we attach to the image X(M) ⊂ Rn at finitely many of its boundary points X(p j ), p j ∈ bM, long and highly oscilating curves with small extrinsic diameter. We may assume that M is a compact domain in a bigger Riemann surface R. By using the technique of exposing boundary points of M (see Theorem 6.7.1), we can precompose X by a suitably chosen conformal diffeomorphism φ : M → φ (M) ⊂ R to get a new conformal minimal immersion X = X ◦ φ : M → Rn which is close to X outside of small neighbourhoods of the  contains long strips boundary points p j , while at the same time the image X(M) close to the chosen curves, so the intrinsic boundary distance to the points p j in the metric X∗ ds2 becomes suitably big. In the next and the main step we apply a Riemann-Hilbert deformation in order to increase the boundary distance elsewhere, while at the same time keeping the effect of the first modification. We now embark on the details of the proof. We may assume that M is a smoothly bounded compact domain in an open Riemann surface R and, by Proposition 3.3.2 and Theorem 3.6.1, that X extends ˚ to a nonflat conformal minimal immersion X : R → Rn . Up to enlarging K in M, we assume that it is a smoothly bounded compact domain which is a deformation ˚ and, as we may in view of (7.16), that retract of M, that {p0 } ∪ Λ ⊂ K, distX (p0 , bK) > μ.

(7.18)

By slightly modifying the smooth map Y : bM → Rn if necessary, we also assume that Y (p) = X(p) for all p ∈ bM. (7.19) Finally, for simplicity of exposition we assume that bM is connected, being aware that the same argument applies to each of its connected components. Fix a number ε0 > 0 to be specified later. By (7.15), there are an integer l ≥ 3 and a family of compact connected arcs {α j ⊂ bM : j ∈ Zl = Z/lZ} such that ! j∈Zl

α j = bM,

(7.20)

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7 The Calabi-Yau Problem for Minimal Surfaces

and for every j ∈ Zl the arcs α j and α j+1 have a common endpoint p j and are otherwise disjoint, α j ∩ αa = ∅ for all a ∈ Zl \ { j − 1, j, j + 1}, |Y (p) −Y (q)| < ε0

for all {p, q} ⊂ α j ,

(7.21)

and |X(p) −Y (q)| < δ and |X(p) − X(q)| < ε0 for all {p, q} ⊂ α j .

(7.22)

For each j ∈ Zl , denote by   π j : Rn → span X(p j ) −Y (p j ) ⊂ Rn

(7.23)

the orthogonal projection onto the real line span{X(p j ) −Y (p j )}; see (7.19). In the first step of the proof, we perturb X near the points {p j : j ∈ Zl } to find a conformal minimal immersion in CMI1 (M, Rn ) which is close to X in the C 1 (K) topology and the intrinsic distance between p0 and each p j is large in a suitable way. Lemma 7.3.4. Let δ > 0 and ε0 > 0 be as in (7.15), (7.21), and (7.22). Given a number ε1 > 0, there is a conformal minimal immersion X0 ∈ CMI1 (M, Rn ) satisfying the following conditions. (A1) X0 |K is ε1 -close to X|K in the C 1 (K) topology. (A2) |X0 (p) −Y (q)| < δ and |X0 (p) − X(q)| < ε0 for all {p, q} ⊂ α j , j ∈ Zl . (A3) X0 − X vanishes to order d at every point in Λ . (A4) FluxX0 = FluxX . (A5) For each j ∈ Zl there is an open neighbourhood U j of p j in M, with U j ∩ K = ∅, such that if γ ⊂ M is an arc with the initial point in K and the final point in U j and {Ja }a∈Zl is a partition of γ by Borel measurable subsets, then

∑ length πa (X0 (Ja )) > η.

a∈Zl

Proof. Choose a family of pairwise disjoint Jordan arcs {γ j ⊂ R : j ∈ Zl } such that each γ j contains p j as an endpoint, γ j ∩ M = {p j }, and the compact set S := M ∪

!

γj ⊂ R

j∈Zl

is admissible (see Definition 1.12.9). Fix a nowhere vanishing holomorphic 1-form θ on R. We now extend X smoothly to the arcs γ j ( j ∈ Zl ) so as to obtain a generalized conformal minimal immersion (X, f θ ) ∈ GCMI1 (S, Rn ) satisfying the following two conditions. (a1) |X(p) −Y (q)| < δ and |X(p) − X(q)| < ε0 for all (p, q) ∈ (γ j−1 ∪ α j ∪ γ j ) × α j , j ∈ Zl . (a2) If j ∈ Zl and {Ja }a∈Zl is a partition of γ j by Borel measurable subsets, then

7.3 C 0 Small Perturbations Enlarging the Intrinsic Diameter

313

∑ length πa (X(Ja )) > 2η.

a∈Zl

(Here, πa is the projection (7.23).) To ensure these conditions, it suffices to define the map X on each arc γ j such that X(γ j ) has small extrinsic diameter and is highly oscillating in the direction of the vector X(pa ) −Y (pa ) = 0 for every a ∈ Zl . By Theorem 3.6.1 (the Mergelyan theorem for conformal minimal immersions) we obtain a conformal minimal immersion X : R → Rn having these properties and approximating X in C 1 (S). We replace X by X and drop the tilde. Let V ⊂ R be a small open neighbourhood of S. For each j ∈ Zl we denote by q j the endpoint of γ j different from p j . By (a1) and (a2), there are open neighbourhoods W j  W j of p j and V j of γ j in V \ K, j ∈ Zl , satisfying the following conditions. (a3) V j ∩ M  W j  W j  V \ K. (a4) |X(p) −Y (q)| < δ for all (p, q) ∈ (W j−1 ∪V j−1 ∪ α j ∪W j ∪V j ) × α j , j ∈ Zl . (a5) If γ j ⊂ W j ∪V j is an arc with the initial point in W j and the final point q j , and if {Ja }a∈Zl is a partition of γ j by Borel measurable subsets, then

∑ length πa (X(Ja )) > 2η.

a∈Zl

(a6) The compact sets W j ∪V j ( j ∈ Zl ) are pairwise disjoint. By Theorem 6.7.1 (on exposing of boundary points of a bordered Riemann surface) there is a smooth conformal diffeomorphism φ : M → φ (M) ⊂ V satisfying the following conditions.

(a7) φ is as close as desired to the identity in the smooth topology on M \ j∈Zl W j . (a8) φ is tangent to the identity map to order d at each point in Λ . (a9) For every j ∈ Zl we have φ (M ∩W j ) ⊂ W j ∪V j and φ (p j ) = q j ∈ φ (bM). It is not hard to check from properties (a1)–(a9), (7.21), and (7.22) that, if the approximation in (a7) is close enough, the conformal minimal immersion X0 := X ◦ φ ∈ CMI1 (M, Rn ) satisfies Lemma 7.3.4. (See [14, Proof of Lemma 4.3] for further details.)   We now continue with the proof of Lemma 7.3.3. Fix a number ε1 > 0 to be specified later. Let X0 ∈ CMI1 (M, Rn ) and {U j : j ∈ Zl } be given by Lemma 7.3.4. We assume without loss of generality that the sets U j are simply connected, smoothly bounded, and pairwise disjoint. Note that X0 satisfies conditions (i), (iii), (iv), and (v) in Lemma 7.3.3; see properties (A1)–(A4). Moreover, taking into account also (7.16) and (A5), we see that length(X0 ◦ γ) > μ + η for any path γ in M connecting p0 and bM ∩ j∈Zl U j . To conclude the proof of Lemma 7.3.3, we shall now perturb X0 outside j∈Zl U j , preserving what has already been achieved so far. It is at this point when the Riemann-Hilbert deformation technique for minimal surfaces enters the picture.

314

7 The Calabi-Yau Problem for Minimal Surfaces

Fix a number ε2 > 0 to be specified later, an annular neighbourhood A ⊂ M \ K of bM, and a smooth retraction ρ : A → bM. In view of (A2) there is a family of closed, smoothly bounded, pairwise disjoint discs D j in M \ K, j ∈ Zl , satisfying |X0 (p) −Y (q)| < δ

for all (p, q) ∈ D j × α j

(7.24)

and the following conditions (see Figure 7.3).

(B1) j∈Zl D j ⊂ A. (B2) D j ∩bM is a compact connected Jordan arc in α j \{p j−1 , p j } with an endpoint in U j−1 and the other endpoint in U j . (B3) ρ(D j ) ⊂ α j \ {p j−1 , p j } and |X0 (ρ(p)) − X0 (p)| < ε2 for all p ∈ D j , j ∈ Zl . Moreover, for each j ∈ Zl we choose a pair of compact connected Jordan arcs β j  I j  D j ∩ α j with an endpoint in U j−1 and the other endpoint in U j (see Figure 7.3), and a pair of vectors u j , v j ∈ Rn such that

Fig. 7.3 The sets U j , D j , β j , and I j .

uj , vj ,

X(p j ) −Y (p j ) |X(p j ) −Y (p j )|

 is an orthonormal set in Rn .

(7.25)

For the latter, recall that X − Y does not vanish anywhere on bM; see (7.19). Consider a continuous function r : bM → [0, 1] such that r = 1 on

!

βj

and r = 0 on bM \

j∈Zl

Define the function σ :



j∈Zl I j × D

!

Ij.

(7.26)

j∈Zl

→ C by

σ (p, ξ ) = ηξ ,

(p, ξ ) ∈

!

I j × D,

j∈Zl

and let κ : bM × D → Rn be the continuous map given by

(7.27)

7.3 C 0 Small Perturbations Enlarging the Intrinsic Diameter



315



if p ∈ bM \ j∈Zl I j ; if p ∈ I j , j ∈ Zl . (7.28) By Theorem 6.6.2 (the Riemann-Hilbert method for minimal surfaces) there are an arbitrarily small open neighbourhood Ω j ⊂ D j of I j in M ( j ∈ Zl ) and a conformal minimal immersion X ∈ CMI1 (M, Rn ) satisfying the following conditions. κ(p, ξ ) =

X0 (p), X0 (p) + ℜσ (p, r(p)ξ )u j + ℑσ (p, r(p)ξ )v j ,

 (C1) dist(X(p), κ(p, T)) < ε2 for all p ∈ bM.  (C2) dist(X(p), κ(ρ(p), D)) < ε2 for all p ∈ Ω := j∈Zl Ω j . (C3) X is ε2 -close to X0 in the C 1 norm on M \ Ω . (C4) X − X0 vanishes to order d at every point in Λ . (C5) FluxX = FluxX0 . Moreover, (C2), the definition of κ, and (B3) ensure that (C6) π j ◦ X is 2ε2 -close to π j ◦ X0 in the C 0 norm on Ω j for all j ∈ Zl . (Here, π j is the projection (7.23).) From the definition of κ (7.28) and conditions (A1)–(A5), (B1)–(B3), (C1)–(C6), (7.15)–(7.27) it is not hard to infer that the immersion X ∈ CMI1 (M, Rn ) satisfies the conclusion of Lemma 7.3.3 provided that the constants ε0 > 0, ε1 > 0, and ε2 > 0 are chosen sufficiently small. We shall explain how to verify conditions (i) and (ii); conditions (iii), (iv), and (v) are seen in a straightforward way. To check (i), fix a point p ∈ bM and let us see that   |X(p) −Y (p)| < δ 2 + η 2 . If p ∈ bM \ Ω then conditions (C3) and (A2) ensure that   |X(p) −Y (p)| ≤ |X(p) − X0 (p)| + |X0 (p) −Y (p)| < ε2 + δ
0 is chosen sufficiently small. Assume on the contrary that p ∈ bM ∩ Ω , and hence p ∈ bM ∩ Ω j for some j ∈ Zl . By (C1), (7.26), (7.27), and the definition of κ we obtain that    X(p)  − X0 (p) + ηr(p) (ℜξ u j + ℑξ v j )  < ε2 for some ξ ∈ T. (7.29) On the other hand, by (7.25), Pythagoras’ theorem, and the fact that 0 ≤ r ≤ 1 (see (7.26)), we have that    X(p j ) −Y (p j ) + ηr(p) (ℜξ u j + ℑξ v j ) = |X(p j ) −Y (p j )|2 + η 2 r(p)2  ≤ |X(p j ) −Y (p j )|2 + η 2 , and hence

316

7 The Calabi-Yau Problem for Minimal Surfaces

    X0 (p) + ηr(p) (ℜξ u j + ℑξ v j ) −Y (p) ≤  (A2),(7.21) < |X0 (p) − X(p j )| + |X(p j ) −Y (p j )|2 + η 2 + |Y (p j ) −Y (p)|  |X(p j ) −Y (p j )|2 + η 2 + 2ε0 . Combining this inequality with (7.29) we obtain    |X(p) −Y (p)| < |X(p j ) −Y (p j )|2 + η 2 + 2ε0 + ε2 < δ 2 + η 2 , where the second inequality holds by virtue of (7.15) provided that ε0 > 0 and ε2 > 0 are chosen sufficiently small. This implies condition (i). On the other hand, (A1) and (C3) give that X is (ε1 + ε2 )-close to X in the C 1 norm on K, and so, by (7.18) and assuming that ε1 > 0 and ε2 > 0 are chosen sufficiently small, we have that distX (p0 , bK) ≈ distX (p0 , bK) > μ. Thus, in order to check condition (ii) it suffices to see that distX (K, bM) is bounded below by a constant as close to η as desired provided that ε1 > 0 and ε2 > 0 are small. Choose an arc γ ⊂ M \ K˚ with the initial point in bK and the final point in  bM, and otherwise disjoint from K ∪ bM, and let us estimate length(X(γ)). If γ ∩ U j = ∅ for some j ∈ Zl , then (A5), (C3), and (C6) ensure that  length(X(γ)) > η. If on the contrary γ ⊂ M \ j∈Zl U j , then there are j ∈ Zl and a subarc γ of γ with the endpoints q ∈ M \ Ω and p ∈ β j such that γ ⊂ Ω j \ (U j−1 ∪U j ). Thus, by (7.25), (7.26) and (7.29), there is ξ ∈ T such that  |X(p) − X0 (p)| ≥ ηr(p)|ℜξ u j + ℑξ v j | − ε2 = η − ε2 . Therefore, using also (A2), (B3), and (C3), we obtain that   γ)) ≥ |X(q)  − X(p)|  length(X(γ)) ≥ length(X(  > |X(p) − X0 (p)| − ε0 − 2ε2 ≥ η − ε0 − 3ε2 . This implies condition (ii) provided that ε1 , ε2 > 0 are sufficiently small.  

7.4 The Main Results on the Calabi-Yau Problem In this section we present our main results on the conformal and the asymptotic Calabi-Yau problem for conformal minimal surfaces in Rn (n ≥ 3), as well as some related results in other geometries — for holomorphic curves and higher dimensional complex submanifolds in complex Euclidean spaces (Yang’s problem),

7.4 The Main Results on the Calabi-Yau Problem

317

holomorphic null curves in Cn for n ≥ 3, holomorphic Legendrian curves in an arbitrary complex contact manifold, and superminimal surfaces in self-dual Einstein four-manifolds. Proofs are given in the following section. We begin with the following result, due to the authors and B. Drinovec Drnovˇsek [14, Theorem 1.1], which provides an optimal solution to the Calabi-Yau problem for an arbitrary bordered Riemann surface. Interpolation at finitely many points (see condition (iv)) is included here for the first time. Theorem 7.4.1 (The Calabi-Yau problem for bordered Riemann surfaces). Assume that M is a compact bordered Riemann surface with smooth boundary and X0 : M → Rn is a conformal minimal immersion of class CMI1 (M, Rn ) for some n ≥ 3. Given a finite set Λ ⊂ M˚ and numbers ε > 0 and d ∈ N, there is a continuous map X : M → Rn satisfying the following conditions. (i) |X(p) − X0 (p)| < ε for all p ∈ M. (ii) X|M˚ : M˚ → Rn is a complete conformal minimal immersion. (iii) X|bM : bM → Rn is a topological embedding. In particular, X(bM) consists of finitely many pairwise disjoint Jordan curves. (iv) The difference X − X0 vanishes to order d at every point in Λ . (v) We have that FluxX = FluxX0 . (vi) If n ≥ 4 and X0 has simple double points on Λ , then X has simple double points. (vii) If n ≥ 5 and the map X0 |Λ : Λ → Rn is injective, then X is injective. Remark 7.4.2. In Theorem 7.4.1 it suffices to assume that the boundary bM is of class C k,α for some k ∈ N and 0 < α < 1. Indeed, by Corollary 1.10.9 and Theorem 1.10.10 there is a conformal diffeomorphism φ : M → M  of class C k,α onto a l  ˚ bordered surface of the form M = R\ i=1 Di , where D1 , . . . , Dl are closed, pairwise disjoint, smoothly bounded discs in a compact Riemann surface R. Precomposing a conformal minimal immersion M  → Rn of class C 1 (M  ) by φ then yields a  conformal minimal immersion M → Rn of class C 1 (M).  Theorem 7.4.1 shows in particular that every finite collection of smooth Jordan curves in Rn spanning a connected minimal surface can be uniformly approximated by families of Jordan curves spanning complete connected minimal surfaces. Hence, it can be viewed as an approximate solution of the Plateau problem by complete minimal surfaces. (For information on the Plateau problem we refer to Lawson [216, Chap. II], Osserman [282], or to any of numerous other sources on the subject.) By the isoperimetric inequality (see Section 2.7), minimal surfaces spanning rectifiable Jordan curves are not complete; hence, the boundary curves of surfaces in Theorem 7.4.1 are nonrectifiable at every point. However, using ideas from [244] we show that X can be chosen such that these curves have Hausdorff dimension 1. The following generalization of Theorem 7.4.1 to bordered surfaces of finite genus and with countably many boundary curves was obtained by Alarc´on and Forstneriˇc [20] in 2019.

318

7 The Calabi-Yau Problem for Minimal Surfaces

Theorem 7.4.3 (The Calabi-Yau problem for Riemann surfaces of finite genus and countably many boundary curves). Let R be a compact connected Riemann surface, and let M be a domain in R of the form M = R\

∞ !

Di ,

(7.30)

i=0

where {Di }i∈Z+ is a countable family of pairwise disjoint closed discs in R with boundaries of class C r for some r > 1. For every n ≥ 3 there exists a continuous map X : M → Rn such that X : M → Rn is a complete conformal minimal immersion and X(bM) = i X(bDi ) is the union of pairwise disjoint Jordan curves of Hausdorff dimension 1. If n = 4 then X : M → R4 can be chosen an immersion with simple double points, and if n ≥ 5 then X : M → Rn can be chosen an embedding. Remark 7.4.4. Our proof of Theorem 7.4.3 (see Sect. 7.5) gives several additions, similar to those in Theorem 7.4.1. In particular, the conformal minimal immersion X can be chosen with vanishing flux. Alternatively, one can prescribe the flux of X on any given finite family of homology classes in H1 (M, Z); however, we do not know whether X can be chosen with arbitrarily prescribed flux map. If we do not insist on controlling the flux, then we can choose a conformal minimal immersion Xi : Mi = R \ ij=0 D˚ j → Rn (i ∈ Z+ ) and find for any given ε > 0 a map X as in the theorem which is uniformly ε-close to Xi on M. With these additions, Theorem 7.4.3 includes Theorem 7.4.1 as a special case where M is of the form (7.30) with  finitely many complementary discs Di ; see Remark 7.4.2.  Remark 7.4.5. The analogues of Theorems 7.4.1 and 7.4.3, and also of Theorem 7.4.9 presented below, also hold for domains M of the form (7.30) in a compact nonorientable conformal surface, R. Indeed, there is an orientable double sheeted  and the preimage M  = ι −1 (M) is a covering ι : R → R by a Riemann surface R,  domain in R of the same type as M, i.e., with at most countably many boundary components which are not points. Hence, the methods from [29] provide an ι → Rn as in the above theorems invariant conformal minimal immersion X : M which passes down to a conformal minimal immersion X : M → Rn with the desired properties. (See Section 2.4 for more details on this reduction.) The appropriate version of Lemma 7.3.1 in the nonorientable situation is furnished by [29, Lemma 6.7]; by using this lemma, our proofs apply without changes.   The following related question was asked by D. Hoffman and B. White in a private communication with the second named author (Stanford, Feb. 11, 2019). Problem 7.4.6. Is there a closed Jordan curve C in R3 such that every minimal disc in R3 with boundary C is complete? We emphasize that the class of domains in Theorem 7.4.3 contains the conformal classes of all Riemann surfaces of finite genus with at most countably many ends, none of which are point ends. Indeed, the uniformization theorem of Z.-X. He and

7.4 The Main Results on the Calabi-Yau Problem

319

O. Schramm [188, Theorem 0.2] from 1993 solving Koebe’s conjecture says that every open Riemann surface, M  , of finite genus and at most countably many ends is conformally equivalent to a circle domain in a compact Riemann surface R, that is, a domain of the form ! M = R \ Di (7.31) i

whose complement is the union of at most countably many connected components Di , each of which is a closed geometric disc or a point. Here, a geometric disc in a Riemann surface R is a topological disc whose lifts in the universal covering R of R (which is the disc, the Euclidean plane, or the Riemann sphere) are round  The ends Di of M which are points are called point ends, while the discs in R. others are called disc ends. An annular end is a disc end which does not contain any limit points of other ends. A puncture end, or simply a puncture, is an end which is conformally isomorphic to the punctured disc; it corresponds to an isolated boundary point of a domain (7.31). The conformal type of an end is independent of a particular representation of a given open Riemann surface as a circle domain, and hence the above notions are well defined for open Riemann surfaces in this class. The theorem of He and Schramm includes as a special case open Riemann surfaces of finite topological type (i.e., those with finitely generated first homology group H1 (M, Z)) and says that every such surface is conformally equivalent to a domain in a compact Riemann surface whose complement consists of finitely many closed geometric discs and points. This was known earlier, see Stout [328, Theorem 8.1] and Corollary 1.10.9. In light of these results, Theorem 7.4.3 gives the following immediate corollary. Corollary 7.4.7. Every open Riemann surface of finite genus and with at most countably many ends, none of which are point ends, is the conformal structure of a complete bounded immersed minimal surface in R3 , and of a complete bounded embedded minimal surface in R5 . An open Riemann surface of finite type admits a bounded complete conformal minimal immersion into Rn for any n ≥ 3 if and only if it has no point ends. The second statement follows from the fact that a bounded harmonic function extends harmonically across an isolated point. On the other hand, it was proved by Colding and Minicozzi [93, Corollary 0.13] that a complete embedded minimal surface of finite topology in R3 is necessarily proper in R3 , hence unbounded; this was extended to surfaces of finite genus and having countably many ends by Meeks, P´erez, and Ros [251, Theorem 1.3]. In light of these results, Corollary 7.4.7 exposes a major dichotomy between the immersed and the embedded conformal Calabi-Yau problem in dimension n = 3. The situation concerning the conformal Calabi-Yau problem for Riemann surfaces more general than those in Theorem 7.4.3 is not understood. We mention the following open problems in this direction.

320

7 The Calabi-Yau Problem for Minimal Surfaces

Problem 7.4.8. (A) Let M be a domain of the form M = R\K in a compact Riemann surface R, where K is a nonempty compact subset of R. Assume that M admits a nonconstant bounded harmonic function h : M → (a, b) ⊂ R which does not extend to a bounded harmonic function on any bigger domain in R. Does M admit a bounded complete conformal minimal immersion into R3 ? (B) Is there a complete bounded minimal surface in R3 whose underlying complex structure is C \ K where K is a Cantor set in C? Is this true for every Cantor set satisfying the assumption in (A)? Recall (see [203, 16, 17, 18]) that every nonconstant bounded harmonic function h : M → (a, b) ⊂ R on an open Riemann surface M is a component function of a complete conformal minimal immersion X = (X1 , X2 , h) : M → R3 whose range is therefore contained in the slab {x = (x1 , x2 , x3 ) ∈ R3 : a < x3 < b}. Hence, the above questions are very natural. A particular case of Problem 7.4.8 (A) concerns surfaces with point ends on which disc ends cluster. We wish to thank Antonio Ros for having asked what could be said about the Calabi-Yau problem in this case (private communication, Granada, May 17, 2019). This seems a difficult problem, and the answer depends on how the disc ends approach the set of point ends. We have the following positive result from [20, Theorem 1.5] under the assumption that a compact set containing all point ends is at infinite distance from the interior due to the clustering of disc ends on it. Theorem 7.4.9. Assume that R is a compact Riemann surface and M is an open domain in R of the form M = R \ (D ∪ E), (7.32)

where E is a compact set in R and D = ∞ i=0 Di is the union of a countable family of pairwise disjoint closed geometric discs Di ⊂ R \ E. Fix a point p0 ∈ M and set Mi = R \ ij=0 D j for every i ∈ N. If lim distMi (p0 , E) = +∞,

i→∞

(7.33)

then there exists a continuous map X : M → R3 such that X|M : M → R3 is a complete 3 conformal minimal immersion and X|bD : bD = ∞ i=0 bDi → R is a topological embedding. In particular, M is the complex structure of a complete bounded minimal surface in R3 . The analogous result holds if R is a nonorientable conformal surface. The distance distMi (p0 , E) is measured with respect to any Riemannian metric on the ambient compact surface R. In particular, the set E may consist of point ends on which the disc ends Di cluster, and it may even be a Cantor set. The assumption (7.33) means that the disc ends form a labyrinth around E. Theorem 7.4.9 is proved in a similar way as Theorem 7.4.3. It remains an open problem to decide what can happen at point ends at finite distance from the interior. Most likely, there is no clear answer in this case. We now give an example of a domain in CP1 satisfying the requirements in Theorem 7.4.9. The example is inspired by the labyrinth constructed by L. Jorge and F. Xavier in [203] (see Figure 7.1).

7.4 The Main Results on the Calabi-Yau Problem

321

Example 7.4.10. Let 0 < a < b < 1 be a pair of numbers and let λ > 0. Choose numbers a < s0 < s1 < · · · < sk < b and for each j = 1, . . . , k define δ j > 0 and K j ⊂ bD \ aD as in (7.1) and (7.2). Up to a slight enlargement of each K j , we can assume that they are smoothly bounded pairwise disjoint closed discs. Setting K = kj=1 K j , we have distD\K (aT, bT) > λ provided that k ≥ 1 is chosen sufficiently large; assume that this holds and call Ka,b,λ = K. Pick a sequence 1 > b1 > a1 > b2 > a = 0. Set R = CP1 , D0 = CP1 \ D, and denote by D1 , D2 , . . . a2 > · · · with limi→∞ ∞ i the components of j=1 Ka j ,b j ,1 , ordered so that |zi | > |z j | for all zi ∈ Di and z j ∈ D j for any pair of indices i < j. It is clear that the domain M = R \ (D ∪ E) with E = {0}  satisfies (7.33) for Mi = R \ ij=0 D j and any point p0 ∈ M.  The idea in the previous example can also be used to show the following. Proposition 7.4.11. Let E be a proper compact subset of a compact Riemann surface R. Then there is a sequence of closed, smoothly bounded, pairwise disjoint discs Di ⊂ R \ E (i ∈ Z+ ) such that, setting Mi = R \ ij=0 D j and letting p0 ∈ M :=  ∞  R\ i=0 Di ∪ E , condition (7.33) holds. Hence, the Riemann surface M is the complex structure of a complete bounded minimal surface in R3 . Proof. Choose a point p0 ∈ R \ E and a Morse exhaustion function ρ : R \ E → R with ρ(p0 ) < 0. There is a sequences 0 < a1 < b1 < a2 < b2 < · · · converging to +∞ such that ρ has no critical values in [a j , b j ] for every j ∈ N. Hence, the set A j = {p ∈ M : a j ≤ ρ(p) ≤ b j } is a union of finitely many pairwise disjoint annuli for each j (see Theorem 1.4.7 (a)). By placing sufficiently many closed pairwise disjoint geometric discs in the interior of each connected component of A j , similarly to what was done in Example 7.4.10, the length (measured with respect to a fixed Riemannian metric on R) of any path crossing A j and avoiding the discs is at least 1. Doing this for every j ∈ N yields a countable sequence of pairwise disjoint closed discs Di ⊂ R \ (E ∪ {p0 }) such that the length of any path from p0 to E which avoids  all the discs Di is infinite and (7.33) holds.  Analogous results hold for several other classes of maps from bordered Riemann surfaces, with similar proofs. Here is a summary statement to this effect. Theorem 7.4.12 (Calabi-Yau property of null curves and Legendrian curves). Let M be a compact bordered Riemann surface. Any holomorphic immersion Z0 : M → Cn for n ≥ 2 can be uniformly approximated by continuous maps Z : M → Cn , injective if n ≥ 3, such that Z : M˚ → Cn is a complete holomorphic immersion (embedding if n ≥ 3) and Z(bM) = i Z(bDi ) is the union of pairwise disjoint Jordan curves Z(bDi ) of Hausdorff dimension one. If n ≥ 3 and Z0 : M → Cn is a holomorphic null immersion, then in addition Z can be chosen such that Z : M˚ → Cn is a complete holomorphic null embedding. Similarly, if n ≥ 3 is odd and Z0 : M → Cn is a holomorphic Legendrian immersion, then Z as above can be chosen such that Z : M˚ → Cn is a complete holomorphic Legendrian embedding. The analogous result holds for holomorphic Legendrian curves in an arbitrary complex contact manifold (X, ξ ). The analogue of Theorem 7.4.3 also holds in these geometries.

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7 The Calabi-Yau Problem for Minimal Surfaces

We refer to the papers [23, 14, 31] where the appropriate analogues of Lemma 7.3.1 were developed. The case of holomorphic curves is also discussed in Section 6.1, and holomorphic null curves are treated in Sections 6.2–6.5. The conclusion of Theorem 7.4.12 concerning holomorphic immersions and embeddings into Cn generalizes to an arbitrary complex manifold X in place of Cn by using standard complex analytic tools along with the RiemannHilbert modification technique. Questions in Problem 7.4.8 are likewise open for holomorphic curves, holomorphic null curves, and holomorphic Legendrian curves. Let us comment on the part of Theorem 7.4.12 concerning Legendrian curves. A complex contact manifold is a complex manifold X of necessarily odd dimension with a completely noninvolutive holomorphic hyperplane subbundle ξ of the complex tangent bundle T X; such ξ is called a complex contact structure on X. The standard complex contact structure on the Euclidean space C2n+1 (n ∈ N) with complex coordinates (x1 , y1 , . . . , xn , yn , z) is the kernel ξstd = ker αstd of the holomorphic 1-form αstd = dz + ∑nj=1 x j dy j . By the complex version of Darboux’s theorem [31, Theorem A.2], any complex contact structure ξ ⊂ T X on a complex manifold X 2n+1 is locally of this form in suitable local holomorphic coordinates at any point of X. A holomorphic map F : M → (X, ξ ) is said to be Legendrian if it is tangent to ξ at every point. The Calabi-Yau property for holomorphic Legendrian curves in (C2n+1 , ξstd ) was proved by the authors in [31, Theorem 1.2]. By using also the Darboux neighborhood theorem for noncompact holomorphic Legendrian curves [25, Theorem 1.1], the same result follows for Legendrian curves in an arbitrary complex contact manifold (see [25, Theorem 1.3] and also [132, Corollary 1.10] for a precise statement). Theorem 7.4.12 contributes to the collection of results, inspired by a question by P. Yang [351, 352] from 1977, on the existence and boundary behaviour of bounded complete immersed or embedded complex submanifolds in complex Euclidean spaces. Yang’s problem is an analogue of the Calabi-Yau problem for the special class of minimal surfaces given by complex curves (and higher dimensional complex submanifolds). The first examples in the line of Yang’s question were constructed by P. Jones in 1979 [201]. For recent developments, especially those concerning Yang’s problem for embedded complex submanifolds, we refer to the papers [8, 21, 27, 34, 35, 45, 158] and the 2019 survey [26] by the first two authors. In particular, the following is a compilation of results by J. Globevnik [158], A. Alarc´on [8, Corollary 1.2], and A. Alarc´on and F. Forstneriˇc [27]. Theorem 7.4.13. For every pair of integers 1 ≤ q < n there exists a holomorphic submersion f : Bn → Cq whose fibres are complete complex submanifolds of Bn . In particular, Bn can be foliated by complete properly embedded holomorphic discs. The methods used in the proof of this theorem are entirely different from those in the proof of our main results on the Calabi-Yau problem for minimal surfaces. Complex curves are much more special than arbitrary minimal surfaces and there are many additional tools for their construction at one’s disposal, in particular those coming from the Anders´en-Lempert theory of holomorphic automorphisms of Cn (see [140, Chapter 4]) and the theory of holomorphic submersions from

7.4 The Main Results on the Calabi-Yau Problem

323

Stein manifolds (see [134] and [140, Chapter 9]). These stronger tools allow for the construction of embedded complete complex curves of any given topology in the ball of C2 (see [34, 35]). No comparable tools are available to construct more general embedded minimal surfaces in Rn below dimension n = 5 where embeddings are not generic. A particular problem is that the Riemann-Hilbert deformation method may introduce double points of immersions for n < 5. Let us mention a recent application of the last part of Theorem 7.4.12 to superminimal surfaces in self-dual or anti-self-dual Einstein four-manifolds (X, g). These are oriented smooth Riemannian four-manifolds with very special metrics which play a major role in mathematics and theoretical physics. Recall (see M. Atiyah, N. J. Hitchin and I. M. Singer [55, p. 427]) that the Weyl tensor W = W + + W − is the conformally invariant part of the curvature tensor of a Riemannian four-manifold (X, g), meaning that it only depends on the conformal class of the metric. The manifold is called self-dual if W − = 0 and anti-self-dual if W + = 0. In particular, W = 0 if and only if the metric is conformally flat. A Riemannian manifold (X, g) is an Einstein manifold if the Ricci tensor of g is proportional to the metric, Ricg = kg for some constant k ∈ R. The curvature tensor of such g reduces to the constant scalar curvature (the trace of the Ricci curvature, hence 4k when dim X = 4) and the Weyl tensor W (see [55, p. 427]). The Einstein condition is equivalent to the metric being a solution of the vacuum Einstein field equations with a cosmological constant. Self-dual Einstein four-manifolds are important as gravitational instantons in quantum theories of gravity. A classical reference for Einstein manifolds is the monograph [65] by A. L. Besse. Superminimal surfaces form an interesting class of minimal surfaces in any four dimensional Riemannian manifold. Their characteristic feature is that the curvature ellipse at any point of the surface, determined by the second fundamental forms associated to unit normal vectors to the surface at that point, is a circle or a point. We refer to [131] for the precise definition and a survey of the literature on the subject. The following result combines Theorems 1.2 and 5.3 in [131]; the special case when X is the four-sphere S4 with the spherical metric was first obtained by A. Alarc´on, F. Forstneriˇc and F. L´arusson in [28, Theorem 7.5]. Theorem 7.4.14 (Calabi-Yau property of superminimal surfaces). Let (X, g) be an oriented four dimensional Einstein manifold whose Weyl tensor W = W + +W − satisfies W + = 0 or W − = 0. Given a compact bordered Riemann surface M and a conformal superminimal immersion f0 : M → X of class C 5 (M, X) of ± spin (with the choice of the sign depending on W ± = 0), we can approximate f0 uniformly on M by continuous maps f : M → X such that f : M˚ → X is a complete conformal superminimal immersion and f : bM → X is a topological embedding. The analogue of Theorem 7.4.3 also holds in this context. Remark 7.4.15. In [131] this result was stated for a conformal superminimal immersion f0 : M → X of class C 3 (M, X), but the Mergelyan approximation theorem for Legendrian curves in [132, Theorem 1.2] only supports the statement  with f0 ∈ C 5 (M, X) given above. 

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7 The Calabi-Yau Problem for Minimal Surfaces

This result follows from the last part of Theorem 7.4.12 by using Penrose twistor bundles; see [286, 287]. Here is a brief sketch; see [131] for the details. To every oriented smooth Riemannian four-manifold (X, g) one associates in a natural way a couple of twistor bundles π ± : Z ± → X; these are smooth fibre bundles with fibre CP1 whose total spaces (Z ± , J ± ) are real six dimensional almost complex manifolds, and the fibres of π ± are holomorphic rational curves in Z ± . The LeviCivita connection of (X, g) determines complex horizontal subbundles ξ ± ⊂ T Z ± projecting by dπ ± isomorphically onto the tangent bundle of X. Given a Riemann surface, M, the Bryant correspondence says that every conformal superminimal immersion M → X of ± spin lifts to a unique horizontal J ± -complex immersed curve M → Z ± , and vice versa. (The spin of an immersed superminimal surface gets reversed under a reversal of the orientation on X.) Furthermore, due to the above mentioned property of the twistor projections π ± , complete Legendrian curves in Z ± project onto complete conformal superminimal surfaces in X, and vice versa. This correspondence was discovered by R. Bryant [73] in 1982 in the case when X is the four-sphere S4 with the spherical metric, in which case both twistor spaces Z ± can be naturally identified with the three dimensional complex projective space CP3 such that the horizontal distibutions ξ ± are holomorphic contact bundles on CP3 . (By a result of C. Le Brun and S. Salamon [219], the complex contact structure on CP3 is unique up to holomorphic contactomorphisms and is given in suitable linear coordinates on any affine chart C3 ⊂ CP3 by the contact 1-form α = dz + xdy − ydx.) The Bryant correspondence was extended to twistor bundles over any oriented Riemannian four-manifold (X, g) by T. Friedrich [149] in 1984. The picture is completed by a couple of classical integrability theorems. According to M. Atiyah, N. J. Hitchin and I. M. Singer [55, Theorem 4.1], the twistor space (Z ± , J ± ) of a smooth oriented Riemannian four-manifold (X, g) is an integrable complex manifold if and only if the Weil tensor W = W + + W − of the metric g satisfies W + = 0 or W − = 0, respectively. Assuming that this holds, a result of S. Salamon [308, Theorem 10.1] (see also J. Eells and S. Salamon [120, Theorem 4.2]) says that the horizontal bundle ξ ± (with the respective choice of sign) is a holomorphic hyperplane subbundle of T Z ± if and only if g is an Einstein metric, and if this holds then ξ ± is a holomorphic contact bundle if and only if the (constant) scalar curvature of g is nonzero. Theorem 7.4.14 follows by combining this information with Theorem 7.4.12. If the (constant) scalar curvature of g is nonzero, we apply Theorem 7.4.12 for Legendrian curves in complex contact manifolds. If on the other hand the scalar curvature of g vanishes, then the horizontal bundle ξ ± is involutive and defines a holomorphic foliation on Z ± by complex hypersurfaces. Any horizontal curve lies in a leaf of this foliation, and hence we can apply Theorem 7.4.12 for immersed complex curves in complex manifolds of dimension at least two. These developments expose the following fundamental problem concerning minimal surfaces in non-Euclidean geometries.

7.5 Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9

325

Problem 7.4.16. Does the Calabi-Yau property (i.e., the analogues of Theorems 7.4.1 and 7.4.3) hold for minimal surfaces in every Riemannian manifold (X, g) of dimension at least 3? Although a connection to complex analysis is missing for a general metric, it is conceivable that the Riemann-Hilbert deformation method could be developed for conformal harmonic maps from bordered Riemann surfaces without a bypass to holomorphic maps. When dim X = 4, one might try to use the Eells-Salamon correspondence (see [120, Corollary 5.3]) between nonconstant conformal harmonic maps M → X and nonvertical J2 -holomorphic curves M → Z ± in the Penrose twistor bundles over X, where the (nonintegrable!) almost complex structure J2 is obtained from the standard one considered in [55] (and used in Theorem 7.4.14) by reversing the orientation on the fibres CP1 of the twistor projections Z ± → X.

7.5 Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9 The main ingredient in the proofs is Lemma 7.3.1 which enables us to increase the intrinsic diameter of an immersed conformal minimal surface M → Rn by an arbitrarily big amount while keeping the deformation arbitrarily C 0 small on M. Although the proofs follow a similar line of arguments, we prefer to begin by considering the geometrically simpler situation in Theorem 7.4.1 involving only finitely many boundary curves. In this proof, completeness of the limit map X : M → Rn is ensured by controlling the intrinsic diameter of an inductively chosen ˚ increasing sequence of compact domains Ki ⊂ M˚ exhausting M. Similar arguments are used in the proofs of Theorems 7.4.3 and 7.4.9, but with an important difference: the compacts Ki exhausting M are now chosen in advance and are only used to control the interior approximation and general position properties, while completeness of the limit map is obtained by combining the use of Lemma 7.3.1, which enables us to increase the distance to finitely many boundary curve at every induction step, and Lemmas 7.2.1 and 7.2.2, which provide a lower bound on the conformal radius under a uniformly small perturbation of the map. Proof of Theorem 7.4.1. In addition to the stated assumptions, we may assume that the following conditions hold; see Proposition 3.3.2 and Theorem 3.4.1.  • M is a compact smoothly bounded domain in an open Riemann surface M. r n • X0 ∈ CMI (M, R ) for any given integer r ≥ 1, and X0 is full. (By Theorem 3.6.1  Rn ), but this is beside the point.) we may assume that X0 ∈ CMI(M, n • The map X0 |bM : bM → R is an embedding. • If n = 4 and X0 has simple double points on Λ , then X0 : M → Rn has simple double points. • If n ≥ 5 and X0 |Λ is injective, then X0 : M → Rn is an embedding.  and let d : M  ×M →R Choose a nowhere vanishing holomorphic 1-form θ on M 2  denote the distance function on the Riemannian surface (M, |θ | ). Choose a compact

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7 The Calabi-Yau Problem for Minimal Surfaces

smoothly bounded domain K0 ⊂ M˚ with Λ ⊂ K˚ 0 and a point p0 ∈ K˚ 0 \ Λ . Set ε0 = ε/2. Using Lemma 7.3.1 inductively we obtain sequences of full conformal minimal ˚ and immersions Xi ∈ CMIr (M, Rn ), smoothly bounded compact domains Ki ⊂ M, numbers εi > 0 satisfying the following conditions for all i ∈ N. (1i ) (2i ) (3i ) (4i ) (5i ) (6i ) (7i ) (8i ) (9i ) (10i )



Ki−1 ⊂ K˚ i and max{d(p, bM) : p ∈ bKi } < 1/i. Hence, M = ∞ i=0 Ki . Xi − Xi−1 C r (Ki−1 ) < εi−1 . |Xi (p) − Xi−1 (p)| < εi−1 for all p ∈ M. distXi (p0 , bKk ) > k for all k ∈ {0, . . . , i}. FluxXi = FluxX0 . The difference Xi − X0 vanishes to order d at every point in Λ . The map Xi |bM : bM → Rn is injective. If n = 4 and X0 has simple double points on Λ , then Xi has simple double points. If n ≥ 5 and X0 |Λ : Λ → Rn is injective, then Xi : M → Rn is an embedding. We have 0 < εi < 12 min {εi−1 , δi , ς i }, where     ∂ Xi 1 δi := i min  (p) > 0, 2 p∈M θ

 1 1 ς i := 2 inf |Xi (p) − Xi (q)| : p, q ∈ bM, d(p, q) > > 0. i i

Alternatively, if n ≥ 5 and X0 |Λ : Λ → Rn is injective, then ς i is defined by the same formula but using all pairs of points p, q ∈ M with d(p, q) > 1i . (11i ) In addition to (10i ), εi > 0 is small enough such that every map Y : Ki → Rn of class C r (Ki ) with Y − Xi C r (Ki ) < 2εi is a full injective immersion (in case (9i )), or a full immersion with simple double points (in case (8i )). Note that the triple Ξ0 = (X0 , K0 , ε0 ) meets conditions (40 ), (50 ), (70 ), (80 ), and (90 ), whereas the remaining conditions are void. Assume inductively that for some i ∈ N there exist triples Ξ0 , . . . , Ξi−1 satisfying the above conditions. Lemma 7.3.1 furnishes the next map Xi ∈ CMIr (M, Rn ) and compact set Ki ⊂ M˚ satisfying conditions (1i )–(9i ). Indeed, the lemma together with the general position theorem (see Theorem 3.4.1) gives a map Xi ∈ CMIr (M, Rn ) satisfying conditions (5i )–(9i ) and such that Xi − Xi−1 C 0 (M) is arbitrarily small (hence conditions (2i ) and (3i ) hold), while distXi (p0 , bM) is arbitrarily big. Choosing the compact set Ki ⊂ M˚ big enough we can ensure that conditions (1i ) and (4i ) hold as well. Since X0 , . . . , Xi−1 are immersions, the number δi in (10i ) is positive. Condition (7i ) ensures that the number ς i in (10i ) is positive as well. (If condition (9i ) holds then the number ς i is positive also when using the alternative definition explained in (10i ).) Therefore, there exists εi > 0 satisfying conditions (10i ) and (11i ). This concludes the induction step and hence the construction of the sequence {Ξi }i∈N . By conditions (3i ) and (10i ) the sequence Xi converges uniformly on M to a continuous limit map X = limi→∞ Xi : M → Rn . More precisely, for every i ∈ Z+ and p ∈ M we have that

7.5 Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9

327

∞

∞

k=i

k=i

|X(p) − Xi (p)| ≤ ∑ |Xk+1 (p) − Xk (p)| < ∑ εk < 2εi . In particular, since 2ε0 < ε, X is ε-close to X0 in the C 0 norm on M; this gives condition (i) in the statement of the theorem. Moreover, we see as in the proofs of Theorems 3.6.1 and 3.9.1 that X|M˚ : M˚ → Rn is a full conformal minimal immersion satisfying all conditions in the theorem except perhaps (iii) and (vii). We shall now explain how to get condition (iii); the proof of (vii) is exactly the same by using the alternative definition of the number ς i in (10i ), taking into account all pairs of points p, q ∈ M with d(p, q) > 1i . Pick a pair of distinct points p, q ∈ bM and let us verify that X(p) = X(q). Take i0 > 1 such that 1i < d(p, q) for all i ≥ i0 . By (3i+1 ) and (10i ), given i ≥ i0 we have that |Xi (p) − Xi (q)| ≤ |Xi+1 (p) − Xi (p)| + |Xi+1 (q) − Xi (q)| + |Xi+1 (p) − Xi+1 (q)| < 2εi + |Xi+1 (p) − Xi+1 (q)| < ς i + |Xi+1 (p) − Xi+1 (q)| 1 ≤ 2 |Xi (p) − Xi (q)| + |Xi+1 (p) − Xi+1 (q)|, i and hence

  1 |Xi+1 (p) − Xi+1 (q)| > 1 − 2 |Xi (p) − Xi (q)|. i

It follows that |Xi0 +k (p) − Xi0 +k (q)| > |Xi0 (p) − Xi0 (q)|

i0 +k−1 

∏

i=i0

1−

1 i2

 for all k ∈ N.

Taking limits as k goes to infinity we obtain that 1 |X(p) − X(q)| ≥ |Xi0 (p) − Xi0 (q)| > 0, 2 where the latter inequality in ensured by (7i0 ). This establishes (iii) and hence completes the proof of Theorem 7.4.1.   Proof of Theorem 7.4.3. Assume that R is a compact Riemann surface and M is a domain in R of the form (7.30), that is, M = R \ ∞ k=0 Dk where {Dk }k∈Z+ is a countable family of pairwise disjoint closed discs in R. We shall construct a map X : M → Rn satisfying the conclusion of the theorem such that the complete conformal minimal immersion X : M → Rn has vanishing flux and the Jordan curves X(bDk ), k = 0, 1, 2, . . ., have Hausdorff dimension one. For every i = 0, 1, 2, . . . set Mi = R \

i !

D˚ k .

(7.34)

k=0

This is a compact bordered Riemann surface with boundary bMi =

i

k=0 bDk ,

and

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7 The Calabi-Yau Problem for Minimal Surfaces

M0 ⊃ M1 ⊃ M2 ⊃ · · · ⊃

∞ 1

Mi = M.

i=0

Denote by d a Riemannian distance function on R. Pick a point p0 ∈ M. By Theorem 3.4.1 (d) there exists a conformal minimal immersion X0 : M0 → Rn of class C 1 , with vanishing flux and such that X0 |bM0 : bM0 → Rn is injective. Pick numbers ε0 > 0 and τ0 ∈ N. An inductive application of Lemmas 7.2.1 and 7.3.1 furnishes a sequence of full conformal minimal immersions Xi : Mi → Rn , numbers εi > 0, and integers τi > i satisfying the following conditions for every i ∈ N. (ai ) distXi (p0 , bMi ) > i. (bi ) Xi : bMi → Rn is injective. Furthermore, Xi : Mi → Rn has only simple double points if n = 4 and is an embedding if n ≥ 5. (ci ) sup p∈Mi |Xi (p) − Xi−1 (p)| < εi−1 . (di ) For every continuous map X : Mi → Rn with X − Xi C (Mi ) < 2εi we have   inf length(X ◦ γ) : γ ∈ Γqd (M˚ i , p0 ) > distXi (p0 , bMi ) − 1 > i − 1. (Recall that Γqd denotes the set of quasidivergent paths; see p. 306.)   (ei ) We have 0 < εi < 21 min εi−1 , ς i , τi−i , where

 1 1 ς i := 2 inf |Xi (p) − Xi (q)| : p, q ∈ bMi , d(p, q) > > 0. i i (fi ) We have τi > τi−1 and for each k ∈ {0, . . . , i} there is a set Ai,k ⊂ Xi (bDk ) consisting of τii+1 points such that   1 max dist(p, Ai,k ) : p ∈ Xi (bDk ) < i , τi where dist(p, Ai,k ) = min{|p − q| : q ∈ Ai,k } is the Euclidean distance in Rn . (gi ) Xi has vanishing flux. Let us explain the induction step. Assume that for some i ∈ N we have maps X0 , . . . , Xi−1 and numbers ε0 , . . . , εi−1 and τ0 , . . . , τi−1 satisfying these conditions. This holds for i = 1 by using X0 , ε0 , and τ0 , and the above conditions are void except for (a0 ), (b0 ), and (g0 ). Lemma 7.3.1 applied to Xi−1 |Mi furnishes the next conformal minimal immersion Xi : Mi → Rn satisfying conditions (ai ), (bi ), (ci ), and (gi ). Pick τi ∈ N so large that (fi ) is satisfied; it suffices to choose i

τi > τi−1 + ∑ length(Xi (bDk )). k=0

Pick a number εi > 0 satisfying conditions (ei ). Finally, decreasing εi > 0 if necessary we may assume that (di ) holds as well in view of Lemma 7.2.1. The induction may now proceed.

7.5 Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9

329

Conditions (ci ) and (ei ) imply that the sequence Xi converges uniformly on M (7.30) to a continuous map X = limi→∞ : M → Rn satisfying ∞

∞

k=i

k=i

|X(p) − Xi (p)| ≤ ∑ |Xk+1 (p) − Xk (p)| < ∑ εk < 2εi ,

p∈M

(7.35)

for every i ∈ N. By Tietze’s extension theorem, we can extend X from M to a continuous map X : Mi → Rn such that the above inequality holds for all p ∈ Mi . By choosing the number εi > 0 sufficiently small at every step (in addition to the conditions (di ) and (ei )) we can also ensure that the restriction X : M → Rn to the interior is a conformal minimal immersion. This is done as in the proof of Theorem 7.4.1, exhausting M by an increasing sequence of compacts Ki ⊂ M and noting that sup-norm approximation of harmonic maps on M implies approximation in the C r norm on every Ki . We prefer not to clutter the proof with this addition which has already been adequately explained. Conditions (bi ), (ci ), and (ei ) imply that the limit map X : M → Rn is injective on bM = i∈Z+ bDi (see the proof of Theorem 7.4.1), whereas (gi ) ensures that X has vanishing flux. We claim that the limit map X : M → Rn is complete. Indeed, consider any divergent path γ : [0, 1) → M with γ(0) = p0 . There is an increasing sequence 0 < t1 < t2 < · · · < 1 with lim j→∞ t j = 1 such that p := lim j→∞ γ(t j ) ∈ bM. Then, p ∈ bDi0 for some i0 ∈ Z+ , and hence p ∈ bMi for all i ≥ i0 . Thus, γ is a quasidivergent path (see p. 306) in M˚ i for any i ≥ i0 . It follows from (7.35) that X extends to a continuous map X : Mi → Rn satisfying the same estimate (7.35) for all p ∈ Mi . Hence, conditions (ai ), (di ), and (ei ) imply length(X(γ)) > distXi (p0 , bMi ) − 1 > i − 1 for all i ≥ i0 . Letting i → +∞ shows that length(X(γ)) = +∞. Finally, we check that every Jordan curve X(bDk ) (k = 0, 1, 2, . . .) has Hausdorff dimension one. By (7.35), (ei ), and (fi ) we have for each i > k that  2  max dist(p, Ai,k ) : p ∈ X(bDk ) < i . τi

(7.36)

Since the set Ai,k consists of precisely τii+1 points and we have τii+1 (2/τii )1+1/i = 21+1/i ≤ 4 for every integer i > k, the inequality (7.36) implies that the Hausdorff measure H 1 (X(bDk )) is finite, and hence the Hausdorff dimension of X(bDk ) is at most one (cf. [244, Lemma 2.2] or [9, Sect. 4.1]; see also [265] for an introduction to the Hausdorff measure). On the other hand, X(bDk ) is homeomorphic to the circle and hence its Hausdorff dimension is at least one, so it is one.

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7 The Calabi-Yau Problem for Minimal Surfaces

By using the general position theorem for minimal surfaces as in the proof of Theorem 3.4.1, we can also ensure that the limit map X : M → Rn is an immersion with simple double points if n = 4, and that X : M → Rn is an embedding if n ≥ 5. We refer to the proof of Theorem 7.4.1 for the details.  

Proof of Theorem 7.4.9. For every i = 0, 1, 2, . . . let Mi = R \ ik=0 D˚ k be the compact domain (7.34). Choose a point p0 ∈ M and a conformal minimal immersion X0 : M0 → R3 whose restriction to bM0 is an embedding. Then, the given metric on R is comparable on M0 to the metric g0 = (X0 )∗ (ds2 ) induced by X0 , and hence conditions (7.33) hold for the latter metric as well. We shall use the same argument at every step when changing the metric. Condition (7.33) implies that there is i1 ∈ N such that distMi1 ,g0 (p0 , E) > 1. Choose a conformal minimal immersion X1 : Mi1 → R3 which approximates X0 uniformly on Mi1 , its restriction to bMi1 is an embedding, and we have that distMi1 ,X1 (p0 , bMi1 ) > 1

and

distMi1 ,X1 (p0 , E) > 1.

The first condition is achieved by Lemma 7.3.1, while the second one holds by Lemma 7.2.2 provided that X1 approximates X0 sufficiently uniformly closely on Mi1 . Let g1 = (X1 )∗ (ds2 ); this metric is comparable on Mi1 with g0 . Hence, by (7.33) there is an integer i2 > i1 such that distMi2 ,g1 (p0 , E) > 2. By the same argument as before we can find a conformal minimal immersion X2 : Mi2 → R3 which approximates X1 uniformly on Mi2 and satisfies distMi2 ,X2 (p0 , bDi2 ) > 2

and

distMi2 ,X2 (p0 , E) > 2.

Continuing inductively we get sequences of integers i1 < i2 < · · · and conformal minimal immersions Xk : Mik → R3 satisfying distMik ,Xk (p0 , bMik ) > k

and

distMik ,Xk (p0 , E) > k.

(7.37)

Assuming as we may that Xk approximates Xk−1 sufficiently closely uniformly on 7.4.3 that the sequence Mik for every k ∈ N, we can ensure as in the proof of Theorem / Xk converges uniformly on the compact set M  = ∞ i=0 M i to a continuous limit map X : M  → R3 whose restriction to the interior of M  is a complete conformal minimal immersion and X|M : M → Rn satisfies the conclusion of the theorem. (Here, M = M˚  \ E is given by (7.32).) In particular, the distance from p0 ∈ M to bM  ∪ E = i bCi ∪ E in the metric X ∗ (ds2 ) is infinite by (7.37) and Lemma 7.2.2. (See the proof of Theorem 7.4.3 for further details.)   So far we have been constructing bounded minimal surfaces in Rn which are complete with respect to the Euclidean metric. Any such surface is necessarily of

7.5 Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9

331

hyperbolic conformal type. On the other hand, every open Riemann surface can be immersed into Rn for any n ≥ 3 as a (possibly unbounded) minimal surface which is complete with respect to any given Riemannian metric on Rn . Here is a precise result in this direction. Theorem 7.5.1 (Proper conformal minimal surfaces in Rn which are complete in any given Riemannian metric). Let g be an arbitrary Riemannian metric on Rn for some n ≥ 3. Then, every open Riemann surface, M, admits a proper conformal minimal immersion into (Rn , ds2 ) (with simple double points if n = 4, embedding if n ≥ 5) which is g-complete, i.e. the image of any divergent path in M has infinite g-length in Rn . The analogous result holds for immersions and embeddings of M as holomorphic curves in Cn for n ≥ 2, as holomorphic null curves in Cn for n ≥ 3, and as holomorphic Legendrian curves in C2n+1 for any n ≥ 1. Proof. Assume that M is a connected open Riemann surface. Choose a smooth strongly subharmonic Morse exhaustion function ρ : M → R+ and increasing sequences 0 < a1 < b1 < a2 < b2 < · · · such that limi→∞ ai = limi→∞ bi = +∞ and ρ does not have any critical values on [ai , bi ] for all i ∈ N. For each i ∈ N the set Ai = {p ∈ M : ai ≤ ρ(p) ≤ bi } is a union of finitely many pairwise disjoint annuli, and Mi = {ρ ≤ bi } is a compact bordered Riemann surface. (See Section 1.4 and in particular Theorem 1.4.7 (a).) We inductively construct a sequence of conformal minimal immersions Xi : Mi → Rn such that Xi approximates Xi−1 sufficiently closely in C 1 (Mi−1 , Rn ), and any path in M crossing Ai has length bigger than 1 in the metric Xi∗ g on Mi . For the induction step, we first approximate Xi−1 as closely as desired on Mi−1 by a conformal minimal immersion Y : Mi → Rn by using Theorem 3.6.1 (the Mergelyan theorem for conformal minimal surfaces). Next, we apply Lemma 7.3.1 in order to find a conformal minimal immersion Xi : Mi → Rn which approximates Y as closely as desired uniformly on Mi (and hence in the smooth topology on Mi−1 ) and whose conformal radius with respect to the metric Xi∗ (ds2 ) is as big as desired. Since the required deformation takes place in a compact subset of Rn where the metric g is comparable to the Euclidean metric ds2 , we can arrange that the crossing distance for Ai in the metric Xi∗ g is bigger than 1 (or any given positive constant). This completes the induction step. By carrying out this procedure in a correct way, we obtain a sequence Xi converging to a conformal minimal immersion X : M → (Rn , ds2 ) such that the metric X ∗ g on M is complete. In dimensions n = 4 and n ≥ 5 we can also arrange that X is an immersion with simple double points and an embedding, respectively, by using Theorem 3.4.1 (the general position theorem). In order to construct proper immersions into Rn , we replace the use of Theorem 3.6.1 by Theorem 3.10.3 in the inductive construction. A similar argument applies to holomorphic (null, Legendrian) curves. We leave further details to the reader.  

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7 The Calabi-Yau Problem for Minimal Surfaces

7.6 Complete Dense Minimal Surfaces The problem of constructing complete minimal surfaces in R3 which are dense in R3 , or in some domain in R3 , has attracted considerable interest in the literature. The first such example was obtained by H. Rosenberg (unpublished) by Schwarzian reflection on a fundamental domain; his surface is simply connected. In 2004, J. A. G´alvez and P. Mira [155] found complete dense simply connected minimal surfaces in R3 as solutions to certain Bj¨orling problems. In 2014, F. J. L´opez [229] constructed complete dense minimal surfaces in R3 of arbitrary genus and parabolic conformal type by using Runge approximation. Finally, in 2018 A. Alarc´on and I. Castro-Infantes exploited the Mergelyan approximation theorem for minimal surfaces (see Theorem 3.6.1) to prove that every open Riemann surface admits a complete conformal minimal immersion into Rn (n ≥ 3) with dense image [11, Theorem 1.1]. This result is now trivially implied by Theorems 3.6.1 and 3.9.1 which give the following more general hitting result. Corollary 7.6.1. Let M be an open Riemann surface and n ≥ 3 be an integer. For any countable subset C ⊂ Rn there is a complete conformal minimal immersion X : M → Rn with C ⊂ X(M). Likewise, given a countable subset C ⊂ Cn there is a complete holomorphic null curve Z : M → Cn with C ⊂ Z(M). If the set C in the corollary is chosen dense in Rn or Cn then X(M) and Z(M) are dense subsets of Rn and Cn , respectively. Actually, Theorems 3.6.1 and 3.9.1 give considerably more. For instance, given a sequence of points ci ∈ Rn (i ∈ N) and a closed discrete subset Λ = {pi }i∈N ⊂ M, we can ensure that X(pi ) = ci for all i = 1, 2, . . .; that is, we can solve not only the hitting problem but also the interpolation problem. Furthermore, X can be chosen with arbitrary flux and, assuming that the interpolation condition on Λ ⊂ M allows it, to have simple double points if n = 4 and to be an injective immersion if n ≥ 5. Also, we can choose X to be arbitrarily uniformly close on a compact subset of M to a given conformal minimal immersion M → Rn ; in particular, we infer that conformal minimal immersions M → Rn which are complete and have dense images form a dense subset of CMI(M, Rn ) with respect to the compact-open topology. Thus, complete dense minimal surfaces in Rn are abundant. The analogues of all these facts also hold for holomorphic null curves in Cn . In a similar direction, P. Andrade [50] gave an example of a complete minimal surface in R3 , conformally equivalent to C, which is not dense in the whole space but whose closure nevertheless has nonempty interior. The question appears whether a given domain in R3 , or, more generally, in Rn for n ≥ 3, contains complete minimal surfaces as dense subsets. Building on the complex analytic methods developed in Chapters 3 and 6, it was also proved in [11, Theorems 1.1 and 1.2] that every domain D ⊂ Rn contains complete dense orientable minimal surfaces with any given topology or normalized by any given bordered Riemann surface.

7.6 Complete Dense Minimal Surfaces

333

We now prove the following more general hitting result with additional interpolation on a finite subset. Theorem 7.6.2. Let D ⊂ Rn (n ≥ 3) be a domain (i.e., a connected open subset) and let C ⊂ D be a countable subset. Assume that M = M ∪ bM is a compact bordered Riemann surface, Λ ⊂ M is a finite set, and r ∈ N is a positive integer. Then, every conformal minimal immersion X : M → Rn of class CMI1 (M, Rn ) with X(M) ⊂ D can be approximated uniformly on compact sets in M by complete conformal minimal immersions X : M → Rn satisfying the following conditions.  (i) C ⊂ X(M) ⊂ D. (ii) FluxX = FluxX . (iii) The difference X − X vanishes to order r at every point in Λ . (iv) If n = 4 and X has simple double points on Λ , then X has simple double points. (v) If n ≥ 5 and X is injective on Λ , then X is an injective immersion. The analogue of Theorem 7.6.2 for holomorphic null curves holds true and can be obtained by an obvious modification of the proof. A general domain D ⊂ Rn does not contain minimal surfaces with arbitrary conformal structure. In particular, a bounded domain contains no minimal surfaces of parabolic conformal type. So, in general, one can not choose M in the theorem to be any open Riemann surface. Likewise, interpolation on a countable discrete subset is impossible in general even when M is the disc. We shall easily obtain the following corollary to Theorem 7.6.2. Corollary 7.6.3. Let D ⊂ Rn (n ≥ 3) be a domain and C ⊂ D be a countable subset. (i) Every bordered Riemann surface, M, admits a complete conformal minimal immersion X : M → Rn such that C ⊂ X(M) ⊂ D. (ii) Every smooth open orientable surface, M, carries a complete minimal immersion X : M → Rn such that C ⊂ X(M) ⊂ D. In particular, if the set C is chosen dense in D in any of these two cases, then the complete minimal surface X(M) is dense in D as well. We shall prove Theorem 7.6.2 by an inductive application of the following lemma which shows how to perturb a conformal minimal immersion M → D so as to include a chosen point of D into the image. Lemma 7.6.4. Let D, M = M ∪ bM, r, and X : M → Rn be as in Theorem 7.6.2. Given pairwise distinct points a1 , . . . , ak ∈ M (k ∈ N), a neighbourhood Ω ⊂ M of a given point p ∈ bM, a point c ∈ D, and a number ε > 0, there is a conformal minimal immersion X : M → Rn of class CMI1 (M, Rn ) satisfying the following conditions.  (i) X(M) ⊂ D.  (ii) X − XC r (M\Ω ) < ε. (iii) The difference X − X vanishes to order r at a j for all j = 1, . . . , k.

334

7 The Calabi-Yau Problem for Minimal Surfaces

 (iv) c ∈ X(M). (v) FluxX = FluxX . Proof. Assume as we may that M is a smoothly bounded compact Runge domain  Take a smooth embedded arc E ⊂ M  \ M having in an open Riemann surface M. p as an endpoint and otherwise disjoint from M such that S = M ∪ E is a Runge  (Definition 1.12.9). Let q ∈ M  \ M denote the other endpoint admissible subset of M of E and fix a point b in the relative interior of E. Choose a generalized conformal minimal immersion (X0 , f θ ) ∈ GCMI1 (S, Rn ) (see (3.8)) satisfying the following conditions. (a) X0 |M = X. (b) X0 (E) ⊂ D. (c) X0 (b) = c. By Theorem 3.6.1 we may assume that X0 extends to a conformal minimal  → Rn . By condition (b) the compact set X0 (S) is contained in immersion X0 : M  such that the domain D, and hence there is an open neighbourhood U of S in M X0 (U) ⊂ D.

(7.38)

Next, we use the method of exposing boundary points on a compact bordered Riemann surface, explained in Sect. 6.7. Choose an open neighbourhood U   U of the point p ∈ bM, with U  ∩ M ⊂ Ω , and an open neighbourhood V  U of E. By Theorem 6.7.1 there is a smooth conformal diffeomorphism φ : M → φ (M) ⊂ U

(7.39)

satisfying the following conditions. (A) (B) (C) (D)

φ is as close as desired to the identity in the smooth topology on M \U  . φ is tangent to the identity map to order r at each point a1 , . . . , ak . φ (M ∩U  ) ⊂ U ∪V . b ∈ φ (M).

Set X = X0 ◦ φ ∈ CMI1 (M, Rn ). Properties (7.38) and (7.39) imply condition (i) in the lemma. Condition (ii) is ensured by (a) and (A) provided that the approximation in (A) is sufficiently close; recall that M \ Ω ⊂ M \ U  . Conditions (a) and (B) imply (iii), whereas (c) and (D) guarantee (iv). Finally, (v) follows from (a) and  So, X satisfies the conclusion of the lemma.  the definition of X.  Proof of Theorem 7.6.2. Fix a smoothly bounded compact domain K0 ⊂ M that is a strong deformation retract of M and satisfies Λ ⊂ K˚ 0 . Also, fix ε0 > 0. We assume without loss of generality that the countable set C ⊂ D contains X(Λ ). Fix a point p0 ∈ K˚ 0 \ X −1 (C), write C \ X(Λ ) = {c j : j ∈ N}, and set X0 = X : M → Rn . We shall inductively construct a sequence of conformal minimal immersions X j ∈ CMI1 (M, Rn ), smoothly bounded compact domains K j ⊂ M, points b j ∈ K˚ j , and numbers ε j > 0 satisfying the following conditions for every j ∈ N.

7.6 Complete Dense Minimal Surfaces

335

(1 j ) X j (M) ⊂ D. (2 j ) X j − X j−1 C r (K j−1 ) < ε j . (3 j ) K j−1 ⊂ K˚ j and j∈N K j = M. (4 j ) X j − X j−1 vanishes to order r at every point in Λ ∪ {b1 , . . . , b j−1 }. (5 j ) X j (bi ) = ci for all i ∈ {1, . . . , j}. (6 j ) FluxX j = FluxX . (7 j ) distX j (p0 , bKi ) > i for all i ∈ {0, . . . , j}. (8 j ) If n = 4 and X has simple double points on Λ , then X j : M → Rn has simple double points. (9 j ) If n ≥ 5 and X|Λ : Λ → Rn is injective, then X j : M → Rn is an embedding. (10 j ) We have 0 < ε j < ε j−1 /2, and every map Y : K j−1 → Rn of class C r (K j−1 ) such that Y − X j−1 C r (K j−1 ) < 2ε j is a full injective immersion (in case (9 j )) or a full immersion with simple double points (in case (8 j )), and it satisfies Y (K j−1 ) ⊂ D. With such sequences in hand, it is clear that the limit conformal minimal immersion X = lim X j : M → Rn j→∞

exists and satisfies the conclusion of the theorem. Note that conditions (4 j ) and   whereas (2 j ) and (10 j ) guarantee that X(M) ⊂ D; the (5 j ) imply that C ⊂ X(M)  completeness of X is ensured by (7 j ). The basis of the induction is given by K0 and X0 . For the inductive step we assume that we have suitable Xi , Ki , bi , and εi for all i ≤ j − 1 for some j ∈ N. Choose ε j > 0 so small that (10 j ) is satisfied. We first apply Lemma 7.6.4 to obtain a conformal minimal immersion X j ∈ CMI1 (M, Rn ) and a point b j ∈ M for which conditions (1 j ), (2 j ), (4 j ), (5 j ), and (6 j ) hold. Moreover, we choose X j so close to X j−1 in the C r (K j−1 ) norm that distX j (p0 , bKi ) > i for all i ∈ {0, . . . , j − 1}; take into account (7 j−1 ). Next, by Lemma 7.3.1 we may assume that distX j (p0 , bM) > j. Hence, we may choose K j so large that b j ∈ K˚ j and conditions (3 j ) and (7 j ) hold. Finally, Theorem 3.4.1 enables us to assume that X j satisfies conditions (8 j ) and (9 j ). This closes the induction and hence completes the proof of the theorem.   Proof of Corollary 7.6.3. Assertion (i) is trivially implied by Theorem 7.6.2 since every compact bordered Riemann surface M is known to carry a conformal minimal immersion X0 : M → Rn (see e.g. Theorem 3.6.1), which, up to composing with a homothety and a translation, can be assumed to take values in D. Assertion (ii) follows from Lemma 7.6.4 by a similar inductive construction as that in the proof of Theorem 7.6.2, but with some additions. Choose a complex structure on M and an exhaustion by smoothly bounded, Runge compact domains M0  M1  · · · 

! j∈Z+

Mj = M

336

7 The Calabi-Yau Problem for Minimal Surfaces

as in the proof of Theorem 3.6.1. We set K0 := M0 , choose a conformal minimal immersion X0 : K0 → Rn of class CMI1 (K0 , Rn ) with X0 (K0 ) ⊂ D, and inductively construct a sequence {X j , K j , b j , ε j } j∈N satisfying conditions (2 j )–(10 j ) in the proof of Theorem 7.6.2 and also the following ones. (1 j ) X j (K j ) ⊂ D. (11 j ) K j ⊂ M j and K j is a strong deformation retract of M j for all j ∈ N. With such a sequence in hand, the limit map X = lim X j : Ω = j→∞

!

K j → Rn

j∈N

is a complete conformal minimal immersion with C ⊂ X(Ω ) ⊂ D. Since, by (11 j ), Ω is diffeomorphic to M, X satisfies (ii). For the inductive construction, we adapt the proof of Theorem 7.6.2 by using in addition the Mergelyan theorem for conformal minimal immersions (see Theorem 3.6.1) to deal with the critical case when there is a change of topology between M j−1 and M j (see the proof of the referred result for more details). Moreover, we choose each domain K j in the inductive construction such that condition (1 j ) is satisfied. For the details in a similar framework, we refer to [11, Sect. 3.1].   We have already mentioned that the problem of interpolation of conformal minimal immersions M → D ⊂ Rn on closed discrete sequences in M is a much more subtle question than hitting, especially if D is a bounded domain. We pose the following problem in this direction. Problem 7.6.5. Find conditions on a pair of discrete sequences a j ∈ D and b j ∈ BnR ⊂ Rn such that there is a (proper) conformal minimal immersion X : D → B with X(ai ) = bi for all i ∈ N.

Chapter 8

Minimal Surfaces in Minimally Convex Domains

A major problem in minimal surface theory is to understand which classes of domains in Rn admit properly immersed or embedded minimal surfaces of a given conformal type, and how the geometry of the domain influences the behaviour of such surfaces. We refer to [250, Section 3] for background on this topic. In this chapter we show how the Riemann–Hilbert problem for minimal surfaces, developed in Chapter 6, provides a new method for the construction of proper conformal minimal immersions from any bordered Riemann surface, M, into a certain class of domains in Rn for any n ≥ 3 which contains all convex domains, but also many nonconvex ones. Previous constructions of minimal surfaces in convex domains can be found in many papers; see [14] and the references therein. Here we focus on new existence results in not necessarily convex domains. Our presentation is mainly based on the papers [14, 15]. The analogous results hold for nonorientable conformal surfaces as shown in [29, Section 6.4]. In Section 8.1 we introduce p-plurisubharmonic functions and p-convex domains in Rn for any pair of integers n ≥ 3 and p ∈ {1, 2, . . . , n}. The subject of partial positivity was initiated by H. Wu [348] in 1987 and continued in a series of papers by R. Harvey and H. B. Lawson [183, 184, 185]. The central algebraic concept underlying this theory is that of a p-positive quadratic form Q on a finitedimensional vector space V ∼ = Rn for some 1 ≤ p ≤ n = dimV . By definition, Q is (strongly) p-positive if the sum of its smallest p eigenvalues is nonnegative (resp. positive). Equivalently, the restriction of Q to any p-dimensional linear subspace L ⊂ V has nonnegative (resp. positive) trace: trL Q ≥ 0 (resp. trL Q > 0). A C 2 function u : D → R on a domain D ⊂ Rn is said to be p-plurisubharmonic if its Hessian Hessu (x; · ) (a quadratic form on Tx Rn , see (8.1)) is p-positive at any point x ∈ D. Similarly one defines the class of strongly p-plurisubharmonic functions. For p = 1 this notion coincides with the convexity of u, while for p = n this is the usual subharmonicity, Δ u ≥ 0. It turns out that u is p-convex if and only if its restriction u|M to any minimal p-dimensional submanifold M of D is a subharmonic function on M with respect to the metric Laplacian (see Proposition 8.1.2). By smoothing, these notions and results easily extend to upper semicontinuous functions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_8

337

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8 Minimal Surfaces in Minimally Convex Domains

A domain D ⊂ Rn is said to be p-convex if it admits a p-plurisubharmonic exhaustion function (see Definition 8.1.9 and Proposition 8.1.10). Hence, pplurisubharmonic functions and p-convex domains play a similar role in the analysis of p-dimensional minimal submanifolds in Rn as the usual plurisubharmonic functions (see Definition 1.5.8) and pseudoconvex domains play in the analysis of complex submanifolds in Cn and in more general complex manifolds. In particular, we shall see that this class of functions yields the strongest maximum principles for minimal surfaces. Of particular interest is the case p = 2, which is relevant for the analysis of 2dimensional minimal surfaces. Following the terminology introduced in [117], 2plurisubharmonic functions will also be called minimal plurisubharmonic functions, and 2-convex domains will be called minimally convex domains. A domain D ⊂ R3 with C 2 boundary is minimally convex if and only if it is mean-convex, in the sense that the mean curvature of its boundary bD from the interior side is nonnegative at every point (see Remark 8.1.17). As a limiting case, a domain in R3 bounded by an embedded minimal surface is minimally convex. More generally, a smoothly bounded domain D ⊂ Rn for n ≥ 3 is mean-convex if and only if it is (n − 1)-convex. In Section 8.2 we introduce the notion of a null plurisubharmonic function and develop an important analytic tool used in the sequel. The main results of the chapter are discussed in Section 8.3 and proved in Section 8.5 after we prepare the necessary technical ingredients in Section 8.4. In dimension n = 3 we provide essentially optimal results on the existence of proper (and complete if so desired) conformal minimal immersions from an arbitrary bordered Riemann surface into any minimally convex domain in R3 ; see Theorems 8.3.1, 8.3.3, and 8.3.4. We show in Remark 8.3.6 that these results fail for proper minimal surfaces in any wider class of domains in R3 . Similar results are obtained for 2-dimensional conformal minimal surfaces in (n − 2)-convex domains in Rn for any n ≥ 4; see Theorem 8.3.11. We expect that these results also hold for minimal surfaces in the bigger class of (n − 1)-convex domains in Rn , but we lack a sufficiently precise version of the Riemann–Hilbert modification method to prove them. The existing techniques using the Riemann–Hilbert boundary value problem for n > 3 only apply to flat boundary discs lying in parallel planes (see Theorem 6.6.2), but to obtain similar results in (n − 1)-convex domains would require the use of arbitrary families of conformal minimal discs, in analogy to what is done in Theorem 6.4.1 in dimension n = 3. The results in Section 8.3 show that every minimally convex domain in R3 admits many complete properly immersed minimal surfaces of hyperbolic conformal type. The situation is entirely different for proper minimal surfaces of finite total curvature. In Section 8.6 we prove that the only proper, smoothly bounded, minimally convex domains in R3 which contain a complete connected minimal surface of finite total curvature properly immersed in R3 are slabs and halfspaces (see Theorem 8.6.1); by the Hoffman–Meeks halfspace theorem [192] it follows that the minimal surface is then a plane. The essential ingredient in the proof of this result is a new maximum principle at infinity, given by Theorem 8.6.2.

8.1 p-Plurisubharmonic Functions and p-Convex Domains

339

8.1 p-Plurisubharmonic Functions and p-Convex Domains We begin this preparatory section by introducing the classes of p-plurisubharmonic functions and p-convex domains in Rn . We summarize the basic results concerning these classes, referring to the papers by F. R. Harvey and H. B. Lawson [183, 184, 185] and the references therein for a more complete account. In Subsection 8.1.5 we prove a Kontinuit¨atssatz for minimal p-dimensional submanifolds in p-convex domains (see Proposition 8.1.20) and a strong version of the maximum principle for such submanifolds (see Proposition 8.1.21).

8.1.1 p-Plurisubharmonic Functions Let x = (x1 , . . . , xn ) be coordinates on Rn . Given a domain D ⊂ Rn and a C 2 function u : D → R, the Hessian of u at a point x ∈ D is the quadratic form Hessu (x) = Hessu (x; · ) on the tangent space Tx Rn ∼ = Rn , given by Hessu (x; ξ ) =

∂ 2u (x) ξ j ξk , j,k=1 ∂ x j ∂ xk n

ξ = (ξ1 , . . . , ξn ) ∈ Rn .

∑

(8.1)

(The Hessian has already been introduced in Sect. 1.4, cf. (1.36).) The trace of the Hessian is the Laplace operator on Rn (1.37): tr (Hessu ) = Δ u =

n

∂ 2u

∑ ∂ x2 .

j=1

j

The Euclidean metric ds2 = ∑nj=1 dx j ⊗ dx j on Rn induces a Riemannian metric g = gM on any smoothly immersed submanifold M → Rn . A real function u ∈ C 2 (D) is said to be subharmonic on a submanifold M ⊂ D if ΔM (u|M ) ≥ 0, where ΔM is the Laplace operator on M associated to the metric gM (see (1.96)). In particular, if L is an affine p-dimensional subspace of Rn given by p   L = x(ξ ) = a + ∑ ξ j v j ∈ Rn : ξ1 , . . . , ξ p ∈ R , j=1

where a ∈ Rn and v1 , . . . , v p ∈ Rn is an orthonormal set, then u|D∩L is subharmonic if and only if the function ξ → u(x(ξ )) is subharmonic on {ξ ∈ R p : x(ξ ) ∈ D}. Recall that a function u : D → [−∞, ∞) = R ∪ {−∞} is said to be upper semicontinuous if lim sup u(x) ≤ u(x0 ) holds for each point x0 ∈ D. x→x0

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8 Minimal Surfaces in Minimally Convex Domains

An upper semicontinuous function u : D → [−∞, ∞) on a domain D ⊂ Rn which is not identically −∞ on any connected component of D is said to be subharmonic if it satisfies the sub-mean value property: for each x ∈ D and every r > 0 such that the closed ball B(x, r) ⊂ Rn centred at x of radius r is contained in D we have that u(x) ≤

1 Vol(S(x, r))



u dH n−1 ,

(8.2)

S(x,r)

where S(x, r) = bB(x, r) and H k denotes the k-dimensional Hausdorff measure on Rn . (See H. Federer [124] or F. Morgan [265] and note that, on a smooth kdimensional submanifold M ⊂ Rn , H k coincides with the k-dimensional volume measure determined by the restriction of the Euclidean metric to M; cf. (1.91).) The sub-mean value property clearly implies that u is regular upper semicontinuous: lim sup u(x) = u(x0 ) for every x0 ∈ D.

(8.3)

x→x0

Assuming that u satisfies (8.3), it is classical that u is plurisubharmonic if and only if Δ u ≥ 0 holds in the sense of distributions, that is, D u Δ v dV ≥ 0 for every smooth function v with compact support in D. If Δ u = 0 holds in the sense of distributions, then u is a harmonic function and equality holds in (8.2). Definition 8.1.1. Let p ∈ {1, . . . , n} be an integer. An upper semicontinuous function u : D → R ∪ {−∞} on a domain D ⊂ Rn is p-plurisubharmonic if u|L∩D is subharmonic for every p-dimensional affine plane L ⊂ Rn . A 2-plurisubharmonic function is also called a minimal plurisubharmonic function. Note that every p-plurisubharmonic function is regular, i.e. (8.3) holds. The space of all p-plurisubharmonic functions on a domain D ⊂ Rn is denoted by Psh p (D). For p = 2 we shall also use the notation Psh2 (D) = MPsh(D). An n-plurisubharmonic function on a domain D ⊂ Rn is a subharmonic function in the usual sense, and a 1-plurisubharmonic function is simply a convex function. From the characterization of subharmonic functions by the sub-mean value property (8.2) and Fubini’s theorem we easily see that Psh1 (D) ⊂ Psh2 (D) ⊂ · · · ⊂ Pshn (D).

(8.4)

Clearly, the space Psh p (D) is closed under addition and multiplication by nonnegative real numbers. Most of the familiar (to complex analysts) properties of plurisubharmonic functions on domains in Cn also hold for p-plurisubharmonic functions on domains in Rn (see e.g. [183, Section 6]). In particular, every pplurisubharmonic function can be approximated by smooth p-plurisubharmonic functions; see Proposition 8.1.6 for a more precise statement. The following result summarizes [185, Proposition 2.3 and Theorem 2.13].

8.1 p-Plurisubharmonic Functions and p-Convex Domains

341

Proposition 8.1.2. Let 1 ≤ p ≤ n be integers and D be a domain in Rn . The following conditions are equivalent for a function u ∈ C 2 (D). (a) The function u is p-plurisubharmonic on D (see Definition 8.1.1). (b) We have that trL Hessu (x) ≥ 0 for every p-dimensional affine subspace L ⊂ Rn and every point x ∈ D ∩ L. (Here, trL denotes the trace of the restriction to L.) (c) If λ1 (x) ≤ λ2 (x) ≤ · · · ≤ λn (x) are the eigenvalues of Hessu (x) then λ1 (x) + · · · + λ p (x) ≥ 0 for every x ∈ D.

(8.5)

(d) The restriction of u to any minimal p-dimensional submanifold M of D is a subharmonic function on M. Proof. The equivalences (a) ⇔ (b) ⇔ (c) are elementary, and (d) ⇒ (a) is obvious. The nontrivial implication (b) ⇒ (d) follows from the following formula which holds for every smooth submanifold M ⊂ Rn (cf. [182, Proposition 2.10]): ΔM (u|M ) = trM Hessu − Hu.

(8.6)

Here, trM Hessu is the trace of the restriction of the Hessian of u to the tangent bundle of M (a subbundle of T Rn |M ), ΔM is the Laplacian on M with the induced metric, and H is the mean curvature vector field of M (see (2.14)). If M is a minimal submanifold then H = 0 and we get that ΔM (u|M ) = trM Hessu ≥ 0. Since the proof of the formula (8.6) given in [182] takes considerable preparations in the subject of calibrated geometry, we offer here an elementary proof in the special case when M is a minimal 2-dimensional surface; this is the only case that we shall need. The result we want to prove is of local nature, so we may assume that M = f (D) is the image of a conformal minimal embedding f : D → Rn from the unit disc. We denote by z = x + iy the complex coordinate on the disc. (The coordinate on Rn is denoted x = (x1 , . . . , xn ), so this should not cause any confusion.) We identify the partial derivatives fx and fy with the vector fields f∗ ∂∂x and f∗ ∂∂y along M. Since f is conformal, these vector fields are orthogonal and satisfy | fx | = | fy |. Let T=

fx ∧ fy fx ∧ fy = | fx |· | fy | | fx |2

(8.7)

be the unit field of tangent 2-vectors along M. We denote by trTx (Hessu ) the trace of the Hessian of u restricted to the 2-plane span Tx ⊂ Tx Rn . The following is [117, Lemma 5.3]. Lemma 8.1.3. Given a conformal harmonic immersion f = ( f1 , . . . , fn ) : D → Rn and a C 2 function u on a neighbourhood of f (D), we have that dd c (u ◦ f )(z) = trT f (z) (Hessu )· | fx (z)|2 dx ∧ dy for any z ∈ D, where T is the unit 2-vector field along f (D) given by (8.7). Note that | fx (z)|2 dx ∧ dy = dA is the area form of the metric f ∗ (ds2 ) on D.

(8.8)

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8 Minimal Surfaces in Minimally Convex Domains

Proof. We have that d c (u ◦ f ) = ∑nj=1 dd (u ◦ f ) = c

n

∑d

j=1



∂u ◦f ∂xj

 ∂u

∂xj

 ◦ f · d c f j and hence

 ∧d fj = c

n

∑

i, j=1



 ∂ 2u ◦ f · d fi ∧ d c f j . ∂ xi ∂ x j

(We used that dd c f j = 0 since the map f is harmonic.) We also have d fi ∧ d c f j = ( fi,x dx + fi,y dy) ∧ (− f j,y dx + f j,x dy) = ( fi,x f j,x + fi,y f j,y ) dx ∧ dy. Inserting this identity into the previous formula yields  n    ∂ 2u c fi,x f j,x + fi,y f j,y dx ∧ dy ◦f dd (u ◦ f ) = ∑ ∂ x ∂ x i j i, j=1 = (Hessu ( f ; fx ) + Hessu ( f ; fy )) dx ∧ dy = trT f (·) (Hessu ) · | fx |2 dx ∧ dy. We used that | fx | = | fy | and fx · fy = 0 since f is conformal. This gives (8.8).   This completes the proof of Proposition 8.1.2.   It is natural to introduce the following subspace of Psh p (D). Definition 8.1.4. A function u ∈ C 2 (D) on a domain D ⊂ Rn is strongly pplurisubharmonic if trL Hessu (x) > 0 for every p-dimensional affine subspace L ⊂ Rn and every point x ∈ D ∩ L. Equivalently, if λ1 (x) ≤ λ2 (x) ≤ · · · ≤ λn (x) are the eigenvalues of Hessu (x) then λ1 (x) + · · · + λ p (x) > 0 for all x ∈ D. The analogue of Proposition 8.1.2 holds for strongly p-plurisubharmonic functions; in particular, we have the following result. Proposition 8.1.5. A function u ∈ C 2 (D) on a domain D ⊂ Rn is strongly pplurisubharmonic if and only if u|M is strongly subharmonic on every minimal pdimensional submanifold M ⊂ D. For any u ∈ Psh p (D) ∩ C 2 (D) and ε > 0 the function u(x) + ε|x|2 is strongly p-plurisubharmonic. The convolution of a p-plurisubharmonic function u with a radially symmetric approximate identity is a smooth p-plurisubharmonic function v ≥ u on a somewhat smaller domain. The following is an easy consequence. Proposition 8.1.6. Given u ∈ Psh p (D) and a relatively compact subdomain D  D, there is a decreasing sequence of smooth strongly p-plurisubharmonic functions uk ∈ Psh p (D ) ∩ C ∞ (D ) with limk→∞ uk = u on D . The following result is an immediate corollary of Definition 8.1.1 and Propositions 8.1.5 and 8.1.6. Corollary 8.1.7. An upper semicontinuous function u : D → R ∪ {−∞} belongs to Psh p (D) if and only if u|M is subharmonic on every minimal p-dimensional submanifold M of D.

8.1 p-Plurisubharmonic Functions and p-Convex Domains

343

If h : R → R is a C 2 function and u : D → R is a C 2 function on a domain D ⊂ Rn , then for each point x ∈ D and vector ξ = (ξ1 , . . . , ξn ) ∈ Rn we have that Hessh◦u (x; ξ ) = h (u(x)) Hessu (x; ξ ) + h (u(x)) |∇u(x)· ξ |2 .

(8.9)

Hence, if u is (strongly) p-plurisubharmonic and h is (strongly) increasing and convex on the range of u, then h ◦ u is also (strongly) p-plurisubharmonic. A limiting case of p-plurisubharmonic functions are p-harmonic functions (see [185, Definition 2.7]). A C 2 function u : D → R is said to be p-harmonic on D ⊂ Rn if for every point x ∈ D the smallest p eigenvalues of Hessu (x) satisfy λ1 (x) + · · · + λ p (x) = 0. The following is [185, Example 2.8]. Example 8.1.8 (Radial harmonics). (i) (p = 1) The function |x| is 1-harmonic on Rn \ {0}. (ii) (p = 2) The function log |x| is 2-harmonic on Rn \ {0}. (iii) (3 ≤ p ≤ n) The function −1/|x| p−2 is p-harmonic on Rn \ {0}.

8.1.2 p-Convex Hulls and p-Convex Domains The following notions are due to Harvey and Lawson [185, Definitions 3.1 and 3.3]. The terms minimal hull and minimally convex domain were first used in [117]. Definition 8.1.9. Let K be a compact set in a domain D ⊂ Rn and p ∈ {1, . . . , n}. 1. The p-convex hull (or the p-hull) of K in D is the set   p,D = x ∈ D : u(x) ≤ sup u for all u ∈ Psh p (D) . K K

p,Rn . The 2-hull is also called the minimal hull and denoted p = K We write K 2,D M,D = K K

M = K M,Rn . and K

(8.10)

p ; 2-convex compact sets are also A compact set K in D is p-convex if K = K called minimally convex. p,D is compact for every compact set K ⊂ D. 2. A domain D ⊂ Rn is p-convex if K A 2-convex domain is also called a minimally convex domain. In view of the inclusions Psh p (D) ⊂ Psh p+1 (D) (see (8.4)) it follows that n ⊂ · · · ⊂ K 2 ⊂ K 1 = Co(K). K⊂K Simple examples show that these inclusions are strict in general. The following result is due to Harvey and Lawson [185, Theorem 3.4]; the proof is similar to the classical one concerning holomorphically convex domains in Cn .

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8 Minimal Surfaces in Minimally Convex Domains

Proposition 8.1.10. A domain D ⊂ Rn is p-convex for some p ∈ {1, 2, . . . , n} if and only if it admits a smooth strongly p-plurisubharmonic exhaustion function. It was shown by R. E. Greene and H. Wu [166] and J.-P. Demailly [106] that every domain D ⊂ Rn is n-convex. Remark 8.1.11. Note that the Hessian of a minimal strongly plurisubharmonic function on a domain in Rn has at most one negative eigenvalue at every point. Hence, Morse theory implies that a minimally convex domain D has the homotopy type of a 1-dimensional CW complex; in particular, the higher homotopy groups πk (D) for k > 1 all vanish. Similarly, a p-convex domain has the homotopy type of a CW complex of dimension at most p − 1.   The proof of the next result follows closely the case of plurisubharmonic functions; cf. [195, Theorem 5.1.5, p. 117] or [329, Theorem 1.3.8, p. 25]. Proposition 8.1.12. If D is a p-convex domain in Rn and K ⊂ D is a compact pconvex subset then the following hold. (a) There exists a smooth p-plurisubharmonic exhaustion function ρ : D → R+ such that ρ −1 (0) = K and ρ is strongly p-plurisubharmonic on D \ K. (b) For every p-plurisubharmonic function u on a neighbourhood U of K there exists a p-plurisubharmonic exhaustion function u : D → R which agrees with u on K and is smooth strongly p-plurisubharmonic on D \ K. p,D implies that for any point x ∈ D \ K there Proof of (a). The condition K = K exists a smooth strongly p-plurisubharmonic function u on D such that u < 0 on K and u(x) > 0. Pick a smooth function h : R → R+ which equals zero on (−∞, 0] and is strongly increasing and strongly convex on (0, ∞). Then h ◦ u ≥ 0 vanishes on a neighbourhood of K and is strongly p-plurisubharmonic on a neighbourhood V of x in view of (8.9). Hence we can pick a countable collection {(V j , u j )} j∈N , where V j is an open set in D \ K, u j ≥ 0 is a smooth p-plurisubharmonic function on D that vanishes near K and is strongly p-plurisubharmonic on V j , and ∞j=1 V j = D \ K. If the numbers ε j > 0 are chosen small enough then the series v = ∑∞j=1 ε j u j ≥ 0 converges in C ∞ (D). By the construction, v vanishes precisely on K and is strongly p-plurisubharmonic on D \ K. Finally, take ρ = v + h ◦ τ where τ is a smooth pplurisubharmonic exhaustion function on D that is negative on K. Proof of (b). We may assume that U is compact. Choose a smooth function χ on Rn such that χ = 1 on a neighbourhood of K and supp χ ⊂ U. Let the function ρ be as in part (a). The function u = χu +Cρ then satisfies condition (b) if the constant C > 0 is chosen big enough. Indeed, the (very) positive Hessian of Cρ compensates the bounded negative part of the Hessian of χu on the compact support of dχ which is contained in U \ K.  

8.1 p-Plurisubharmonic Functions and p-Convex Domains

345

8.1.3 Domains with Smooth p-Convex Boundaries The following characterizations of p-convexity for domains with smooth boundaries were obtained by Harvey and Lawson [185, Sect. 3] for bounded domains; the proofs were extended to unbounded domains in [15, Theorem 1.2]. Theorem 8.1.13. Let 1 ≤ p < n be integers, and let D ⊂ Rn be a domain with C 3 boundary, not necessarily bounded. The following conditions are equivalent. (a) The domain D is p-convex. (b) There exist a neighbourhood U ⊂ Rn of bD and a C 2 function ρ : U → R such that D ∩U = {ρ < 0}, dρ = 0 on bD ∩U = {ρ = 0}, and trL Hessρ (x) ≥ 0 for every p-plane L ⊂ Tx bD, x ∈ bD.

(8.11)

(c) If x ∈ bD and κ1 ≤ κ2 ≤ . . . ≤ κn−1 are the principal curvatures of bD from the interior side at x, then κ1 + κ2 + · · · + κ p ≥ 0. (d) There exists a neighbourhood U of bD such that the C 2 function − log dist(· , bD) is p-plurisubharmonic on D ∩U. The condition that bD is of class C 3 is needed to ensure that the signed distance function to bD is of class C 2 near bD; the remaining conditions (a)–(c) are pairwise equivalent under the weaker assumption that bD is of class C 2 . Remark 8.1.14. Condition (8.11) is independent of the choice of the defining function ρ. Indeed, at any point x ∈ bD we have that Hessρ (x) = |∇ρ(x)|· IIx on Tx bD,

(8.12)

where IIx denotes the second fundamental form of bD at x (see [185, p. 158, Equation (3.2)]).   Proof. Let D ⊂ Rn be a domain with boundary bD of class C 2 . Assume first that condition (a) holds, i.e., D is p-convex. Then D is also locally p-convex, in the sense that every boundary point x ∈ bD has a neighbourhood U ⊂ Rn such that D ∩U is p-convex (cf. [185, (3.1) and Theorem 3.7]; the cited results also give the converse implication for bounded domains). Furthermore, local p-convexity admits the following differential theoretic characterization (cf. [185, Remark 3.11]): A smoothly bounded domain D ⊂ Rn is locally p-convex at x ∈ bD if and only if there are a neighbourhood U ⊂ Rn of x and a local smooth defining function ρ for D (i.e., D ∩U = {ρ < 0} and dρ = 0 on bD ∩U = {ρ = 0}) such that trL Hessρ (y) ≥ 0 for every tangent p-plane L ⊂ Ty bD, y ∈ bD ∩U. By (8.12), this property is independent of the choice of ρ and is equivalent to condition (c) in Theorem 8.1.13 that the sum of the p smallest principal curvatures of bD is nonnegative. Furthermore, assuming that bD ∈ C 3 and setting δ = dist(· , bD),

346

8 Minimal Surfaces in Minimally Convex Domains

D is locally p-convex if and only if the function − log δ is p-plurisubharmonic on a collar around bD in D (cf. [185, Proposition 3.15 and Summary 3.16]). This justifies the implications (a)⇒(b)⇔(c)⇒(d) in Theorem 8.1.13. It remains to prove that (d)⇒(a). Assume that (d) holds, i.e., the C 2 function − log δ is p-plurisubharmonic on an interior collar U ⊂ D around bD. Choose a smooth function χ : Rn → [0, 1] which equals 0 on an open set V ⊂ D containing D \U and equals 1 on an open set W ⊂ Rn containing Rn \ D. Its differential dχ has support in the set U \ W whose closure is contained in D. The product −χ log δ is then a function of class C 2 (D) which is p-plurisubharmonic near bD and tends to +∞ along bD. Let h : R+ → R+ be a smooth, increasing, strongly convex function. If h is chosen such that its derivative h (t) > 0 grows sufficiently fast as t → +∞, we see from (8.9) that the function given by ρ(x) = −χ(x) log δ (x) + h(|x|2 ),

x∈D

is a strongly p-plurisubharmonic exhaustion function on D, so (a) holds.   The following is an immediate corollary to Theorem 8.1.13 since a minimal hypersurface has everywhere vanishing mean curvature. Corollary 8.1.15. If D ⊂ Rn is a not necessarily bounded domain whose boundary bD is a smooth embedded minimal hypersurface, then D is (n − 1)-convex. In particular, a domain in R3 bounded by a minimal surface is minimally convex. Example 8.1.16. Let D be the domain in R3 given by   D = (x, y, z) ∈ R3 : x2 + y2 > cosh2 z . Since the boundary of D is a minimal surface (a catenoid, see Subsect. 2.8.1), D is minimally convex by Corollary 8.1.15. Clearly, D does not have any convex boundary point and its fundamental group is π1 (D) = Z.   Remark 8.1.17. In the literature on minimal surfaces, a smoothly bounded domain D in Rn is said to be (strongly) mean-convex if the sum of the principal curvatures of bD from the interior side is nonnegative (resp. positive) at each point. This is precisely condition (c) in Theorem 8.1.13 with p = n − 1; hence, a smoothly bounded domain in Rn is mean-convex if and only if it is (n − 1)-convex. In particular, mean-convex domains in R3 coincide with smoothly bounded minimally convex domains. Mean-convex domains have been studied in the literature as natural barriers for minimal hypersurfaces in view of the maximum principle. Proper minimal hypersurfaces in mean-convex domains often arise as solutions to Plateau problems. For instance, W. H. Meeks and S. T. Yau [257] proved that every null homotopic Jordan curve in the boundary of a mean-convex domain D ⊂ R3 bounds an area minimizing minimal disc in D. This method does not seem to provide examples of complete minimal surfaces, or those normalized by a given bordered Riemann surface other than the disc. For a discussion of this subject, see e.g. T. H. Colding and W. P. Minicozzi, II [91, Sect. 6.5].  

8.1 p-Plurisubharmonic Functions and p-Convex Domains

347

8.1.4 Strongly p-Convex Domains The following definition is inspired by condition (b) in Theorem 8.1.13. Definition 8.1.18. A domain D ⊂ Rn with C 2 boundary is strongly p-convex for some p ∈ {1, . . . , n − 1} if it admits a C 2 defining function ρ on a neighbourhood U of bD whose Hessian satisfies the strict inequality in (8.11): trL Hessρ (x) > 0 for every p-plane L ⊂ Tx bD, x ∈ bD.

(8.13)

A strongly 2-convex domain is said to be strongly minimally convex . An analogue of Theorem 8.1.13 holds in this setting. In particular, a domain D  Rn with C 2 boundary is strongly p-convex for some p ∈ {1, . . . , n − 1} if and only if the principal curvatures κ1 ≤ κ2 ≤ . . . ≤ κn−1 of bD at each point x ∈ bD satisfy κ1 + κ2 + · · · + κ p > 0. This is seen directly from the formula (8.12) relating the Hessian Hessρ (x) on Tx bD to the second fundamental form IIx . In particular, condition (8.13) is independent of the choice of the defining function ρ. Note that D is strongly (n − 1)-convex if and only if it is strongly mean-convex (see Remark 8.1.17). A smoothly bounded (weakly) p-convex domain D ⊂ Rn need not admit a local p-plurisubharmonic defining function near a given point of bD. The situation is different for strongly p-convex domains as shown by the following lemma which is a special case of [184, Theorem 5.7] due to Harvey and Lawson. Lemma 8.1.19 (Strongly p-plurisubharmonic defining functions). Assume that D is a strongly p-convex domain in Rn with C r boundary (r ≥ 2). There are a neighbourhood U ⊂ Rn of bD and a strongly p-plurisubharmonic function ρ : U → R of class C r such that D ∩ U = {ρ < 0} and dρ = 0 on bD = {u = 0}. If D is compact then we may choose U to be a neighbourhood of D. Proof. Let ρ : U → R be an arbitrary C r defining function for D on a neighbourhood U of bD. Fix a point x ∈ bD. By (8.13) and continuity we have that trL Hessρ (x) > 0 for every p-plane L ⊂ Tx Rn ∼ = Rn which is sufficiently close to the hyperplane Tx bD. Choose a smooth convex increasing function h : R → R with h(0) = 0 and h (0) = 1. Consider the new defining function ρ = h ◦ ρ : U → R for D. By (8.9) we have Hessρ (x; ξ ) = Hessρ (x; ξ ) + h (0) |∇ρ(x)· ξ |2 , ξ ∈ Rn . Consider this expression on the unit sphere S = {|ξ | = 1} ⊂ Rn . Given a compact subset K ⊂ S \ Tx bD, the number |∇ρ(x)· ξ |2 has a positive lower bound on ξ ∈ K. By choosing h as above such that h (0) > 0 is sufficiently big, we can arrange that Hessρ (x; ξ ) is as big as desired for all ξ ∈ K. Clearly this ensures that trL Hessρ (x) > 0 for every p-plane L ⊂ Rn if h (0) is big enough. For such h, the function ρ is therefore strongly p-plurisubharmonic on a neighbourhood of x in Rn . If bD is compact then we can choose h such that ρ = h ◦ ρ is strongly p-

348

8 Minimal Surfaces in Minimally Convex Domains

plurisubharmonic on a neighbourhood of bD in Rn . If bD is not compact, the same can be arranged by the usual limiting procedure; we leave the details to the reader. Finally, if D is compact then we can extend ρ from a smaller neighbourhood of bD to a strongly p-plurisubharmonic function on a neighbourhood of D as follows. Choose a small ε > 0 such that the set A = {x ∈ D ∩U : ρ(x) ≥ −3ε} is compact. Let τ : R → R be a convex increasing function such that τ(t) = −2ε on t ≤ −3ε and τ(t) = t on t ≥ −ε. Then, τ ◦ ρ is well defined and p-plurisubharmonic on a neighbourhood of D, and it agrees with ρ on the set B = {x ∈ U : ρ(x) ≥ −ε}. Pick a strongly p-plurisubharmonic exhaustion function g : D → R and a smooth cutoff function χ : Rn → [0, 1] which equals 1 on D \ B and has support contained in D. It is then immediate that for δ > 0 small enough the C r function ρ  = τ ◦ ρ + δ χg is well defined and strongly p-plurisubharmonic on a neighbourhood of D, and it agrees with ρ on a neighbourhood of bD.  

8.1.5 A Maximum Principle for Minimal Submanifolds Recall that the restriction of a p-plurisubharmonic function u ∈ Psh p (D) on a domain D ⊂ Rn to any minimal p-dimensional submanifold M ⊂ D is a subharmonic function on M (cf. Propositions 8.1.2 and 8.1.5). Hence, the maximum principle for subharmonic functions implies that for any compact immersed minimal pdimensional submanifold M ⊂ D with boundary bM we have the implication p,D . bM ⊂ K =⇒ M ⊂ K The same conclusion holds for minimal p-dimensional rectifiable currents (see Sect. 9.4). Furthermore, we have the following result, analogous to the classical Kontinuit¨atssatz (the continuity principle) in complex analysis. (Compare with Harvey and Lawson [185, proof of Theorem 3.9, p. 159].) Proposition 8.1.20 (Kontinuit¨atssatz for minimal submanifolds). Assume that D is a p-convex domain in Rn for some p ∈ {1, . . . , n} and {Mt }t∈[0,1) is a continuous family of immersed compact minimal p-dimensional submanifolds of Rn with boundaries bMt . If M0 ⊂ D and t∈[0,1) bMt is contained in a compact subset of D, then t∈[0,1) Mt is also contained in a compact subset of D.

Proof. Let K denote the closure of the set M0 ∪ t∈[0,1) bMt in D. By the hypothesis, p,D ⊂ D of K is also compact. K is compact. Since D is p-convex, the p-hull L = K Consider the set J = {t ∈ [0, 1) : Mt ⊂ L}. By the hypothesis we have that 0 ∈ J. We claim that J = [0, 1). Since the family Mt is continuous in t and L is compact, J is closed. It remains to see that J is also open. Assume that t0 ∈ J; then Mt0 ⊂ L ⊂ D. By continuity it follows that Mt ⊂ D for all t ∈ [0, 1) sufficiently close to t0 , and the  maximum principle implies that Mt ⊂ L for all such t.  The following maximum principle for minimal submanifolds is of independent interest, and it will be used in the proof of Theorem 8.6.1.

8.1 p-Plurisubharmonic Functions and p-Convex Domains

349

Proposition 8.1.21 (A maximum principle for minimal submanifolds). Let D be a proper p-convex domain in Rn and M ⊂ D be a compact, connected, immersed minimal p-dimensional submanifold with boundary bM. Then the following hold: (a) dist(bM, bD) = dist(M, bD). (b) If D has C 3 boundary and there is an interior point x0 ∈ M˚ = M \ bM such that dist(x0 , bD) = dist(bM, bD), then bD contains a translate of M. Moreover, if y0 ∈ bD is such that |x0 − y0 | = dist(x0 , bD), then t∈[0,1) (M + t(y0 − x0 )) ⊂ D and M + (y0 − x0 ) ⊂ bD. (c) If the assumption in (b) holds for p = 2 and n = 3 then M is a piece of a plane. ˚ Pick a point Proof of (a). Assume that dist(x0 , bD) < dist(bM, bD) for some x0 ∈ M. y0 ∈ bD with dist(x0 , bD) = |x0 − y0 | and t0 ∈ R with |x0 − y0 | < t0 < dist(bM, bD). −x0 (t ∈ [0,t0 ]) then violates Proposition The family of translates Mt := M + t |yy0 −x 0 0| 8.1.20. This contradiction proves part (a). Proof of (b). By Theorem 8.1.13 (d) there is a neighbourhood U ⊂ Rn of bD such that the function ρ = − log dist(· , bD) is p-plurisubharmonic on D ∩U. Let x0 ∈ M˚ be such that c := dist(x0 , bD) = dist(bM, bD). Pick a point y0 ∈ bD with |x0 −y0 | = c and consider the family of translates Mt := M +t(y0 −x0 ) for t ∈ [0, 1], with M0 = M. Obviously, Mt ⊂ D for all 0 ≤ t < 1 and the point x0t := x0 +t(y0 − x0 ) ∈ Mt satisfies (1 − t)c = |x0t − y0 | = dist(x0t , bD) ≤ dist(Mt , bD).

(8.14)

Part (a) of the proposition implies that the last inequality is in fact an equality. If t is close enough to 1, there is a compact smoothly bounded disc neighbourhood V ⊂ M of x0 such that the translate Vt = V + t(y0 − x0 ) is contained in U ∩ D. For such t, the function ρ = − log dist(· , bD) restricted to Vt is subharmonic and by (8.14) it assumes the maximum value at the interior point x0t of Vt , so it is constant on Vt . Letting t → 1 this shows that V1 ⊂ bD and for every x ∈ V we have that x + y0 − x0 ∈ bD and

dist(x, bD) = dist(bM, bD).

(8.15)

This argument shows that the set of points x ∈ M satisfying (8.15) is open. Clearly this set is also closed, so it equals M. Thus, M1 = M + y0 − x0 ⊂ bD. Proof of (c). Let x0 and y0 be as in the statement. Then M does not intersect the open ball centred at y0 of radius |x0 − y0 | = dist(M, bD). This implies that y0 = x0 + |x0 − y0 |N(x0 ), where N : V → S2 is a Gauss map of a disc neighbourhood V ⊂ M of x0 . Since (8.15) holds for all x ∈ V , we see that N(x) = N(x0 ) for all x ∈ V . This shows that V , and hence also M, is a piece of a plane and (c) follows.   Remark 8.1.22. Both Propositions 8.1.20 and 8.1.21 are false if the domain D with smooth boundary is not p-convex. More precisely, if the boundary bD fails to be pconvex at some point x ∈ bD, then there exists an embedded minimal p-dimensional submanifold M ⊂ D through the point x such that M \ {x} ⊂ D (see [185, Lemma 3.13]). It is however not clear whether the converse to Proposition 8.1.20 holds for domains with nonsmooth boundaries.  

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8 Minimal Surfaces in Minimally Convex Domains

8.2 Null Plurisubharmonic Functions In this section we introduce the notion of a null plurisubharmonic function and develop an important tool used in the proofs of the main results of this chapter. Let z = (z1 , . . . , zn ) = x + iy be complex coordinates on Cn , with z j = x j + iy j for j = 1, . . . , n. We shall write 0 = (0, . . . , 0) for the origin in Rn or in Cn . Given a C 2 function ρ : Ω → R on a domain Ω ⊂ Cn , we denote by Lρ (z; · ) its Levi form at a point z ∈ Ω (cf. (1.69)): Lρ (z; w) =

∂ 2ρ (z)w j wk , j,k=1 ∂ z j ∂ z¯k n

∑

w = (w1 , . . . , wn ) ∈ Cn .

(8.16)

The proof of the following lemma amounts to a simple calculation. Lemma 8.2.1. Let B = (b j,k ) be a real symmetric n × n matrix, and let w = u + iv ∈ Cn . Then n

2

∑

n

b j,k u j uk = ℜ

j,k=1

∑

n

b j,k w j wk +

j,k=1

∑

b j,k w j wk .

j,k=1

A function ρ : D → R on a domain D ⊂ Rn will also be considered as a function on the tube over D, TD = D × iRn ⊂ Cn , (8.17) which is independent of the imaginary variable: ρ(x + iy) = ρ(x) for all x ∈ D and y ∈ Rn .

(8.18)

Fix a point x ∈ D and a vector u ∈ Rn . The Hessian Hessρ (x) (8.1) has coefficients b j,k :=

∂ 2ρ ∂ 2ρ (x) = 4 (x) ∈ R. ∂ x j ∂ xk ∂ z j ∂ zk

(8.19)

Lemma 8.2.1 shows that for every w = u + iv ∈ Cn we have

1 n 1 Hessρ (x; u) = ℜ ∑ b j,k w j wk + Lρ (x; w). 2 4 j,k=1

(8.20)

Replacing w by −iw = v − iu and noting that Lρ (x; ±iw) = Lρ (x; w) while the first term on the right-hand side of (8.20) changes sign, we obtain Hessρ (x; u) + Hessρ (x; v) = 4Lρ (x; u + iv). In particular, if (u, v) is an orthonormal pair of vectors in Rn and we set L = x + spanR {u, v} ⊂ Rn , it follows that

Λ = x + spanC {u + iv} ⊂ Cn ,

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351

Δ (ρ|L )(x) = 4Lρ (x; u + iv) = Δ (ρ|Λ )(x).

(8.21)

Set a j = ∂∂xρj (x) ∈ R for j = 1, . . . , n and let b j,k be given by (8.19). The identity (8.20) implies that for every point z = x + iy ∈ TD and vector w = u + iv ∈ Cn with components w j = u j + iv j near 0 ∈ Cn we have the Taylor expansion n 1 ρ(z + w) = ρ(x) + ∑ a j u j + Hessρ (x; u) + o(|u|2 ) 2 j=1

n 1 n = ρ(x) + ℜ ∑ a j w j + ∑ b j,k w j wk + Lρ (x; w) + o(|w|2 ). 4 j,k=1 j=1

Denote by Σx ⊂ Cn the local complex hypersurface near the origin 0 ∈ Cn given by  Σx = w = (w1 , . . . , wn ) :

n

 b j,k w j wk = 0 .

(8.22)

z = x + iy ∈ TD , w ∈ Σx .

(8.23)

1

n

∑ a jw j + 4 ∑

j=1

j,k=1

It follows that ρ(z + w) = ρ(x) + Lρ (x; w) + o(|w|2 ),

Recall that A ⊂ Cn denotes the null quadric (2.54). The following classes of functions were introduced in [117, Definitions 2.1 and 2.4]. Definition 8.2.2. Let Ω be a domain in Cn for some n ≥ 3. (a) An upper semicontinuous function u : Ω → R ∪ {−∞} is null plurisubharmonic if for any affine complex line L ⊂ Cn directed by a null vector θ ∈ A∗ the restriction of u to L ∩ Ω is subharmonic. (b) In particular, a function u ∈ C 2 (Ω ) is null plurisubharmonic if and only if Lu (z; w) ≥ 0 for every z ∈ Ω and w ∈ A∗ . (c) A function u ∈ C 2 (Ω ) is null strongly plurisubharmonic if Lu (z; w) > 0 for every z ∈ Ω and w ∈ A∗ . The space of null plurisubharmonic functions on a domain Ω ⊂ Cn is denoted by NPsh(Ω ). Note that for any orthonormal pair (u, v) of vectors in Rn we have u ± iv ∈ A∗ . In view of (8.21) we have the following result for functions u ∈ C 2 (D); the general case for upper semicontinuous functions is obtained by approximating them with smooth ones (see Propositions 8.1.6 and 8.2.7). Lemma 8.2.3. Let D be a domain in Rn (n ≥ 3) and set TD = D × iRn ⊂ Cn . • If u is (strongly) minimal plurisubharmonic on D, then the function U(x + iy) = u(x) is (strongly) null plurisubharmonic on the tube TD .

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8 Minimal Surfaces in Minimally Convex Domains

• Conversely, assume that a function U : TD → R is independent of the variable y = ℑz, and let u(x) = U(x + i0) for x ∈ D. If U is (strongly) null plurisubharmonic on TD , then u is (strongly) minimal plurisubharmonic on D. Clearly we have that Psh(Ω ) ⊂ NPsh(Ω ), and the inclusion is proper as shown by the following simple example. Example 8.2.4. The function u(z1 , z2 , z3 ) = |z1 |2 + |z2 |2 − |z3 |2 is clearly not plurisubharmonic on any open subset of C3 . However, it is null plurisubharmonic on C3 which is seen as follows. Fix a null vector z = (z1 , z2 , z3 ) ∈ A∗ . Then u(ζ z) = |ζ |2 u(z) for ζ ∈ C and we need to check that u(z) ≥ 0. The equation of the null quadric gives z23 = −(z21 + z22 ) and hence |z3 |2 ≤ |z1 |2 + |z2 |2 by the triangle 2 |2 ≥ 0. (Note that u vanishes on some null inequality, so u(z) = |z1 |2 + |z√ 2 | − |z3√ vectors; for example, on z = (i 2/2, i 2/2, 1).) For any point a ∈ C3 the function u(a + ζ z) differs from u(ζ z) by a harmonic term in ζ , so the restriction of u to any affine complex line directed by a null vector is subharmonic.   The next proposition summarizes some properties of null plurisubharmonic functions, analogous to those of plurisubharmonic functions. Proposition 8.2.5. Let Ω be a domain in Cn . (i) If u, v ∈ NPsh(Ω ) and c > 0, then cu, u + v, max{u, v} ∈ NPsh(Ω ). (ii) If u ∈ C 2 (Ω ) then u ∈ NPsh(Ω ) if and only if Lu (z; v) ≥ 0 for every z ∈ Ω and v ∈ A. (iii) The limit of a decreasing sequence of null plurisubharmonic functions on Ω is a null plurisubharmonic function on Ω . (iv) If a family F ⊂ NPsh(Ω ) is locally uniformly bounded above and v(z) = supu∈F u(z) for z ∈ Ω , then the function v∗ (z) = lim supw→z v(w) is null plurisubharmonic on Ω . (v) If u ∈ NPsh(Cn ) is bounded above, then u is constant. (vi) The Levi form of any C 2 null plurisubharmonic function has at most one negative eigenvalue at each point. Properties (i)–(iv) follow in a standard way from the corresponding properties of subharmonic functions. Property (v) follows from the fact that every bounded above subharmonic function on C is constant and any two points of Cn can be connected by a finite chain of affine complex null lines. Property (vi) is seen by observing that every 2-dimensional C-linear subspace of Cn intersects A in a complex line. Remark 8.2.6. Null plurisubharmonic functions provide a natural substitute for plurisubharmonic functions when considering holomorphic null curves (see Definition 2.3.3) as opposed to all holomorphic curves. They are a special case of G-plurisubharmonic functions introduced by Harvey and Lawson who in a series of papers studied plurisubhamonicity in a more general geometric context (see [181, 183, 184, 185], among others).  

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353

The next proposition gives some further properties of null plurisubharmonic functions. In particular, we can approximate them by smooth null plurisubharmonic functions. Proposition 8.2.7. Let Ω be a domain in Cn . 1 (Ω ). (i) If u ∈ NPsh(Ω ) and u ≡ −∞ on Ω , then u ∈ Lloc (ii) If u ∈ NPsh(Ω ) and u ≡ −∞ on Ω , then u can be approximated by smooth null plurisubharmonic functions on domains compactly contained in Ω . (iii) If u ∈ NPsh(Ω ), M is an open Riemann surface and f : M → Ω is a holomorphic null curve, then u ◦ f is subharmonic on M. (iv) If Ω is a pseudoconvex Runge domain in Cn , u ∈ NPsh(Ω ), and K is a compact set in Ω , then there exists a v ∈ NPsh(Cn ) such that v = u on K, v is strongly plurisubharmonic on Cn \ Ω , and v(z) > maxK v for any point z ∈ Cn \ Ω .

Proof. We adapt the usual proof for the plurisubharmonic case (see for example [296, Lemma 4.11, Theorem 4.12, Theorem 4.13] or [196, Chap. 4]). For simplicity of notation we consider the case n = 3; the same arguments apply for any n ≥ 3. We begin by explaining part (i). Choose three C-linearly independent vectors θ1 , θ2 , θ3 in the null cone A ⊂ C3 (2.54). As in the standard case, we assume that u(p) > −∞ for some p ∈ Ω , and we show that for every r > 0 such that Drp := {p + z1 θ1 + z2 θ2 + z3 θ3 : |zi | ≤ r, 1 ≤ i ≤ 3} ⊂ Ω we have u ∈ L1 (Drp ). Since u is bounded above on the compact set Drp , we only need to show that Drp u dV > −∞; the claim then follows as in the standard case. Fix r > 0 as above. For any (z1 , z2 , z3 ) such that |zi | ∈ [0, r] for i = 1, 2, 3 the restriction of u to j−1 zi θi + ζ θ j : |ζ | ≤ r} is subharmonic for each 1 ≤ j ≤ 3. Applying the disc {p + ∑i=1 the sub-mean value property in each variable we obtain u(p) ≤

 2π 2π 2π 0

0

0

u(p + r1 eit1 θ1 + r2 eit2 θ2 + r3 eit3 θ3 ) dt1 dt2 dt3

for all ri ∈ [0, r], i = 1, 2, 3. Multiplying this inequality by r1 r2 r3dr1 dr2 dr3 and integrating with respect to ri ∈ [0, r] for 1 ≤ i ≤ 3 gives u(p) ≤ C Drp u dV , where the positive constant C depends on the choice of the θ j ’s. This proves (i). In part (ii) we proceed as in the usual proof for smoothing plurisubharmonic functions, convolving u by a smooth approximate identity φt satisfying the property φt (∑3i=1 zi θi ) = ϕt (|z1 |, |z2 |, |z3 |). We leave the details to the reader. For functions u ∈ C 2 (Ω ), property (iii) is an immediate consequence of Proposition 8.2.5 (ii). In the general case the same result follows from part (ii) (smoothing) and the fact that the limit of a decreasing sequence of subharmonic functions is subharmonic. It remains to prove property (iv). Since Ω is pseudoconvex and Runge, the  is contained in Ω [195, Theorem 2.7.3, p. 53]. Pick an polynomial hull K  ⊂ U. By [195, Theorem 2.6.11, p. 48] there is a open set U  Ω with K

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8 Minimal Surfaces in Minimally Convex Domains

smooth plurisubharmonic exhaustion function ρ : Cn → R+ that vanishes on a  and is positive strongly plurisubharmonic on Cn \ U. Let neighbourhood of K n χ : C → [0, 1] be a smooth function that equals 1 on U and has support contained in Ω . Then the function v = χu +Cρ for large C > 0 has the stated properties.   From the definition of a null plurisubharmonic function and Proposition 8.2.7 (iii) we obtain the following corollary. This can also be seen from Proposition 8.1.2 (d), Lemma 8.2.3, and the fact that every conformal minimal disc in Rn is the real part of a holomorphic null disc in Cn (see Theorem 2.3.4). Corollary 8.2.8. For an upper semicontinuous function u on a domain Ω ⊂ Cn (n ≥ 3) the following conditions are equivalent. • u is null plurisubharmonic. • For every open Riemann surface M and holomorphic null curve f : M → Ω , the function u ◦ f is subharmonic on M. By using null plurisubharmonic functions, one can introduce and study the class of null pseudoconvex domains in Cn for any n ≥ 3. These are domains D ⊂ Cn admitting a (strongly) null plurisubharmonic exhaustion function. They are natural domains for the existence of proper holomorphic null curves; see Theorem 8.3.13.

8.3 The Main Results In this section we present our main results concerning minimal surfaces in minimally convex domains in Rn for any n ≥ 3. The analogous results in complex analysis, pertaining to proper holomorphic maps from bordered Riemann surfaces to 2-convex complex manifolds, were obtained in [114]; see also [115] for a generalization to higher dimensions. We begin by presenting the results in dimension n = 3 which are the strongest and essentially optimal. The analogous results in higher dimensions are mentioned at the end of the section. The following is [15, Theorem 1.1], except for the last hitting condition which is new. Compare with Theorem 7.6.2 which solves the hitting problem for nonproper maps in arbitrary domains of Rn , n ≥ 3. Theorem 8.3.1. Assume that M is a compact bordered Riemann surface with boundary bM and D is a minimally convex domain in R3 . Then, every conformal minimal immersion X : M → D can be approximated, uniformly on compacts in M˚ = M \ bM, by proper complete conformal minimal immersions X : M˚ → D with  = Flux(X) whose images X(M)  Flux(X) contain any given discrete subset Λ of D. Like all the main results in the book, Theorem 8.3.1 pertains to a fixed conformal structure on the surface M. It is the first general existence and approximation result for (complete) proper minimal surfaces in a class of domains in R3 which contains all convex domains as well as many non-convex ones. Indeed, convexity

8.3 The Main Results

355

has been impossible to avoid with the previously known construction methods. The Riemann–Hilbert method, furnished by Theorem 6.4.1, plays an essential role in the proof. Remark 8.3.2. By following the idea explained in the proof of Theorem 7.5.1, the proper conformal minimal immersions X : M˚ → D in Theorem 8.3.1 can be chosen complete with respect to any given Riemannian metric g on the domain D.   When D = R3 , this result is covered by Theorem 3.10.3. In this special case, the conclusion holds for every open Riemann surface (and not only for bordered surfaces), and one can interpolate on any given discrete subset of M (and not merely hit the given discrete subset of R3 ). However, if D is a bounded domain in Rn or, more generally, a domain with a negative minimal plurisubharmonic exhaustion function ρ : D → (−∞, 0), then the restriction of ρ to any proper minimal surface M in D is a nonconstant negative subharmonic function on M, and hence M is of hyperbolic conformal type (see Definition 8.1.1). Even if M is hyperbolic (e.g. the disc), it is in general impossible to find conformal minimal immersions M → D with prescribed values on a given finite set in M containing at least two points due to Schwarz type (hyperbolicity) obstructions. This shows that the restriction to bordered surfaces and the hitting condition are natural in Theorem 8.3.1. Recall that every open Riemann surface of finite topological type and without point ends is conformally equivalent to a bordered Riemann surface (see Theorem 1.10.6). We do not know how to effectively deal with boundary punctures in the context of Theorem 8.3.1; the analogous problem for complex curves is open as well (see [114]). Our proof of Theorem 8.3.1, given in Section 8.5, shows without additional effort that boundaries of conformal minimal surfaces can be pushed to a minimally convex end of a domain D ⊂ R3 as in the following theorem (see [15, Theorem 1.7]; the analogous result for holomorphic curves is [115, Theorem 1.1]). Theorem 8.3.3. Assume that Ω ⊂ D are open sets in R3 and ρ : Ω → [0, +∞) is a smooth minimal strongly plurisubharmonic function such that for any pair of numbers 0 < c1 < c2 the set Ωc1 ,c2 = {x ∈ Ω : c1 ≤ ρ(x) ≤ c2 } is compact. Let M be a compact bordered Riemann surface. Then, every conformal minimal immersion X : M → D satisfying X(bM) ⊂ Ω can be approximated, uniformly on compacts in M˚ = M \ bM, by complete conformal minimal immersions X : M˚ → D such that  ∈ Ω for every z ∈ M˚ sufficiently close to bM and X(z)  lim ρ(X(z)) = +∞.

z→bM

(8.24)

In a typical application, the set Ω in Theorem 8.3.3 is a collar around a minimally convex boundary component S ⊂ bD. (By Theorem 8.1.13, a smooth boundary component S ⊂ bD is minimally convex if and only if the function − log dist(· , S) is minimal plurisubharmonic near S.) Theorem 8.3.3 furnishes a proper complete conformal minimal immersion M → D whose boundary cluster set is contained in S as shown by (8.24). There is no assumption on the remaining ends of D.

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8 Minimal Surfaces in Minimally Convex Domains

Our next result improves Theorem 8.3.1 for bounded strongly minimally convex domains in R3 (see Definition 8.1.18). The original source is [15, Theorem 1.9]. Theorem 8.3.4. Let D be a bounded strongly minimally convex domain in R3 . Given a compact bordered Riemann surface, M, and a conformal minimal immersion X : M → D, we can approximate X uniformly on compacts in M˚ by continuous maps  ⊂ bD, X : M˚ → D is a proper complete conformal X : M → D such that X(bM)  minimal immersion, Flux(X) = Flux(X), and   − X(z)| ≤ C max dist(X(z), bD) (8.25) sup |X(z) z∈M

z∈bM

for some constant C > 0 depending only on D. The improvement over Theorem 8.3.1 is that X can be continuous up to the  boundary of M, so X(bM) ⊂ bD is a union of finitely many closed curves and we have the estimate (8.25). The corresponding result for smoothly bounded strongly convex domains in Rn for any n ≥ 3 is [14, Theorem 1.2]; see also [10]. We do not know whether X can be a topological embedding on bM. In particular, unlike in  Theorem 7.4.1, we do not claim that X(bM) consists of Jordan curves. Corollary 8.3.5. Every domain D ⊂ R3 having a C 2 strongly minimally convex boundary point contains complete properly immersed minimal surfaces extending continuously up to the boundary and normalized by any bordered Riemann surface. Proof. Assume that x0 ∈ bD is a strongly minimally convex boundary point. Then there are a neighbourhood U of x0 and a strongly minimally convex domain D ⊂ D such that D ∩ U = D ∩ U. (It suffices to intersect D by a small ball around x0 and smooth the corners.) Given a conformal minimal immersion X : M → D from a compact bordered Riemann surface whose image X(M) lies close enough to the   point x0 , the map X : M → D furnished by Theorem 8.3.4 satisfies X(bM) ⊂ bD ∩U in view of the estimate (8.25), and hence the map X : M˚ → D is proper.   The following remark and Examples 8.3.8 and 8.3.9 show that the hypothesis of minimal convexity is essentially optimal in Theorems 8.3.1, 8.3.3, and 8.3.4. Remark 8.3.6. The conclusion of Theorem 8.3.3 fails along a compact smooth boundary component S ⊂ bD which is strongly minimally concave, i.e., such that κ1 (x) + κ2 (x) < 0 holds for every point x ∈ S, where κi (x) (i = 1, 2) denote the principal curvatures of S at x from the interior side. Indeed, Lemma 8.1.19 furnishes an open neighbourhood U of S in R3 and a minimal strongly plurisubharmonic function φ : U → R that vanishes on S and is positive on D ∩ U. The maximum principle applied with φ then shows that there is no minimal surface in D ∩U with boundary in S. The same argument holds locally near a smooth strongly minimally  concave boundary point x0 ∈ bD.  For a complete proper minimal surface there is another restriction on the location of its boundary points. Assume that D ⊂ R3 is a domain with C 2 boundary and

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357

X : D → D is a complete conformal proper minimal immersion from the unit disc extending continuously to D. Then, the boundary curve X(bD) ⊂ bD does not contain any strongly concave boundary points of D (see A. Alarc´on and N. Nadirashvili [46]). This is especially relevant in connection to Theorem 8.3.4. However, we do not know whether X(bD) could contain a strongly minimally concave boundary point; the following remains an open problem. Problem 8.3.7. Let D be smoothly bounded domain in R3 and X : D → D be a complete conformal proper minimal immersion extending continuously to D (hence X(bD) ⊂ bD). Do we have that κ1 (x) + κ2 (x) ≥ 0 for every point x ∈ X(bD)?   We now illustrate by a couple of examples that Theorem 8.3.1 fails in general for domains in R3 which are not minimally convex. Example 8.3.8. We exhibit a simply connected domain of the form D = 2B \ K, where B is the unit ball of R3 and K is a compact set contained in a thin shell around the sphere S = bB, such that the image of every proper minimal disc D → D avoids the ball 21 B ⊂ D. Hence, Theorem 8.3.1 fails in this example even without the completeness condition. Note however that D admits complete properly immersed minimal surfaces normalized by any given bordered Riemann surface in view of Theorem 8.3.3 applied to the strongly convex boundary component 2S ⊂ bD. The example is essentially the one given by F. Forstneriˇc and J. Globevnik [143, Sect. 5] in the context of holomorphic discs in domains in C2 . We cover the unit sphere S ⊂ R3 by small open spherical caps C1 , . . . ,Cm (i.e., every C j is the intersection of S by a half-space defined by an affine plane H j ⊂ R3 ) such that m 1 j=1 Co(C j ) ∩ 2 B = ∅. (Recall that Co denotes the convex hull.) Pick a number r > 1 so close to 1 that S ⊂ mj=1 Co(rC j ). Choose pairwise distinct numbers ρ1 , . . . , ρm very close to r such that the pairwise disjoint spherical caps Γj = ρ jC j satisfy S ⊂ mj=1 Co(Γj ) and mj=1 Co(Γ j ) ∩ 12 B = ∅. Let D = 2B \ mj=1 Γ j . For any proper conformal minimal disc X : D → D, its boundary cluster set Λ (X) (i.e., the set of all limit points lim j→∞ X(ζ j ) ∈ bD along sequences ζ j ∈ D with lim j→∞ |ζ j | = 1) is a connected compact set in bD; hence it is contained in the sphere 2S or in one of the caps Γj . Assume now that X(ζ0 ) ∈ 12 B for some ζ0 ∈ D. If Λ (X) ⊂ Γj for some j ∈ {1, . . . , m}, then X(D) ⊂ Co(Γj ) by the maximum principle, a contradiction since Co(Γj ) does not intersect the ball 12 B. If on the other hand Λ (X) ⊂ 2S, there is a point ζ1 ∈ D with X(ζ1 ) ∈ S. Pick j ∈ {1, . . . , m} such that X(ζ1 ) ∈ Co(Γj ). Since X has no cluster points on Γj , the set U = {ζ ∈ D : X(ζ ) ∈ Co(Γj )} is a nonempty relatively compact domain in D, and X(bU) lies in the affine plane H j which determines the spherical cap Γj . By the maximum principle it follows that X  maps all of U, and hence the whole disc D, into H j , a contradiction.  In the paper [239], F. Mart´ın, W. H. Meeks and N. Nadirashvili constructed bounded domains in R3 which do not admit any proper complete minimal surfaces of finite topology. In the next example, we show that their collection includes a domain D ⊂ R3 which carries no proper minimal discs, irrespective of completeness. A similar result in the holomorphic category is due to A. Dor [113] who constructed a bounded domain D in Cm for any m ≥ 2 without any proper holomorphic discs.

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8 Minimal Surfaces in Minimally Convex Domains

Example 8.3.9. Let S be the cylindrical shell    S = (x, y, z) ∈ R3 : 1 < |(x, y)| = x2 + y2 < 2, 0 < z < 1 . For 0 < t < 1, let Ct := S ∩ {(x, y, z) ∈ R3 : z = t} denote the planar round open annulus obtained by intersecting the cylinder S with the plane z = t. For j ∈ N, denote by Ct, j the planar round compact annulus

 1 1 Ct, j = (x, y, z) ∈ Ct : 1 + ≤ |(x, y)| ≤ 2 − . 2j 2j Note that bCt, j = {(x, y, z) ∈ R3 : |(x, y)| ∈ {1 + 21j , 2 − 21j }, z = t}. Let t1 ,t2 ,t3 , . . . denote the sequence 1 1 2 1 2 3 1 2 n−1 , , , , , , ··· , , , ··· , , ··· . 2 3 3 4 4 4 n n n

Set Γ = j∈N bCt j , j ⊂ S and D = S \Γ . By [239, proof of Theorem 1], D is a domain in R3 and the boundary cluster set Λ (E) ⊂ D \ D of any proper minimal annular end E ⊂ D lies in a horizontal plane of R3 . By the maximum principle, this implies that every proper minimal disc D → D is contained in a horizontal plane, but clearly D does not admit any such discs. More generally, D does not carry any proper minimal surfaces of finite genus with a single end.   Minimal surfaces in Theorems 8.3.1, 8.3.3, and 8.3.4 are images of bordered Riemann surfaces, hence of finite topological type. If one does not insist on the approximation of maps and on the control of the conformal structure on these surfaces, then our methods also give complete proper minimal surfaces of arbitrary topological type as in the following corollary, whose proof is sketched in Section 8.5. For domains D ⊂ Rn (n ≥ 3) that are convex, or have a C 2 smooth strictly convex boundary point, this was established in [14]; for n = 3 see also L. Ferrer, F. Mart´ın, and W. H. Meeks [126]. Corollary 8.3.10. If D is a domain in R3 with a minimally convex end (see Theorem 8.3.3) or a strongly minimally convex boundary point, then every open orientable smooth surface, S, carries a complete proper minimal immersion S → D with arbitrary flux. We conclude by presenting analogous results in higher dimensions. Theorem 8.3.11. The analogues of Theorems 8.3.1 and 8.3.3 hold for (n − 2)convex domains D ⊂ Rn for any n > 3 (in particular, they hold for minimally convex domains). Furthermore, the approximating conformal minimal immersions X : M˚ → D can be chosen to have simple double points if n = 4, and to be embeddings if n ≥ 5. If in addition D is strongly (n − 2)-convex, then the analogue of Theorem 8.3.4 holds true with the same additions as above.

8.4 Lifting Boundaries of Conformal Minimal Surfaces

359

Problem 8.3.12. Does Theorem 8.3.11 hold under the weaker assumption that the domain D ⊂ Rn is (strongly) mean-convex, that is, (n − 1)-convex? Our methods do not suffice to obtain this stronger result when n > 3, the reason being that the Riemann–Hilbert deformation method used in the proof (see Theorem 6.6.2) is currently only available for flat boundary discs lying in parallel planes, and such families of discs with desired geometric properties only exist if D has an (n − 2)-plurisubharmonic exhaustion function. Results analogous to those presented in this section hold true, with similar proofs, for holomorphic curves in Cn (n ≥ 2) and holomorphic null curves in Cn for n ≥ 3. As has already been mentioned in connection to Theorem 7.4.12, the appropriate analogues of Lemma 7.3.1 were developed in [23, 14]; one uses the corresponding version of the Riemann–Hilbert problem from Chapter 6. For example, one can show the following analogue of Theorem 8.3.4; the details are left to the reader. Theorem 8.3.13. Let D be a bounded strongly pseudoconvex domain in Cn for n > 1. Given a compact bordered Riemann surface, M, and a holomorphic immersion Z : M → D, we can approximate Z uniformly on compacts in M˚ by continuous maps  ⊂ bD, Z : M˚ → D is a proper complete holomorphic Z : M → D such that Z(bM) immersion, and the estimate   − Z(z)| ≤ C max dist(Z(z), bD) sup |Z(z) z∈M

z∈bM

for some C > 0 depending on D. The analogous result holds for holomorphic null immersions if D  Cn , n ≥ 3, is a bounded strongly null pseudoconvex domain. Remark 8.3.14. Most results presented in this section pertain to minimal surfaces in minimally convex domains which are normalized by bordered Riemann surfaces. This restriction cannot by entirely removed since a bounded domain in Rn (and, more generally, any domain which admits a nonconstant negative 2plurisubharmonic exhaustion function) does not contain any minimal surface of parabolic conformal type.

8.4 Lifting Boundaries of Conformal Minimal Surfaces We now prepare some technical results which will be used in the proofs of the main theorems stated in the previous section. We focus on the case n = 3; the higherdimensional versions are analogous and geometrically even simpler if we assume that the domain D is (n − 2)-convex. Let ρ : D → R be a smooth minimal strongly plurisubharmonic exhaustion function on a domain D ⊂ R3 . By exploiting the geometric properties of ρ, we shall construct families of small conformal minimal discs in D such that ρ restricted to any one of them has a minimum at the centre of the disc and it increases proportionally to the square of the distance from the centre; see Lemma 8.4.1.

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By using such discs, the Riemann–Hilbert deformation method, given by Theorem 6.4.1, furnishes a sequence of deformations of a conformal minimal immersion M → D such that the image of the boundary bM is pushed to higher levels of ρ (i.e., closer to bD), while at the same time ρ does not decrease much anywhere on the surface and we approximate the given map at each step uniformly on a compact subset of M˚ (see Proposition 8.4.2). The estimates in this push-up method degenerate at critical points of ρ. In order to bypass these points, we use an idea borrowed from similar constructions of proper complex curves; see Lemma 8.4.5. A combination of these tools provides the main induction step in the proof, given by Lemma 8.4.6. The procedure is similar in spirit to the construction in [114] of proper complex curves in certain complex manifolds. We now explain the details of this program. We extend ρ to a function on the tube TD = D × iR3 ⊂ C3 which is independent of the imaginary variable; see (8.18). By Lemma 8.2.3 the extended function is null strongly plurisubharmonic on TD . For every point x ∈ D we denote by Σx ⊂ C3 the quadratic complex hypersurface given by (8.22):  Σx = w = (w1 , w2 , w3 ) :

3

 ∂ 2ρ (x)w j wk = 0 . j,k=1 ∂ x j ∂ xk

∂ρ

3

∑ ∂ x j (x)w j + 4 ∑

j=1

Let P denote the critical locus of ρ. We assume in the sequel that x ∈ D \ P; then Σx is nonsingular at 0 ∈ Σx and its tangent space is  T0 Σx = w ∈ C3 :

3

∂ρ



∑ ∂ x j (x)w j = 0

.

(8.26)

j=1

By shrinking Σx around 0 we may assume that it is nonsingular. The intersection of the null quadric A (2.54) with any complex hyperplane 0 ∈ Λ ⊂ C3 consists of two complex lines which may coincide forcertain Λ . However, for a 2-plane Λ = w = (w1 , w2 , w3 ) ∈ C3 : ∑3j=1 a j w j = 0 with real coefficients a1 , a2 , a3 ∈ R not all equal to 0 (such as those in (8.26)) the intersection A ∩ Λ consists of two distinct complex lines as is seen by a simple calculation. Identifying the tangent space Tz C3 with C3 , we consider the null quadric A as a subset of Tz C3 for any z ∈ C3 . By what has been said above, for any point z ∈ Σx sufficiently close to 0 the intersection A ∩ Tz Σx is a union of two distinct complex lines. This defines on Σx a couple of holomorphic null direction fields, and hence (by integration) a couple of one-dimensional complex analytic foliations by holomorphic null curves. In particular, for any point x ∈ D \ P we have two distinct embedded holomorphic null discs Nx1 , Nx2 ⊂ Σx passing through 0. Although there is no well defined global ordering of these two null discs when x runs over D \ P, such an ordering clearly exists on every simply connected subset. By the definition of Σx and the Taylor expansion (8.23) of ρ we have that ρ(z + w) = ρ(z) + Lρ (x; w) + o(|w|2 ),

w ∈ Σx .

8.4 Lifting Boundaries of Conformal Minimal Surfaces

361

This holds in particular for all w ∈ Nx1 ∪ Nx2 ⊂ Σx . Since ρ is null strongly plurisubharmonic on the tube TD , the Levi form Lρ (x) is positive on the null lines T0 Nx j for j = 1, 2. It follows that for every compact set L ⊂ D \ P there are constants C = C(L) > 0 and δ = δ (L) > 0 such that ρ(z + w) ≥ ρ(z) +C|w|2

for all x ∈ L, w ∈ Nx1 ∪ Nx2 , |w| ≤ δ .

(8.27)

By projecting the discs Nx1 , Nx2 to R3 we get a corresponding family of conformal minimal discs with the analogous properties. We summarize the above discussion in the following lemma. Lemma 8.4.1. Let D be a domain in R3 , and let ρ : D → R be a C 2 minimal strongly plurisubharmonic function with the critical locus P. For every compact set L ⊂ D\P there exist a constant c = cL > 0 and families of embedded holomorphic null discs σxj = αxj + iβxj : D → C3 (x ∈ L, j = 1, 2), depending locally C 1 smoothly on the point x ∈ L and satisfying the following conditions. (a) σxj (0) = 0. (b) {x + αxj (ζ ) : ζ ∈ D} ⊂ D.   (c) The function D ζ → ρ x + αxj (ζ ) is strongly convex and satisfies   ρ x + αxj (ζ ) ≥ ρ(x) + c|ζ |2 ,

ζ ∈ D.

(8.28)

The conformal minimal discs αxj : D → R3 furnished by Lemma 8.4.1 will be used to push the boundary X(bM) of a given conformal minimal immersion X : M → D to a higher level set of ρ, except near the critical points of ρ which shall be avoided by another method to be explained in the sequel. Proposition 8.4.2 (Lifting boundaries of conformal minimal surfaces). Let D be a domain in R3 and ρ : D → R be a C 2 minimal strongly plurisubharmonic function with the critical locus P. Given a compact set L ⊂ D \ P, there are constants ε0 > 0 and C0 > 0 such that the following holds. Let M be a compact bordered Riemann surface and X : M → D be a conformal minimal immersion of class C 1 (M). Given a continuous function ε : bM → [0, ε0 ] supported on the set J = {ζ ∈ bM : X(ζ ) ∈ L}, an open set U ⊂ M containing supp(ε) in its relative interior, and a constant δ > 0, there exists a conformal minimal immersion Y : M → D satisfying the following conditions. (1) |ρ(Y (ζ )) − ρ(X(ζ )) − ε(ζ )| < δ for every ζ ∈ bM. (2) ρ(Y (ζ )) ≥ ρ(X(ζ )) − δ for every ζ ∈ M. (3) Y − XC 1 (M\U) < δ . √ (4) Y − XC 0 (M) ≤ C0 ε0 . (5) Flux(Y ) = Flux(X). ˚ (6) Y agrees with X to a given finite order at a given finite set of points in M. Proof. By Proposition 3.3.2 we may assume that X is of class C ∞ (M).

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8 Minimal Surfaces in Minimally Convex Domains

Pick a compact set L0 ⊂ D \ P which contains L in its interior. Let cL0 be the constant furnished by Lemma 8.4.1 for the set L0 , and choose a number ε0 such that 0 < ε0 < cL0 . Set J0 = {ζ ∈ bM : X(ζ ) ∈ L0 }. By approximation, we may assume that the function ε : bM → [0, ε0 ] in Proposition 8.4.2 is smooth and supported in the relative interior of J0 ∩U. Assume first that the support of ε does not contain any boundary curves of M; the general case will be obtained by two consecutive applications of this special case. Choose finitely many closed pairwise disjoint arcs I1 , I2 , . . . , Im ⊂ J0 ∩U whose union I = mj=1 I j contains supp(ε) in its relative interior. Note that X(I) ⊂ L0 . Since I is simply connected, Lemma 8.4.1 furnishes a family of conformal minimal discs αX(ζ ) : D → R3 , depending smoothly on ζ ∈ I, such that αX(ζ ) (0) = 0 and   ρ X(ζ ) + αX(ζ ) (ξ ) ≥ ρ(X(ζ )) + cL0 > ρ (X(ζ )) + ε0 ,

ζ ∈ I, |ξ | = 1. (8.29)

Without loss of generality we may assume that δ < 3ε0 . Let ε˜ : I → [δ /3, ε0 ] be obtained by smoothing the function max{ε, δ /3}; in particular, we assume that ε˜ = ε on the set where ε ≥ δ /2 and δ /3 ≤ ε˜ < δ /2 on the complementary set. It follows that |ε(ζ ) − ε˜ (ζ )| < δ /2 for all ζ ∈ I. The properties of the discs αx , furnished  imply that for every fixed ζ ∈ I the function  by Lemma 8.4.1, D ξ → ρ X(ζ ) + αX(ζ ) (ξ ) is strongly convex with the minimum ρ (X(ζ )) at ξ = 0 and with no other critical points. It follows that the set     Dζ := ξ ∈ D : ρ X(ζ ) + αX(ζ ) (ξ ) < ρ(X(ζ )) + ε˜ (ζ ) (8.30) contains the origin, is simply connected (a disc), it is compactly contained in D in view of (8.29) and the choice of ε˜ , and the discs Dζ depend smoothly on the point ζ ∈ I. Choose a smooth family of diffeomorphisms φζ : D → D ζ (ζ ∈ I) which are holomorphic on D and satisfy φζ (0) = 0. Let α : I × D → R3 be defined by α(ζ , ξ ) = αX(ζ ) (φζ (ξ )),

ζ ∈ I, ξ ∈ D.

(8.31)

It follows from (8.29), (8.30), and (8.31) that ρ (X(ζ ) + α(ζ , ξ )) = ρ(X(ζ )) + ε˜ (ζ ),

ζ ∈ I, |ξ | = 1.

(8.32)

Pick a smooth function r : I → [0, 1] such that r(ζ ) = 1 when ε(ζ ) ≥ δ /2 and the support of r is contained in the relative interior of J0 ∩ U. We now apply Theorem 6.4.1 (the Riemann–Hilbert deformation method) to the conformal minimal immersion X : M → D, the map α given by (8.31), and the function r. It is straightforward to verify that the resulting conformal minimal immersion Y : M → D given by Theorem 6.4.1 satisfies conditions (1)–(3) in Proposition 8.4.2 provided that the number η > 0 in Theorem 6.4.1 is chosen small enough. (The main point is the condition (8.32) and the definition of the functions ε˜ and r.) To obtain (4), note that we clearly have a uniform estimate |αxj (ξ )| ≤ b|ξ | (ξ ∈ D, x ∈ L, j = 1, 2) for some constant b > 0. From (8.31) we get |α(ζ , ξ )| ≤ b|φζ (ξ )| for ζ ∈ I and ξ ∈ D. Together with (8.28) and (8.30) we obtain

8.4 Lifting Boundaries of Conformal Minimal Surfaces

363

ε0 ≥ ε˜ (ζ ) ≥ ρ(X(ζ ) + α(ζ , ξ )) − ρ(X(ζ )) ≥ c|φζ (ξ )|2 ≥

c |α(ζ , ξ )|2 b2

√ √ which in turn gives |α(ζ , ξ )| ≤ C0 ε0 with C0 = b/ c for all ζ ∈ I and ξ ∈ D. On bM \ I the same estimate follows by the choice of the function r which is supported in I. By increasing C0 slightly, this gives (4) provided that the approximation in Theorem 6.4.1 (see (6.6) and (i)) is close enough. Finally, the preservation of flux (condition (5)) and interpolation condition (6) are guaranteed by Theorem 6.4.1. This completes the proof provided the support of the function ε does not contain any boundary curves of M. Otherwise, write ε = ε1 + ε2 where each of the two functions ε1 , ε2 : bM → [0, ε0 ] satisfies the conditions of the special case considered above. By first deforming X to Y1 using the function ε1 and subsequently deforming Y1 to Y = Y2 using the function ε2 , the resulting conformal minimal immersion Y satisfies the conclusion of Proposition 8.4.2 provided that the approximations are sufficiently close at each step.   We now explain how to avoid critical points of a Morse exhaustion function ρ : D → R by adapting the method from [140, Sections 3.10–3.11]. A similar construction for holomorphic curves was developed in [114]. Definition 8.4.3. A critical point x0 of a C 2 function ρ is nice if ρ agrees locally near x0 with its second order Taylor polynomial based at x0 . Lemma 8.4.4. Every Morse function ρ can be approximated arbitrarily closely in the fine C 2 topology by a Morse function ρ with the same critical locus and with nice critical points. Furthermore, ρ can be chosen to agree with ρ outside an arbitrarily small neighbourhood of the critical locus. Proof. Assume that x0 is an (isolated) critical point of ρ and ρ(x) = Q(x) + η(x),

lim

x→x0

η(x) = 0. |x − x0 |2

Choose a smooth increasing function χ : R → [0, 1] such that χ(t) = 0 for t ≤ 1 and χ(t) = 1 for t ≥ 2. Given ε > 0, we consider the function ρε (x) = Q(x) + χ(ε −1 |x − x0 |) η(x). Then ρε = Q on |x − x0 | ≤ ε and ρε = ρ on |x − x0 | ≥ 2ε. As ε → 0, the C 2 norm of ρ(x) − ρε (x) = (1 − χ(ε −1 |x − x0 |)) η(x) tends to zero. If ε > 0 is chosen small enough then ρε satisfies the conclusion of the lemma at the critical point x0 . The same modification can be performed simultaneously at all critical points of ρ.   A minimal strongly plurisubharmonic function has no critical points of index bigger than 1 (see Remark 8.1.11). Critical points of index zero are local minima

364

8 Minimal Surfaces in Minimally Convex Domains

and are not approached by the boundary X(bM) when applying Proposition 8.4.2. Assume now that x0 is a nice Morse critical point of ρ with Morse index 1. The subsequent analysis is local near x0 , so we assume that x0 = 0 ∈ R3 , ρ(x0 ) = 0, and ρ(x) = ρ(x1 , x2 , x3 ) = −a1 x12 + a2 x22 + a3 x32 + η(x),

(8.33)

where −a1 < 0 < a2 ≤ a3 and the function η vanishes on a neighbourhood of the origin. Note that a1 < a2 since ρ is minimal strongly plurisubharmonic. Choose a number c0 > 0 small enough such that η vanishes on the set   (8.34) Pc0 := (x1 , x2 , x3 ) ∈ R3 : a1 x12 ≤ c0 , a2 x22 + a3 x32 ≤ 4c0 . The straight line arc E ⊂ R3 defined by   E = (x1 , 0, 0) ∈ R3 : a1 x12 ≤ c0

(8.35)

is a local stable manifold of the critical point 0 of ρ. Set λ = a2 /a1 > 1. Choose a number μ ∈ R with 1 < μ < λ and set t0 = c0 (1 − 1/μ)2 .

(8.36)

Note that 0 < t0 < c0 (1 − 1/λ )2 < c0 . The following is [140, Lemma 3.10.1, p. 94], adapted to the situation at hand. Lemma 8.4.5. Assume that 0 is the only critical value of the function ρ (8.33) in the set {−c0 < ρ < 3c0 }. Then there exists a minimal strongly plurisubharmonic function τ : D ∩ {ρ < 3c0 } → R satisfying the following conditions. (a) {ρ ≤ −c0 } ∪ E ⊂ {τ ≤ 0} ⊂ {ρ ≤ −t0 } ∪ E (here E is given by (8.35)). (b) {ρ ≤ c0 } ⊂ {τ ≤ 2c0 } ⊂ {ρ < 3c0 }. (c) There is a t1 ∈ (t0 , c0 ) such that τ = ρ + t1 holds outside the set Pc0 (8.34). (d) The function τ has no critical values in the interval (0, 2c0 ]. The sublevel sets {τ < c} for c > 0 in a neighbourhood of the origin are shown in [140, Fig. 3.5, p. 100] (in a similar setting of strongly plurisubharmonic functions). Proof. The choice of the number t0 (8.36) implies that there is a smooth convex increasing function h : R → [0, +∞) satisfying the following conditions. (i) (ii) (iii) (iv)

h(t) = 0 for t ≤ t0 . For t ≥ c0 we have that h(t) = t − t1 with t1 = c0 − h(c0 ) ∈ (t0 , c0 ). For t0 ≤ t ≤ c0 we have that t − t1 ≤ h(t) ≤ t − t0 . ˙ ≤ 1 and 2t h(t) ¨ + h(t) ˙ < λ. For all t ∈ R we have that 0 ≤ h(t)

The construction of such a function is entirely elementary (cf. [140, pp. 99–100]; its graph is shown on [140, Fig. 3.4, p. 99]). Let τ : R3 → R be given by τ(x) = −h(a1 x12 ) + a2 x22 + a3 x32 + η(x).

(8.37)

8.4 Lifting Boundaries of Conformal Minimal Surfaces

365

Setting t = a1 x12 , a calculation shows that on the set Pc0 ⊂ {η = 0} (8.34) we have −

  ∂ 2 τ(x) ∂ 2 τ(x) ¨ + h(t) ˙ 2t h(t) < 2a = 2a = , 1 2 ∂ x12 ∂ x22

where the inequality holds by property (iv) of h (recall that λ = a2 /a1 ). This shows that τ is minimal strongly plurisubharmonic on Pc0 . The other properties of τ follow immediately from the properties of h (cf. [140, Lemma 3.10.1]). Condition (c) shows that τ is minimal strongly plurisubharmonic also on the complement of Pc0 . Condition (d) obviously holds on Pc0 , while on the complement of Pc0 it follows from (c) and the assumptions on ρ.   Combining Proposition 8.4.2 and Lemma 8.4.5 we now prove the following lemma which provides the main induction step in the proof of Theorem 8.3.1. Lemma 8.4.6. Let ρ be a minimal strongly plurisubharmonic function on a domain D ⊂ R3 , and let a < b be real numbers such that the set Da,b = {x ∈ D : a < ρ(x) < b}

(8.38)

is relatively compact in D. Given numbers 0 < η < b − a, ε > 0, δ > 0, a conformal ˚ a number minimal immersion X : M → D such that X(bM) ⊂ Da,b , a point p0 ∈ M, ˚ there exists a conformal minimal immersion d > 0, and a compact set K ⊂ M, Y : M → D satisfying the following conditions. (a) Y (bM) ⊂ Db−η,b (equivalently, b − η < ρ(Y (ζ )) < b for every ζ ∈ bM). (b) ρ(Y (ζ )) ≥ ρ(X(ζ )) − δ for every ζ ∈ M. (c) Y − XC 1 (K) < ε. (d) distY (p0 , bM) > d. (e) FluxY = FluxX . ˚ (f) Y agrees with X to a given finite order at a given finite set of points in M. Proof. If the domain Da,b (8.38) does not contain any critical points of ρ, then a finite number of applications of Proposition 8.4.2 furnishes a conformal minimal immersion Y : M → D satisfying all conditions except perhaps (d). This last condition can be achieved by an arbitrarily C 0 small deformation of Y , using Lemma 7.3.1, which allows one to increase the interior boundary distance by an arbitrarily large amount while staying arbitrarily C 0 -close to the given map. Suppose now that x1 , . . . , xm are the (nice) critical points of ρ in the domain Da,b (8.38). We may assume that the numbers c j = ρ(x j ) are pairwise distinct, and we enumerate the points so that a < c1 < c2 < · · · < cm < b. Decreasing η > 0 if necessary we may assume that cm < b − η. By general position we may also assume that X(bM) does not contain any of the points x j . Pick c0 > 0 such that Lemma 8.4.5 applies to the critical point x1 of the function ρ − c1 (which vanishes at x1 ) with the constant c0 . Applying Proposition 8.4.2 finitely many times, we can find a conformal minimal immersion X1 : M → D such that X1 (bM) ⊂ Dc1 −c0 ,b and X1

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8 Minimal Surfaces in Minimally Convex Domains

satisfies conditions (b), (c), (e) and (f) in Lemma 8.4.6 with X1 in place of Y , and for some new given pair of constants ε1 and δ1 in place of ε and δ . If x1 is a local minimum of ρ (a critical point of index 0) then a new connected component of the sublevel set {ρ < t} appears at x1 when t passes the value t = c1 . In this case, we can apply Proposition 8.4.2 in order to push the boundary curve X1 (bM) into Dc1 ,b , the point being that the image of M cannot enter this new component when applying the lifting method furnished by Proposition 8.4.2. It remains to consider the case when x1 is a critical point of index 1. By Theorem 3.4.1 we can assume that the boundary curve X1 (bM) avoids the local stable manifold E (see (8.35)) of the point x1 . Let τ be the function furnished by Lemma 8.4.5 for the critical point x1 of ρ − c1 and the constant c0 . Applying Proposition 8.4.2 with the function τ finitely many times, we lift the boundary X1 (bM) above the critical level ρ = c1 and obtain a new conformal minimal immersion Y1 : M → D satisfying Y1 (bM) ⊂ Dc1 ,b . (Note that τ is noncritical on the set where the lifting procedure takes place.) As before, Y1 can be chosen to satisfy conditions (b), (c), (e), and (f) in Proposition 8.4.2 with Y1 in place of Y and X1 in place of X, and for some new given pair of constants ε1 > 0 and δ1 > 0. We now repeat the same procedure with the next critical point x2 . First, we apply Proposition 8.4.2 to Y1 in order to push Y1 (bM) into Dc2 −c0 ,b for a suitably small constant c0 > 0, chosen such that Lemma 8.4.5 applies to ρ − c2 at x2 . Call the new conformal minimal immersion X2 . Next, we proceed as in step 1 to lift the image of bM across the level ρ = c2 by applying Lemma 8.4.5 with X replaced by X2 . This furnishes a conformal minimal immersion Y2 : M → D with Y2 (bM) ⊂ Dc2 ,b and satisfying the conditions in the lemma with respect to X2 . The process continues in an obvious way, and in finitely many steps we find a conformal minimal immersion Y : M → D satisfying Y (bM) ⊂ Db−η,b (condition (a)) and also conditions (e) and (f). It follows from the construction and Lemma 8.4.1 that the number of steps needed in the process depends only on the geometry of ρ and may be determined in advance. Hence, we can fulfil conditions (b) and (c) by choosing the positive numbers ε j , ε j , δ j , δ j ( j = 1, . . . , m) sufficiently small at every step. Finally, condition (d) is achieved by appealing to Lemma 7.3.1.  

8.5 Proofs of Theorems 8.3.1, 8.3.3, 8.3.4, and 8.3.11 In this section we prove the main results stated in Section 8.3. Proof of Theorem 8.3.1. Let X = X0 : M → D be a conformal minimal immersion and ˚ Given ε > 0, we shall find a complete proper conformal K be a compact set in M. minimal immersion X : M˚ → D satisfying  X − XK = sup |X(p) − X(p)| < ε p∈K

8.5 Proofs of Theorems 8.3.1, 8.3.3, 8.3.4, and 8.3.11

367

and FluxX = FluxX . A map with these properties will be found as the limit X = lim j→∞ X j of a sequence of conformal minimal immersions X j : M → D constructed  M) ˚ will by an inductive application of Lemma 8.4.6. The hitting condition Λ ⊂ X( be obtained in a similar way as in the proof of Theorem 3.6.1, and the required modifications in the proof are explained at the end. Let ρ : D → R be a smooth minimal strongly plurisubharmonic Morse exhaustion function (see Definition 8.1.1). Denote by P = {x1 , x2 , . . .} ⊂ D the (discrete) critical locus of ρ, where the points x j are enumerated so that ρ(x1 ) < ρ(x2 ) < · · · . By Lemma 8.4.4 we may assume that every x j is a nice critical point of ρ. Pick increasing sequences c1 < c2 < c3 . . . and d1 < d2 < d3 . . . such that supM ρ ◦ X < c1 , lim j→∞ c j = +∞, and lim j→∞ d j = +∞. Also, choose a decreasing sequence δ1 > δ2 > · · · > 0 with δ = ∑∞j=1 δ j < ∞. Fix a point p0 ∈ K˚ and set X0 = X. We shall construct a sequence of smooth conformal minimal immersions X j : ˚ and M → D, an increasing sequence of compacts K = K0 ⊂ K1 ⊂ · · · ⊂ ∞j=1 K j = M, a decreasing sequence of positive numbers ε j > 0 such that the following conditions hold for every j = 1, 2, . . .. (i j ) c j < ρ ◦ X j < c j+1 on M \ K j . (ii j ) ρ ◦ X j > ρ ◦ X j−1 − δ j on M. (iii j ) X j − X j−1 C 1 (K j−1 ) < ε j .

(iv j ) distX j (p0 , M \ K j ) > d j . (Here, distX is the distance in the metric X ∗ (ds2 ).) (v j ) Flux(X j ) =Flux(X j−1 ).  (vi j ) ε j < 21 min ε j−1 , dist(X j−1 (M), bD), inf p∈K j−1 |dX j−1 (p)| .

To begin the induction, set ε0 = ε/2 and K = K0 . Assume inductively that for some j ∈ N we have found maps X0 , . . . , X j−1 , numbers ε0 , . . . , ε j−1 , and compact sets K0 , . . . , K j−1 such that the above conditions hold. Pick a number ε j > 0 satisfying condition (vi j ). Applying Lemma 8.4.6 with the data X j−1 , K j−1 , ε j , d j furnishes a conformal minimal immersion X j : M → D satisfying condition (i j ) on the boundary bM, conditions (ii j ), (iii j ), (v j ), and distX j (p0 , bM) > d j . Next, pick a compact set K j ⊂ M˚ such that K j−1 ⊂ K˚ j and conditions (i j ) and (iv j ) hold. (It suffices to take K j big enough.) This completes the induction step. Condition (vi j ) implies that ∑∞ k= j+1 εk < ε j for every j = 0, 1, . . .; in particular, ∞ ∑k=0 εk < 2ε0 = ε. Condition (iii j ) ensures that the sequence X j converges uniformly on compacts in ∞j=1 K j = M˚ to a harmonic map X = lim j→∞ X j : M˚ → D. Conditions (iii j ) and (vi j ) show that for every j = 0, 1, . . . we have that X − X j C 1 (K j ) ≤

∞

∞

k= j

k= j

∑ Xk+1 − Xk C 1 (K j ) < ∑ εk+1 < 2ε j+1 < ε j .

(8.39)

 j) ⊂ In particular, X − XC 0 (K) < ε. The estimate (8.39) and (vi j+1 ) show that X(K  M) ˚ ⊂ D. Since inf p∈K j |dX j (p)| > 2ε j+1 by D; since this holds for all j, we have X( (vi j+1 ), it follows from (8.39) that X is a conformal minimal immersion on K j . As this holds for all j ∈ N, X : M˚ → D is a conformal minimal immersion. By (v j ) we

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8 Minimal Surfaces in Minimally Convex Domains

 = Flux(X). Finally, conditions (i j )–(iii j ) ensure that X is proper have that Flux(X) into D, while conditions (iii j ) and (iv j ) show that X is complete.  It remains to explain how to ensure that the image X(M) contains a given discrete set Λ ⊂ D. We may choose the exhaustion function ρ and the sequence c j as above such that no point of Λ is a critical point of ρ, and no point of Λ lies on any of the level sets {ρ = c j }. We now explain the initial step in which we find a conformal minimal immersion X0 : M → D which satisfies supM ρ ◦ X0 < c1 , has the same ˚ contains the finite set Λ0 := Λ ∩ {ρ < c1 } = flux as X, and whose image X0 (M) {q1 , . . . , qm }. To simplify the matter, we assume by the general position theorem (see Theorem 3.4.1) that X(M) ∩ Λ = ∅. We may assume that M is a smoothly bounded domain in an open Riemann surface R. Choose distinct points p1 , . . . , pm ∈ bM and attach to M short smooth pairwise disjoint arcs E1 , . . . , Em ⊂ R such that E j ∩ M = {p j } for j = 1, . . . , m and the intersection is transverse. Note that S = M ∪ mj=1 E j is then an admissible set in R (see Definition 1.12.9). We extend X to a smooth immersion on each arc E j such that the extended map X : S → D is a generalized conformal minimal immersion (see Definition 3.1.2) with image contained in the set {ρ < c1 }, and for every j = 1, . . . , m we have that X(b j ) = q j for some point b j in the relative interior of the arc E j . By the Mergelyan theorem for generalized conformal minimal immersions (see Proposition 3.3.2), we can approximate X by a conformal minimal immersion (still denoted X) on a neighbourhood U ⊂ R of S without changing the flux and the values at the points b1 , . . . , bm . By Theorem 6.7.1 there is a conformal diffeomorphism φ : M → φ (M) ⊂ U whose image contains ˚ and which is close to the identity outside a the points b1 , . . . , bm in the interior φ (M) small neighbourhood of the points p1 , . . . , pm . The composition X0 := X ◦φ : M → D is then a conformal minimal immersion with the desired properties. In particular, the ˚ conditions imply that for every j = 1, . . . , m we have φ (a j ) = b j for some a j ∈ M, and hence X0 (a j ) = q j . Thus, X0 hits Λ0 = Λ ∩ {ρ < c1 } in the interior points a1 , . . . , am of M. We now apply the induction step explained above to find a conformal minimal immersion X1 : M → D which satisfies conditions (i1 )–(vi1 ) and agrees with X0 at the points a1 , . . . , am . Next, we modify X1 such that these conditions are preserved ˚ also contains the finite set Λ1 := Λ ∩ {c1 < ρ < c2 }. This is done as in and X1 (M) the initial step explained above which produced X0 from X. Write m1 = m and let am1 +1 , . . . , am2 ∈ M˚ be the points which are mapped by X1 to the points in Λ1 . Clearly this procedure can be continued inductively, and it yields a discrete set A = {a j } ⊂ M˚ such that the limit X : M˚ → D is a proper complete conformal minimal  immersion satisfying Λ ⊂ X(A).   Proof of Theorem 8.3.3. The proof is the same as that of Theorem 8.3.1 modulo the obvious modifications, replacing conditions pertaining to the distance from bD (see condition (vi j ) above) by the corresponding conditions pertaining to the distance from the end of the domain Ω on which the function ρ tends to +∞.   Proof of Theorem 8.3.4. By Lemma 8.1.19 there is a minimal strongly plurisubharmonic function ρ on an open set D ⊃ D such that D = {x ∈ D : ρ(x) < 0} and dρ = 0 on bD = {ρ = 0}. Pick η > 0 such that the set {ρ < η} is relatively compact

8.5 Proofs of Theorems 8.3.1, 8.3.3, 8.3.4, and 8.3.11

369

in D and dρ = 0 on the compact set L = {x ∈ D : −η ≤ ρ(x) ≤ η}.

(8.40)

Let C0 > 0 be a constant satisfying the conclusion of Proposition 8.4.2 for the data D , ρ, L. In view of Theorem 8.3.1 we may assume that the given conformal minimal immersion X0 : M → D satisfies c0 = c0 (X0 ) := inf ρ(X0 (p)) > −η. p∈bM

(8.41)

˚ For every j = 0, 1, 2, . . . we set (Equivalently, X0 (bM) ⊂ D ∩ L.) c j = 2− j c0 ,

η j = c j+1 − c j = 2− j−1 |c0 |.

Pick an increasing sequence 0 < d1 < d2 < · · · with lim j→∞ d j = +∞ and a decreasing sequence δ j > 0 with δ = ∑∞j=1 δ j < ∞. By following the proof of Theorem 8.3.1, using also the estimate (4) in Proposition 8.4.2 with the constant C0 introduced above, we find a sequence of conformal minimal immersions X j : M → D ( j = 1, 2, . . .), an increasing sequence of compacts K = K0 ⊂ K1 ⊂ · · · ⊂ ∞j=1 K j = ˚ and a decreasing sequence of numbers ε j > 0 such that the following conditions M, hold for all j = 1, 2, . . .. ρ ◦ X j > c j on M \ K j . ρ ◦ X j > ρ ◦ X j−1 − δ j on M. X j − X j−1 C 1 (K j−1 ) < ε j . distX j (p0 , M \ K j ) > d j . Flux(X j ) = Flux(X j−1 ). ε j < 12 min{ε j−1 , dist(X j−1 (M), bD), inf p∈K j−1 |dX j−1 (p)|}.  (vii j )X j − X j−1 C 0 (M) ≤ √C0 j |c0 |.

(i j ) (ii j ) (iii j ) (iv j ) (v j ) (vi j )

2

These conditions correspond to those in the proof of Theorem 8.3.1, except that condition (i j ) is adjusted to the present setting and the additional condition (vii j ) follows from the estimate (4) in Proposition 8.4.2. (By Lemma 7.3.1, condition (iv j ) can be achieved by a deformation which is arbitrarily small in the C 0 (M) norm, and the error made by this deformation is absorbed by the constant C0 in (vii j ).) Set C1 = C0 ∑∞j=1 √1 j . Condition (vii j ) ensures that the sequence X j converges 2

uniformly on M to a continuous map X : M → D satisfying X − X0 C 0 (M) ≤  C1 |c0 |. On the set L given by (8.40) the function |ρ| is proportional to the distance from bD, so the number |c0 |, defined by (8.41), is proportional to max p∈bM dist(X0 (p), bD). This gives the estimate (8.25) in Theorem 8.3.4 for a suitable choice of the constant C > 0 which depends only on the geometry of ρ on L. We can see as in the proof of Theorem 8.3.1 that X|M˚ : M˚ → D is a proper complete conformal minimal immersion.  

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8 Minimal Surfaces in Minimally Convex Domains

Proof of Corollary 8.3.10. We follow the procedure in the proofs of Theorems 8.3.1 and 8.3.3, except that we also construct a conformal structure on S during the induction process. Alternatively, we fix a complex structure J on S and then find a domain M ⊂ S diffeotopic to S and a complete proper conformal minimal immersion X from the Riemann surface (M, J) to D; precomposing X by a diffeomorphism h : S → M gives a minimal surface X ◦ h : S → D satisfying the corollary for the complex structure h∗ J on S. The main addition in this construction is that at every step of the induction we have to enlarge the domain in S on which we are working, possibly adding to it a handle in order to ensure a suitable change of its topology. We then extend our map across the handle to obtain a generalized conformal minimal immersion on the resulting admissible set, and then apply the Mergelyan theorem for conformal minimal immersions (see Proposition 3.3.2) to obtain the next map and the next domain in the sequence. For the details of this construction, we refer to [14, proof of Theorem 1.4 (b) and Corollary 1.5].   Proof of Theorem 8.3.11. Let D ⊂ Rn (n > 3) be an (n − 2)-convex domain with a smooth strongly (n − 2)-plurisubharmonic exhaustion function ρ : D → R. The level sets Sc = {ρ = c} for noncritical values c ∈ R of ρ are strongly (n − 2)-convex hypersurfaces, which means in particular that at every point x0 ∈ Sc there is a 2plane L ⊂ Tx0 Sc on which Hessρ is strongly positive (see Definition 8.1.18). Hence, we can apply the same methods as in the proof of Theorem 8.3.1 but using suitably shaped flat discs Δx ⊂ Rn for points x ∈ D near x0 , lying in affine 2-planes parallel to L. For such families of discs, the Riemann–Hilbert method furnished by Theorem 6.6.2 yields modifications of a conformal minimal immersion X : M → D pushing the boundary curve X(bM) to a higher level of ρ (see Proposition 8.4.2). The rest of the proof goes through as before, except that Lemma 8.4.5 must be adapted to the case when ρ is a Morse strongly (n − 2)-plurisubharmonic. (Such a function has critical points of indices at most n − 3.) Details in the case of a convex domain D ⊂ Rn are given in [14, proof of Theorems 1.2 and 1.4].   On the other hand, if the level hypersurfaces of ρ are merely (n − 1)-convex, this approach would require approximate solutions of the Riemann–Hilbert boundary value problem for families of nonflat conformal minimal discs in dimensions n > 3. We are currently unable to prove optimal results for (n − 1)-convex domains. A final comment is in order. Most results discussed thus far in this chapter pertain to minimal surfaces in minimally convex domains which are normalized by bordered Riemann surfaces. This restriction cannot be entirely removed since a bounded domain in Rn (and, more generally, any domain which admits a nonconstant negative 2-plurisubharmonic exhaustion function) does not contain any minimal surface of parabolic conformal type.

8.6 A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3

371

8.6 A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3 Theorems 8.3.1, 8.3.3, and 8.3.4 show that every bounded minimally convex domain D in R3 admits many complete properly immersed minimal surfaces of hyperbolic conformal type. Indeed, any conformal minimal immersion x : M → D from a compact bordered Riemann surface M can be approximated uniformly on compacts in M˚ = M \ bM by proper complete conformal minimal immersions M˚ → D. In contrast, we will now prove the following rigidity result for complete minimal surfaces of finite total curvature which are proper in R3 . Theorems 8.6.1 and 8.6.2 were originally obtained in [15, Sect. 4]. Theorem 8.6.1. Let S ⊂ R3 be a complete, connected, properly immersed minimal surface of finite total curvature possibly with compact boundary bS ⊂ S. If D ⊂ R3 is a smoothly bounded minimally convex domain containing S, then D = R3 or S is a plane; in the latter case, the connected component of D containing S is a slab, a halfspace, or R3 . Although the surface S in the above theorem may have compact boundary bS, completeness implies that it also has at least one noncompact proper end. By a slab in R3 , we mean a domain bounded by two parallel planes. In particular, if Ω is a connected component of R3 \ S where S is a nonflat properly embedded minimal surface of finite total curvature in R3 , then Theorem 8.6.1 shows that Ω is a maximal minimally convex domain, in the sense that the only smoothly bounded minimally convex domain D containing Ω is R3 itself. Theorem 8.6.1 is an application of the following maximum principle at infinity. Theorem 8.6.2. Assume that S ⊂ R3 is a complete, connected, immersed minimal surface of finite total curvature with nonempty compact boundary bS. If D ⊂ R3 is a smoothly bounded minimally convex domain containing S, then dist(S, R3 \ D) = dist(bS, R3 \ D). The particular case of Theorem 8.6.2 when S is compact (with boundary) is already ensured by Proposition 8.1.21. The main difficulty in the general case is that one must deal with the contact of S and bD at infinity, a rather delicate task. Maximum principles at infinity are the key to many celebrated classification results in the theory of minimal surfaces; see for instance W. H. Meeks and H. Rosenberg [256] and the references therein. In the proof of Theorem 8.6.2 we exploit the geometry of complete minimal surfaces of finite total curvature along with the Kontinuit¨atssatz for conformal minimal surfaces (see Proposition 8.1.20). Our proofs strongly use the hypothesis that D has smooth boundary. Problem 8.6.3. Do Theorems 8.6.1 and 8.6.2 still hold if D is a minimally convex domain whose boundary is not smooth? Before proving Theorem 8.6.2, we show how it implies Theorem 8.6.1 by a Kontinuit¨atssatz type argument as in Proposition 8.1.20.

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8 Minimal Surfaces in Minimally Convex Domains

Proof of Theorem 8.6.1, assuming Theorem 8.6.2. Assume that Dc = R3 \ D = ∅ and let us prove that S is a plane. Choose a relatively compact disc ! ⊂ S and set S = S \ !. By Theorem 8.6.2 we have that dist(S , Dc ) = dist(bS , Dc ), and hence dist(S, Dc ) = dist(!, Dc ). Thus, there exist points x0 ∈ S˚ and y0 ∈ bD such that |x0 − y0 | = dist(S, Dc ), and we infer from Proposition 8.1.21 that S is a plane. Without loss of generality we may assume that S = {(x, y, z) ∈ R3 : z = 0}. Set W+ = {(x, y, z) ∈ R3 : z > 0}. We claim that if the set W+ \ D is nonempty then it is a halfspace. Indeed, assume that W+ \ D = ∅ and let us prove first that d+ := dist(S, bD ∩W+ ) = dist(S,W+ \ D) > 0. Consider the family of vertical negative half-catenoids   Σa = (x, y, z) ∈ R3 : x2 + y2 = a2 cosh2 (z/a), z ≤ 0 ,

(8.42)

0 < a ≤ 1.

Let A+ denote the cylinder   A+ := (x, y, z) ∈ R3 : x2 + y2 ≤ 2, 0 ≤ z ≤ τ+ , where τ+ > 0 is chosen small enough such that ((0, 0, τ+ ) + Σ1 ) ∩ {z ≥ 0} ⊂ A+ ⊂ D. The Kontinuit¨atssatz for minimal surfaces (cf. Proposition 8.1.20) implies that Σa+ := ((0, 0, τ+ ) + Σa ) ∩ {z ≥ 0} ⊂ D for all 0 < a ≤ 1. Indeed, Σa+ are minimal surfaces with boundaries in A+ ∪ S ⊂ D, and Σ1+ ⊂ A+ ⊂ D. It is easily seen that 0 0, thereby proving (8.42). If there is a point (x0 , y0 , d+ ) ∈ bD then, arguing as in the proof of Proposition 8.1.21 and using that D is connected, we infer that the plane Π+ := {z = d+ } lies in bD ∩ {z > 0} = b(W+ \ D), and hence W+ \ D is a halfspace. Otherwise, Π+ ⊂ D and we may reason as above (replacing S by Π+ ) to see that dist(Π+ ,W+ \ D) > 0 in contradiction to (8.42). A symmetric argument guarantees that {(x, y, z) ∈ R3 : z < 0} \ D is either empty or a halfspace. This proves Theorem 8.6.1, assuming that Theorem 8.6.2 holds.   The proof of Theorem 8.6.2 also follows from a Kontinuit¨atssatz argument; however, the construction of a suitable family of minimal surfaces is much more delicate. The surfaces will be multigraphs obtained as solutions of suitable Dirichlet problems for the minimal surface equation over finite coverings of annuli in R2 ; see Lemma 8.6.4. Before going into the construction, we introduce some notation.

8.6 A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3

For each pair of numbers 0 ≤ R0 < R ≤ +∞ we set   AR0 ,R := (x, y) ∈ R2 : R0 < |(x, y)| < R ,

373

AR0 = AR0 ,+∞ .

Endow A0 = R2 \ {(0, 0)} with the Euclidean metric and denote by πn : An0 −→ A0 the n-sheeted isometric covering for every n ∈ N. We also set AnR0 ,R := πn−1 (AR0 ,R ) for all 0 ≤ R0 < R < +∞, cR := {(x, y) ∈ R2 : |(x, y)| = R},

AnR0 = πn−1 (AR0 ),

cnR := πn−1 (cR ),

R > 0.

(8.43) (8.44)

Obviously, bAnR0 = cnR0 and bAnR0 ,R = cnR0 ∪ cnR , 0 < R0 < R < +∞, n ∈ N. A function u ∈ C 2 (AnR0 ,R ) is said to satisfy the minimal surface equation in AnR0 ,R if

∇u =0 div  1 + |∇u|2

in AnR0 ,R ;

(8.45)

equivalently, if {(p, u(p)) : p ∈ AnR0 ,R } is a minimal surface (a minimal multigraph). Compare with the minimal graph equation in (2.99) and (2.100). Given φ ∈ n C 0 (bAnR0 ,R ), a function u ∈ C 2 (AnR0 ,R ) ∩ C 0 (AR0 ,R ) is said to be a solution of the Dirichlet problem for the minimal surface equation in AnR0 ,R with boundary data φ if u satisfies (8.45) and the boundary condition u|bAnR ,R = φ . 0

We shall need the following lemma. Lemma 8.6.4. Let 0 < R0 < R1 , K ∈ (0, 1), n ∈ N, and let AnR0 ,R and AnR0 be defined by (8.43). There exists a number ε > 0, depending only on R0 , R1 , and K, such that the following holds. If R ≥ R1 , δ ∈ [0, ε], n

(a) v : AR0 → R is a real analytic solution of the minimal surface equation in AnR0 , n

(b) |∇v| < K/2 in AR0 , and (c) we set φR,δ : bAnR0 ,R → R, φR,δ = v in cnR0 and φR,δ = v + δ in cnR , then the Dirichlet problem for the minimal surface equation in AnR0 ,R with the boundary data φR,δ has a unique solution uR,δ . Furthermore, uR,δ satisfies the following conditions. n

(i) v ≤ uR,δ ≤ v + δ on AR0 ,R . (ii) uR,δ depends continuously on (R, δ ) ∈ [R1 , +∞) × [0, ε]. n (iii) {uR,δ }R>R1 → v on compact subsets of AR0 for all δ ∈ [0, ε] as R → +∞. The number ε > 0 in the lemma will only depend on suitable barrier functions ν p0 at boundary points p0 ∈ bAnR0 ,R adapted to our problem. In turn, the construction of these barrier functions only depends on the constants R0 , R1 , and K. Proof. For the existence part of the lemma we use Perron’s method for the minimal surface equation; for background see for instance [157, 275].

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8 Minimal Surfaces in Minimally Convex Domains

 n  Take arbitrary numbers R > R0 > 0 and δ ≥ 0. If w ∈ C 0 AR0 ,R , D is a convex n disc in AR0 ,R , and wD is the solution of the minimal surface equation in D which D the equals w on bD (such exists by the classical Rado Theorem), we denote by w n n D = w in AR0 ,R \ D and w D = wD in D. continuous function in AR0 ,R given by w  n  A function w ∈ C 0 AR0 ,R is said to be a subsolution of the Dirichlet problem n for the minimal surface equation in AR0 ,R , defined by φR,δ given in (c), if w ≤ φR,δ n D in D (hence w ≤ w D everywhere in AR0 ,R ) for all discs D in bAnR0 ,R and w ≤ w − as above. We denote by FR,δ the family of all such subsolutions. Likewise, w D in D (hence in is said to be a supersolution if w ≥ φR,δ in bAnR0 ,R and w ≥ w n n + AR0 ,R ) for all discs D in AR0 ,R ; the space of supersolutions is denoted FR,δ . Note − + that v|An ∈ FR,δ and (v + δ )|An ∈ FR,δ for all R > R0 and δ > 0, where v R0 ,R R0 ,R is the function given in item (a) in the statement of the lemma; hence these are − + nonempty families. If w1 ∈ FR,δ is a subsolution and w2 ∈ FR,δ is a supersolution then w1 ≤ w2 by the maximum principle. If on the other hand both w1 and w2 are − − then max{w1 , w2 } ∈ FR,δ is a subsolution as well; likewise, subsolutions in FR,δ + + . Define given w1 , w2 ∈ FR,δ we have that min{w1 , w2 } ∈ FR,δ uR,δ : AnR0 ,R → R,

uR,δ (p) = sup w(p). − w∈FR,δ

(8.46)

It follows that uR,δ is a solution of the minimal surface equation in AnR0 ,R and w1 ≤ uR,δ ≤ w2

− + for all w1 ∈ FR,δ and w2 ∈ FR,δ .

(8.47)

in AnR0 ,R for all R > R0 and all δ ≥ 0.

(8.48)

In particular, we have v ≤ uR,δ ≤ v + δ

Lemma 8.6.5. Given numbers R1 > R0 > 0 and K ∈ (0, 1), there exists an ε > 0 depending only on R0 , R1 , and K such that the following holds. If R ≥ R1 and 0 ≤ δ ≤ ε, then the function uR,δ given by (8.46) is a solution to the Dirichlet problem for the minimal surface equation with boundary data φR,δ in AnR0 ,R , that is to say, lim uR,δ (p) = φR,δ (p0 ) for all points p0 ∈ bAnR0 ,R . p→p0

Proof. Choose R ≥ R1 and a point p0 ∈ bAnR0 ,R . Let us distinguish cases. Case 1: p0 ∈ cnR0 (see (8.44)). Let us prove the existence of a number ε1 > 0, depending only on R0 , R1 , and K, for which the following statement holds. Given n δ ∈ [0, ε1 ] there exists a function ν p0 ∈ C 0 (AR0 ) such that ν p0 (p0 ) = v(p0 ) = + φR,δ (p0 ) and ν p0 |An ∈ FR,δ . Indeed, write πn (p0 ) = (x0 , y0 ) ∈ cR0 . Set R0 ,R

CR0 := {(x, y, z) ∈ R3 : |(x, y)| < R0 } and

8.6 A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3

375

JK := {(x, y, z) ∈ R3 : 0 ≤ z − v(p0 ) = K|(x − x0 , y − y0 )|}. Pick μ > 0 small enough in terms of R0 , R1 , and K such that the set     γ := (bCR0 ∩ JK ) ∩ {0 ≤ z − v(p0 ) ≤ μ} ∪ (JK \CR0 ) ∩ {z − v(p0 ) = μ} is a Jordan curve. It follows that γ has one-to-one orthogonal projection γ0 to the plane {z = 0} ≡ R2 . Ensure also that γ0 bounds a topological disc U ⊂ R2 with U ⊂ AR0 ,R1 ∪ cR0 . Thus, πn−1 (U) consists of n disjoint isometric copies of  ⊂ An be the connected component of πn−1 (U) containing p0 . Denote by U; let U 0 φ : bU → R the unique continuous function such that {(p, φ (p)) : p ∈ bU = γ0 } = γ. Further, the domain U satisfies an exterior sphere condition with radius R0 (cf. [345, Definition 1.4 (i)]) and hence, if μ > 0 is sufficiently small in terms of R0 , R1 , and K, the Dirichlet problem for the minimal surface equation in U with boundary data φ has a unique solution f : U → R satisfying ( f ◦ πn )(p0 ) = φ (p0 ) = v(p0 ) and  \ {p0 }; see [345] and take into account condition (b) in the statement f ◦ πn > v in U of the lemma and γ ⊂ JK . It follows that      \ cnR = inf v(p0 ) + μ − v(p) : p ∈ bU  \ cnR > 0. inf ( f ◦ πn )(p) − v(p) : p ∈ bU 0 0 Choose a number ε1 > 0 smaller than this infimum. Note that ε1 can be chosen not to depend on v; take into account (b). Further, since μ depends on R0 , R1 , and K but n not on p0 ∈ cnR0 , the same holds for ε1 . It now suffices to define ν p0 : AR0 → R by  ν p0 = Since ν p0 |An

R0 ,R

min{ f ◦ πn , v + ε1 } v + ε1

 on U, n  on AR0 \ U.

+ ∈ FR,δ , δ ∈ [0, ε1 ], and ν p0 (p0 ) = v(p0 ) = φR,δ (p0 ), the bounds

(8.47) and (8.48) trivially ensure that lim p→p0 uR,δ (p) = φR,δ (p0 ). Case 2: p0 ∈ cnR . Let us now prove the existence of a number ε2 > 0, depending only on R0 , R1 , and K, for which the following statement holds. Given δ ∈ [0, ε2 ], there n − such that ν p0 (p0 ) = v(p0 ) + δ = φR,δ (p0 ). exists a ν p0 ∈ C 0 (AR0 ,R ) ∩ FR,δ Indeed, consider a small disc V ⊂ R2 centred at the origin and with radius less than R1 − R0 , set UR := (πn (p0 ) +V ) ∩ AR0 ,R , R ⊂ AR ,R denote the connected component of πn−1 (UR ) containing p0 ; and let U 0 R → UR is an isometry. Choose a linear function f : R2 → R obviously, πn |UR : U R \ {p0 }, and pick a number ε2 satisfying ( f ◦ πn )(p0 ) = v(p0 ) and f ◦ πn < v on U such that  \ cnR }. 0 < ε2 < inf{v(p) − ( f ◦ πn )(p) : p ∈ bU n

The existence of such a number follows from condition (b), and it can be chosen independent of v and R (it only depends on R0 , R1 , and K). Given δ ∈ [0, ε2 ], it n suffices to define the function ν p0 : AR0 ,R → R by

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8 Minimal Surfaces in Minimally Convex Domains

 ν p0 =

max{ f ◦ πn + δ , v + δ − ε2 } v + δ − ε2

R , on U n R . on AR0 ,R \ U

− As above, since ν p0 ∈ FR,δ and ν p0 (p0 ) = v(p0 ) + δ = φR,δ (p0 ), (8.47) and (8.48) guarantee that lim p→p0 uR,δ (p) = φR,δ (p0 ).  The number ε := min{ε1 , ε2 } clearly satisfies the lemma. 

We continue the proof of Lemma 8.6.4. In view of Lemma 8.6.5 it remains to check that, given numbers δ ∈ [0, ε] and R ≥ R1 , the solution uR,δ given by (8.46) is unique and satisfies conditions (i), (ii), and (iii). Uniqueness follows directly from the maximum principle. Condition (i) is ensured by (8.48). Since the boundary data φR,δ depend continuously on (R, δ ) ∈ [R1 , +∞) × [0, ε], the same holds for the solutions uR,δ , thereby proving (ii). In order to prove (iii), fix a number δ ∈ [0, ε] and take any divergent sequence {R j } j∈N ⊂ [R1 , +∞). By standard compactness results (see for instance [49]), the sequence {uR j ,δ } j∈N converges uniformly on compact n subsets of AR0 to a solution u of the minimal surface equation in AnR0 with boundary data u = v on cnR0 . Furthermore, (8.48) gives that v ≤ u ≤ v + δ , and hence u = v by the maximum principle at infinity (see for instance [256]). This proves (iii) and concludes the proof of Lemma 8.6.4.   Proof of Theorem 8.6.2. Assume that S is not compact, for otherwise the result follows directly from Proposition 8.1.21. Up to passing to the two-sheeted orientable covering we may assume that S = X(M), where M is a noncompact Riemann surface with compact boundary bM = ∅ and X : M → R3 is a complete conformal minimal immersion of finite total curvature. Note that bS = X(bM). The assumptions on S imply that M is of finite topology and parabolic conformal type (in particular, its ends are annular conformal punctures), and the Gauss map N : M → S2 of X extends conformally to the ends; see Theorem 4.1.1 and note that its proof is local around each end. Given an annular end E ∼ = D \ {0}, let nE denote the limit normal vector of X(E) at infinity and ΠE the linear plane in R3 orthogonal to nE . Let πE : R3 → ΠE denote the orthogonal projection. By the Jorge–Meeks Theorem 4.1.3, the minimal immersion X is a proper map and all the ends are finite sublinear multigraphs. This means that for any annular end E of M there exists an open solid circular cylinder C, with axis parallel to nE , such that the following conditions hold. (i) E ∩ X −1 (C) is compact and contains bE. (ii) (πE ◦ X)|E\X −1 (C) : E \ X −1 (C) → ΠE \C is a finite covering. (iii) lim j→∞ |X(p1 )| nE , X(p j ) = 0 for any divergent sequence {p j } j∈N ⊂ E. j Let wE denote the winding number of X(E) at infinity, which equals the degree of the covering (πE ◦ X)|E\X −1 (C) in (ii). Recall that bS = ∅. If D = R3 then there is nothing to prove. Assume that Dc = R3 \ D = ∅ and let

8.6 A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3

377

d := dist(S, Dc ) < +∞. It suffices to check that there exist points x0 ∈ S and y0 ∈ bD such that |x0 − y0 | = d.

(8.49)

Indeed, assume for a moment that this holds. If x0 ∈ bS, we are done. Otherwise, x0 ∈ S \ bS, and we infer from Proposition 8.1.21 that the surface S is flat and y0 − x0 + S ⊂ bD. It follows that d = dist(bS, Dc ). This concludes the proof of Theorem 8.6.2 provided that (8.49) holds. We now prove the assertion (8.49). We reason by contradiction and assume that dist(x, Dc ) > d

for all x ∈ S.

(8.50)

This implies that there is an annular end E conformally isomorphic to D \ {0} such that dist(X(E), Dc ) = d. Set Et := t nE + X(E) for all t ≥ 0. In particular, E0 = X(E). Condition (8.50) ensures that !

Et ⊂ D.

(8.51)

t∈[0,d]

Set CR := {(x, y, z) ∈ R3 : |(x, y)| < R}. Up to a rigid motion, a shrinking of E, and taking R0 > 0 large enough we may assume that nE = (0, 0, −1), X(bE) ⊂ bCR0 and dist(Ed , Dc ) = 0.

(8.52)

Given δ ≥ 0, we set γR0 (δ ) := bEd+δ = (d + δ )nE + bE0 ⊂ bCR0 .

Since ( t∈[0,d] Et ) ∩CR is compact for all R ≥ R0 (see (i)), condition (8.51) provides numbers ε > 0 and R1 > R0 such that !

(Et ∩CR1 ) ⊂ D.

(8.53)

t∈[0,d+ε]

In particular, γR0 (δ ) ⊂ D for all δ ∈ [0, ε]. Set γR (δ ) := Ed+δ ∩ bCR for all δ ∈ [0, ε] and R > R0 . For simplicity, write n for the winding number wE of X(E) as a multigraph over

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8 Minimal Surfaces in Minimally Convex Domains

AR0 = {(x, y) ∈ R2 : |(x, y)| > R0 } and denote by φR,δ : bAnR0 ,R → R the unique analytic function satisfying   γR0 (δ ) ∪ γR (0) = (p, φR,δ (p)) : p ∈ bAnR0 ,R for all δ ∈ [0, ε] and R > R0 . n

(See p. 372 for the notation.) Likewise, let v : AR0 → R denote the unique analytic function such that n Ed = {(p, v(p)) : p ∈ AR0 }. n

Increasing R0 if necessary we may assume that |∇v| < 1/4 on AR0 ; see condition (iii) above. If ε > 0 is chosen small enough, Lemma 8.6.4 guarantees that the Dirichlet problem for the minimal surface equation in AnR0 ,R with boundary data φR,δ has a unique solution uR,δ for all R ≥ R1 and δ ∈ [0, ε]. Furthermore, we have that v − δ ≤ uR,δ ≤ v in AnR0 ,R for all R ≥ R1 and δ ∈ [0, ε].

(8.54)

Fix a pair of numbers δ ∈ [0, ε] and R ≥ R1 , and set n

TR,δ := {(p, uR,δ (p)) : p ∈ AR0 ,R }. Conditions (8.53) and (8.54) guarantee that TR1 ,δ ⊂ D, whereas (8.51) and (8.53) ensure that bTR,δ = γR0 (δ ) ∪ γR (0) ⊂ D for all R ≥ R1 . Thus, the Kontinuit¨atssatz for minimal surfaces (see Proposition 8.1.20) implies that TR,δ ⊂ D for all R ≥ R1 . Further, by Lemma 8.6.4 we have TR,δ → Ed+δ uniformly on compact subsets as R → +∞, and hence Ed+δ ⊂ D. Since this holds for arbitrary δ ∈ [0, ε], we infer that t∈[0,d+ε] Et ⊂ D, and hence t∈[0,d+ε) Et ⊂ D. This contradicts (8.52), thereby proving (8.49) and hence completing the proof.  

Chapter 9

Minimal Hulls, Null Hulls, and Currents

In this chapter we study minimal hulls of compact sets in Rn and null hulls of compact sets in Cn for any n ≥ 3. These hulls were defined in the previous chapter in terms of the maximum principle for minimal plurisubharmonic and null plurisubharmonic functions, respectively (see Definition 8.1.9). They provide natural barriers for minimal surfaces or null curves with boundaries in the given compact set, in analogy to the role played by the holomorphic hull of a compact set in a complex manifold with respect to its complex subvarieties. In dimension n = 3, the minimal hull coincides with the mean-convex hull which has been studied in the literature on minimal hypersurfaces. The hulls under consideration are particular cases of those studied by F. R. Harvey and H. B. Lawson [183, 184, 185]. In Section 9.1 we discuss the general concept of a hull and review some of the main questions and results. In Section 9.2 we describe the minimal hull of any compact set K ⊂ Rn (n ≥ 3) in terms of bounded sequences of conformal minimal discs whose boundaries converge to K in measure; a similar result holds for the null hull of a compact set in Cn in terms of sequences of holomorphic null discs (see Theorems 9.2.1 and 9.2.6). These results are inspired by the analogous characterization of polynomial hulls in Cn by sequences of analytic discs, due to E. M. Poletsky [291] and S. Q. Bu and W. Schachermayer [74]. The main ingredient in our analysis is the Riemann–Hilbert modification technique for minimal discs and null discs, developed in Chapter 6. We also provide a formula for the largest minimal plurisubharmonic minorant of an upper semicontinuous function on a domain D ⊂ Rn for any n ≥ 3 (see Theorem 9.2.9). In Section 9.3 we show that the null hull of a closed rectifiable curve is either trivial or a holomorphic null curve. In Sections 9.4 and 9.5 we recall some basic facts about currents and the notion of the minimal current hull introduced by F. R. Harvey and H. B. Lawson in [185]. In Section 9.6 we consider Green currents supported on minimal discs, and in Section 9.7 we show that currents of this type characterize the minimal hull of any compact set in Rn . The analogous characterization is given for the null hull of a compact set in Cn by limits of null-positive Green currents. Most of the new results presented in this chapter were obtained in the paper [117] in dimension n = 3, and in [15, Sect. 5] in dimensions n > 3. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4_9

379

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9 Minimal Hulls, Null Hulls, and Currents

9.1 The Role of Hulls in Analysis and Geometry When discussing hulls, one typically considers dual families of objects: a family F of functions on a topological space X, and a family G of geometric objects in X. To fix the discussion, let F be a family of regular upper semicontinuous functions f : X → [−∞, ∞) on a manifold X, i.e., such that f (x) = lim supy→x f (y) holds for every x ∈ X. The F -hull of a compact subset K ⊂ X is defined by   F = x ∈ X : f (x) ≤ sup f for all f ∈ F . K (9.1) K

Note that supK f < ∞ since f is upper semicontinuous. (One often makes additional assumptions on F , but this is inessential for the present discussion.) A compact set F is F . The manifold X is called F -convex if K K ⊂ X is called F -convex if K = K compact for every compact K ⊂ X; equivalently, if X is exhausted by an increasing sequence of compact F -convex sets. These hulls satisfy several obvious properties. In particular, the following hold for every compact set K ⊂ X. F is a closed subset of X. (a) K  F F provided K F is compact. =K (b) K F F ⊂ K F . If in addition F1 is dense in F2 in the compact-open (c) F1 ⊂ F2 =⇒ K 2 1 F .  topology then KF = K 1

2

Assume now that G is a family of geometric objects with boundaries in X (for example, submanifolds, subvarieties, or their measure-theoretic generalizations discussed in Section 9.4) such that the restriction f |C satisfies the maximum principle for every pair f ∈ F and C ∈ G , in the sense that the maximum value is attained at a boundary point of C. (In our main examples, these restrictions will F for every be subharmonic functions on bordered Riemann surfaces.) Then, C ⊂ K C ∈ G with boundary bC ⊂ K, and it is natural to ask the following question: F described by the geometric objects in G ? To what extent is the hull K Ideally one would like to find for a given family of functions, F , a dual family F of any compact set K is precisely the union of objects, G , such that the hull K of supports of all C ∈ G with bC ⊂ K. However, this is impossible in general even in the simplest example of the convex hull Co(K) of a compact set K in Rn , the hull defined by the family of all affine linear functions on Rn . A more reasonable expectation is to obtain the hull in finitely or countably many steps, each time adding to the set from the previous step the supports of all objects C ∈ G with boundaries in it. For example, the convex hull of a compact set in Rn is obtained in finitely many steps by using the family G of affine line segments in Rn . Unfortunately, the most interesting hulls which arise in analysis and geometry can only rarely be fully described in this way by classical geometric objects. On the positive side, such a description is often possible if one appropriately relaxes the understanding of objects and their boundaries in the sense of geometric measure theory, i.e., as varifolds or currents (see Section 9.4).

9.1 The Role of Hulls in Analysis and Geometry

381

Let us illustrate this on the most important example in complex analysis, the O (X) of a compact set K in a complex manifold holomorphically convex hull K X. This is the hull with respect to the set of absolute values of functions in the algebra O(X) of holomorphic functions on X (1.150), which we have already encountered in numerous places in the book. On X = Cn this is the classical polynomial hull. If f ∈ O(X) then | f | ∈ Psh(X) is a plurisubharmonic function, Psh(X) of K (see Definition O (X) contains the plurisubharmonic hull K and hence K 1.5.8). It is a nontrivial result that on any Stein manifold X these two hulls coincide: Psh(X) (see L. H¨ormander [195, Theorem 4.3.4] or E. L. Stout [329, O (X) = K K Theorem 1.3.11]). A natural dual family G consists of compact complex curves in X with boundaries in K. Indeed, the restriction of a plurisubharmonic function to a complex curve is a subharmonic function (hence satisfying the maximum principle), and complex curves form the biggest family of surfaces with this property. In the special case when X is a Riemann surface, the holomorphic hull of a compact set K ⊂ X is the union of K and all its holes (see Lemma 1.12.3). This is an example of the ideal situation alluded to above when the hull is completely described by geometric objects in X with boundaries in K. On the other hand, polynomial hulls in Cn for n > 1 in general cannot be described by holomorphic curves with boundaries in the given set, not even by a repeated application of this procedure. Indeed, G. Stolzenberg [324] constructed in  \ K = ∅, such that 1963 a compact set K ⊂ C2 with nontrivial polynomial hull, K  K does not contain the image of any nonconstant holomorphic disc D → C2 . In this  cannot even be described by positive rectifiable currents T with example, the hull K supp dT ⊂ K; see the discussion following Problem 9.5.3. The first successful description of the polynomial hull by analytic objects was given by E. M. Poletsky [291, 292] and S. Q. Bu and W. Schachermayer [74] in the early 1990s. They showed that a point z ∈ Cn belongs to the polynomial hull of a compact set K ⊂ Cn if and only if there is a uniformly bounded sequence of holomorphic discs f j : D → Cn ( j ∈ N) centred at f j (0) = z whose boundaries converge to K in measure, meaning that for any ε > 0 and open neighbourhood U ⊃ K, all of the boundary f j (bD) except for a set of measure < ε lies in U for every big enough j (see (9.2)). A sequence of this type is called a Poletsky sequence. The nontrivial part is the only if statement; the if part is a simple consequence of the maximum principle for bounded holomorphic functions. This shows that analytic discs with smooth boundaries in Cn constitute a suitable family G in our problem, provided that we consider sequences of discs and relax the boundary condition in a probabilistic way. This result was generalized to the plurisubharmonic hull of a compact set in any complex manifold by J.-P. Rosay [302], and more generally in any irreducible and locally irreducible complex space by B. Drinovec Drnovˇsek and F. Forstneriˇc [116]. It is interesting to note that even for a connected compact set K ⊂ Cn , the theorem fails in general if one insists that the boundaries f j (bD) converge to K uniformly as j → ∞. An explicit example was given by E. Porten [293], answering a question raised in [118] where a positive result of this type was obtained for connected compact circular sets in Cn .

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9 Minimal Hulls, Null Hulls, and Currents

Another description of polynomial hulls by analytic objects was given by  of a J. Duval and N. Sibony [119] in 1995. They showed that the polynomial hull K n compact set K ⊂ C is the union of supports of positive currents T of bidimension (1, 1) on Cn such that dd c T ≤ 0 on Cn \ K. They found such currents T as limits of currents of the form g[C] where C is a compact complex curve with boundary in Cn , [C] is the current of integration on C (see (9.19)), and g is a Green function on C with finitely many poles. For any such T we have that dd c T = μ − δz where μ is a probability measure on K (a Jensen measure for the point z) and δz is the point  furthermore, any Jensen measure for z arises in this way (see [119, mass at z ∈ K; Theorem 4.7]). In 2017, N. Sibony [317] extended this to the description of hulls by directed currents in a large class of Pfaffian systems; see Remark 9.7.11. A link between these two characterizations of the polynomial hull was discovered by E. F. Wold [347] in 2011. He showed that for any Poletsky sequence of discs f j : D → Cn , the images T j = ( f j )∗ G of the Green current G on the unit disc (see Section 9.6) form a sequence of positive (1, 1)-currents with uniformly bounded masses on Cn , and hence a subsequence converges to a Duval–Sibony type current. This gives a description of the polynomial hull by currents of a rather special type. M , We are primarily interested in the minimally convex hull, or minimal hull, K n of a compact set K ⊂ R for n ≥ 3; see (8.10). This is the hull with respect to the family MPsh(Rn ) of minimal plurisubharmonic (also called 2-plurisubharmonic) M,D is defined for any compact set K in a functions on Rn . The analogous hull K n domain D ⊂ R ; it is compact when D is minimally convex (see Definition 8.1.9 and Proposition 8.1.10). In R3 , minimally convex domains coincide with meanconvex domains (see Remark 8.1.17), and the minimal hull is the smallest compact mean-convex barrier containing K, also called the mean-convex hull of K. The main technique used in the literature for finding the mean-convex hull of a given compact set K is the mean curvature flow of hypersurfaces, with K as an obstacle, introduced by K. Brakke [71]. For a discussion of this subject we refer to the monographs by G. Bellettini [60] and T. H. Colding and W. P. Minicozzi [94]. Here we give a description of a different kind. It follows from Proposition 8.1.2 (d) and the maximum principle for subharmonic functions that for any conformal minimal immersion f : M → Rn from a compact bordered Riemann surface we have M . f (bM) ⊂ K =⇒ f (M) ⊂ K In light of the earlier discussion, the main question is whether the converse holds: M Problem 9.1.1. Let K be a compact set in Rn for some n ≥ 3. Is every point p ∈ K contained in a compact minimal surface S ⊂ Rn (possibly with singularities, i.e., a minimal rectifiable varifold) with bS ⊂ K? N of a compact set in The analogous question may be asked for the null hull K Cn (n ≥ 3) with respect to the family of compact bordered holomorphic null curves in Cn (see Definition 9.2.2). In light of Stolzenberg’s example [324] mentioned above, we suspect that the answers to these questions are negative in general. See also the related

9.2 Minimal Hulls and Null Hulls

383

Problem 9.5.3. On the other hand, we prove the analogues of the Poletsky–Bu– Schachermayer theorem for minimal hulls and null hulls; see Theorems 9.2.1 and 9.2.6. In Section 9.7, these results are used to characterize the respective hulls by Green currents, in the spirit of the Duval–Sibony–Wold characterization of polynomial hulls.

9.2 Minimal Hulls and Null Hulls In this section we characterize the minimal hull of a compact set K in Rn for any n ≥ 3 (see Definition 8.1.9) by uniformly bounded sequences of conformal minimal discs whose boundaries converge to K in measure; see Theorem 9.2.1. The analogous result for the null hull of a compact set in Cn is given by Theorem 9.2.6. Let |E| denote the Lebesgue measure of a measurable subset E ⊂ R. The following result was proved in [117, Theorem 1.3] in dimension n = 3, and in [15, Corollary 5.6] for any dimension n ≥ 3. Theorem 9.2.1. Let K be a compact set in Rn for some n ≥ 3, and let B ⊂ Rn be a bounded open convex set containing K. A point p ∈ B belongs to the minimal hull M of K if and only if there exists a sequence of conformal minimal discs f j : D → B K such that for all j = 1, 2, . . . we have that   f j (0) = p and {t ∈ [0, 2π] : dist( f j (eit ), K) < 1/ j} ≥ 2π − 1/ j. (9.2) M,D of a compact set K in The analogous description holds for the minimal hull K any minimally convex domain D ⊂ Rn (see Def. 8.1.9) by sequences of conformal M,D ⊂ B. minimal discs contained in an open relatively compact subset B  D with K Before proceeding to the proof, we describe the analogous result for the null hull of a compact set K ⊂ Cn (n ≥ 3); this is the hull (9.1) with respect to the family F = NPsh(Cn ) of null plurisubharmonic functions on Cn . Explicitly: Definition 9.2.2. Let K be a compact set in Cn (n ≥ 3). The null hull of K is the set  N = z ∈ Cn : v(z) ≤ max v for all v ∈ NPsh(Cn )}. (9.3) K K

Since the restriction of a null plurisubharmonic function on Cn to a holomorphic null curve A ⊂ Cn is a subharmonic function on A, the maximum principle for subharmonic functions implies that for any bounded holomorphic null curve A ⊂ Cn N . Since every affine linear function on with boundary bA ⊂ K we have A ⊂ K Cn is plurisubharmonic (indeed, pluriharmonic) and we obviously have Psh(Cn ) ⊂ NPsh(Cn ), the observation (c) on p. 380 shows that  ⊂ Co(K). N ⊂ K K⊂K

(9.4)

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9 Minimal Hulls, Null Hulls, and Currents

 only rarely equals the convex hull Co(K). We now give a The polynomial hull K N = K.  simple example of a compact set K ⊂ C3 with K  = {(0, 0, ζ ) : |ζ | ≤ 1}. HowExample 9.2.3. Let K = {(0, 0, eit ) : t ∈ R}. Clearly, K 2 2 ever, since the function u(z1 , z2 , z3 ) = |z1 | + |z2 | − |z3 |2 is null plurisubharmonic (cf. Example 8.2.4) and it equals −|z3 |2 on the coordinate axis {(0, 0)} × C, we see N = K. (See Theorem 9.3.1 for a more general result.)   that K  N ⊂ K The following lemma is an immediate consequence of the inclusion K  (9.4), the standard fact that K ⊂ Ω for any compact set K in a pseudoconvex Runge domain Ω ⊂ Cn (see [195, Theorem 2.7.3]), and Proposition 8.2.7 (iv) showing that the restriction map NPsh(Cn ) → NPsh(Ω ) on such Ω has dense image. Lemma 9.2.4. Let Ω be a pseudoconvex Runge domain in Cn . For any compact set N ⊂ Ω and K ⊂ Ω we have that K N = K N,Ω = {z ∈ Ω : v(z) ≤ max v for all v ∈ NPsh(Ω )}. K K

(9.5)

The next proposition is proved as in the classical case of plurisubharmonic functions (see [195, Theorem 2.6.11, p. 48] or [329, Theorem 1.3.8, p. 25]). N ⊂ Cn (n ≥ 3), there Proposition 9.2.5. Given a compact null convex set K = K is a smooth null plurisubharmonic exhaustion function ρ : Cn → R+ such that {ρ = 0} = K and ρ is strongly null plurisubharmonic on Cn \ K. The following result is an analogue of Theorem 9.2.1 for null hulls. The original source is [117, Theorem 1.4] for n = 3 and [15, Corollary 5.5] for n > 3. Theorem 9.2.6. Let K be a compact set in a pseudoconvex Runge domain Ω ⊂ Cn , N (9.3) if and only if there is a n ≥ 3. A point z ∈ Ω belongs to the null hull K sequence of holomorphic null discs Fj : D → Ω whose images are contained in a relatively compact subset of Ω such that for all j = 1, 2, . . . we have that   Fj (0) = z and {t ∈ [0, 2π] : dist(Fj (eit ), K) < 1/ j} ≥ 2π − 1/ j. (9.6) N , is there Problem 9.2.7. If K ⊂ Cn (n ≥ 3) is a compact connected set and z ∈ K a uniformly bounded sequence of holomorphic null discs Fj : D → Cn satisfying Fj (0) = z and max{dist(Fj (ζ ), K) : |ζ | = 1} < 1/ j for all j ∈ N? In the case of polynomial hulls, such a sequence does not exist in general, as shown by E. Porten [293] (see the discussion in the previous section). Remark 9.2.8. There is a simple relationship between the minimal hull and the null hull. Let π : Cn → Rn denote the projection π(x + iy) = x. By Lemma 8.2.3, a minimal plurisubharmonic function u on Rn lifts to a null plurisubharmonic function u ◦ π on Cn . This implies that for any compact set L ⊂ Cn we have that   2 N ⊂ π(L) . π L M

(9.7)

9.2 Minimal Hulls and Null Hulls

385

The inclusion may be strict: take L ⊂ C3 to be a smooth embedded Jordan curve whose projection K = π(L) ⊂ R3 is also a smooth Jordan curve. Then, K bounds a M . However, if for some point minimal surface M which is therefore contained in K p ∈ L the tangent line Tp L does not belong to the null quadric A (2.54), then by N = L. See Corollary 9.7.8 for a more precise result.   Corollary 9.3.3 we have L Theorems 9.2.1 and 9.2.6, and the analogous result for the polynomial hull, are corollaries to a certain disc formula for the respective classes of functions which we now describe. Assume that ω is a domain in Rn or Cn . Let F (ω) denote one of the spaces Psh(ω), NPsh(ω), or MPsh(ω); the latter two denote the spaces of null plurisubharmonic and minimal plurisubharmonic functions on ω, respectively (see Definitions 8.2.2 and 8.1.1). Let D(ω) denote the dual set of discs f : D → ω: • holomorphic discs if ω ⊂ Cn and F (ω) = Psh(ω), • holomorphic null discs if ω ⊂ Cn (n ≥ 3) and F (ω) = NPsh(ω), and • conformal minimal discs if ω ⊂ Rn (n ≥ 3) and F (ω) = MPsh(ω). Let D0 (ω) be the set of linear discs in D(ω), i.e. f (ζ ) = a + ζ θ (ζ ∈ D, a, θ ∈ Cn ) in the first case, with θ ∈ A∗ in the second case, and f = ℜg with g a holomorphic null disc in the third case. Given a point z ∈ ω we set D(ω, z) = { f ∈ D(ω) : f (0) = z},

D0 (ω, z) = { f ∈ D0 (ω) : f (0) = z}.

We have that D0 (ω, z) = D(ω, z) ∩ D0 (ω). Note that an upper semicontinuous function u : ω → R ∪ {−∞} belongs to F (ω) if and only if it satisfies the following submeanvalue property: u(z) ≤

 2π

u( f (eit ))

0

dt 2π

for all z ∈ ω and f ∈ D0 (ω, z).

(9.8)

If u ∈ F (ω) then the same property holds for all discs f ∈ D(ω, z). Theorem 9.2.9. Let ω, F (ω) and D(ω) be as above. If φ : ω → R ∪ {−∞} is an upper semicontinuous function then the function u : ω → R ∪ {−∞}, defined by  2π

u(z) =

inf

f ∈D(ω,z)

0

φ ( f (eit ))

dt , 2π

z ∈ ω,

(9.9)

belongs to F (ω) or is identically −∞; moreover, u is the supremum of functions in F (ω) which are bounded above by φ . Remark 9.2.10. The disc functional Pφ in (9.9), which assigns to any disc f in one of the above classes the average f −→ Pφ ( f ) =

 2π 0

φ ( f (eit ))

dt ∈ R ∪ {−∞}, 2π

of the values of φ on the boundary circle f (T), is called the Poisson functional associated to φ . In the complex case, Theorem 9.2.9 says that the infimum of the

386

9 Minimal Hulls, Null Hulls, and Currents

Poisson functional over all holomorphic discs in a domain ω ⊂ Cn with a given centre z ∈ ω equals the maximum at z of all plurisubharmonic functions on ω which are bounded above by φ . This fundamental result of E. A. Poletsky [291, 292] and S. Q. Bu and W. Schachermayer [74] was generalized by J.-P. Rosay [302, 301] to all complex manifolds (see also F. L´arusson and R. Sigurdsson [214, 215]), and by Drinovec Drnovˇsek and Forstneriˇc [116, Theorem 1.1] to all irreducible and locally irreducible complex spaces.   The disc formula in Theorem 9.2.9 is the main ingredient in the proof of all results on hulls mentioned above. Before proving it, let us show how it implies Theorem 9.2.1 and the analogous results for the polynomial hull and the null hull. Proof of Theorems 9.2.1 and 9.2.6. We first consider Theorem 9.2.1. Since the set M ⊂ Co(K) ⊂ B. Assume that for B ⊂ Rn is convex and contains K, we have that K some point p ∈ B there exists a sequence of conformal minimal discs f j : D → B with f j (0) = p satisfying (9.2). Let U j = {x ∈ Rn : dist(x, K) < 1/ j},

E j = {t ∈ [0, 2π] : f j (eit ) ∈ / U j }.

(9.10)

Then |E j | ≤ 1/ j by (9.2). Pick u ∈ MPsh(Rn ) and set M j = supU j u and M0 = supB u. Note that lim j→∞ M j = supK u as u is upper semicontinuous. Since u ◦ f j is subharmonic by Corollary 8.1.7, we have that u(p) = u( f j (0)) ≤



  dt    dt M0 u f j (eit ) u f j (eit ) + ≤ + Mj. 2π 2π j Ej [0,2π]\E j

M . Passing to the limit as j → ∞ gives u(p) ≤ supK u which shows that p ∈ K n To prove the converse, let U j ⊂ R given by (9.10). The function φ : Rn → R, which equals −1 on U j and 0 on Rn \ U j , is upper semicontinuous. Let u ∈ MPsh(Rn ) be the associated minimal plurisubharmonic function defined by (9.9). M . Then, −1 ≤ u ≤ 0 on Rn and u = −1 on K. Hence, u(p) = −1 for any point p ∈ K Fix such a point p. Theorem 9.2.9 furnishes a sequence of conformal minimal discs f j : D → B such that for all j = 1, 2, . . . we have f j (0) = p and  2π  0

 dt 1 < −1 + . 2π 2π j

φ f j (eit )

By the definition of φ this implies |{t ∈ [0, 2π] : f j (eit ) ∈ U j }| ≥ 2π − 1/ j, so (9.2) holds. This completes the proof. It is obvious that the same argument applies when Rn is replaced by a minimally convex domain D ⊂ Rn and B is replaced by an open M,D . relatively compact subset B  D containing K The same argument shows that Theorem 9.2.9, applied with F (ω) = NPsh(ω) and D(ω) the space of holomorphic null discs D → ω ⊂ Cn , implies Theorem 9.2.6. Likewise, the theorem of Poletsky [291, 292] and Bu and Schachermayer [74] is obtained by applying the same argument with F (ω) = Psh(ω) and D(ω) the space of holomorphic discs D → ω.  

9.2 Minimal Hulls and Null Hulls

387

In the proof of Theorem 9.2.9 we shall need two lemmas. Lemma 9.2.11. Let D0 (ω) ⊂ D(ω) be any of the classes of discs in Theorem 9.2.9. Let v : ω → R ∪ {−∞} be upper semicontinuous and define u : ω → R ∪ {−∞} by u(z) = T (v)(z) :=

inf g∈D0 (ω,z)

 2π 

 dt , 2π

v g(eit )

0

z ∈ ω.

(9.11)

Then u is upper semicontinuous. Furthermore, given a disc f ∈ D(ω) and a number ε > 0, there is a disc f˜ ∈ D(ω) with f˜(0) = f (0) such that  2π  0

 2π   dt  dt < + ε. v f˜(eit ) u f (eit ) 2π 2π 0

(9.12)

Proof. Assume first that ω ⊂ Cn and we are considering (null) holomorphic discs; the proof for conformal minimal discs will follow immediately as indicated below. To show that the function u given by (9.11) is upper semicontinuous, pick a sequence zk ∈ ω converging to z0 ∈ ω and let θ ∈ Cn (or θ ∈ A∗ ) be such that z0 + Dθ = {z0 + ζ θ : |ζ | ≤ 1} ⊂ ω. For big k0 ∈ N the set L = ∞ k=k0 (zk + Dθ ) is relatively compact in ω. Then, v is bounded above on L and satisfies v(z0 + ζ θ ) ≥ lim sup v(zk + ζ θ ) for all ζ ∈ D. k→∞

Fatou’s lemma implies that  2π 

v z0 + eit θ

0

 2π   dt  dt ≥ lim sup ≥ lim sup u(zk ). v zk + eit θ 2π 2π 0 k→∞ k→∞

By (9.11), the infimum of the left-hand side over all vectors θ equals u(z0 ), so we obtain u(z0 ) ≥ lim supk→∞ u(zk ). Therefore, u is upper semicontinuous. It remains to prove (9.12). Since u is upper semicontinuous, there is a continuous function u˜ on ω such that u ≤ u˜ and  2π 

 2π   dt  dt ε < + . u f (eit ) 2π 2π 4 0

u˜ f (eit )

0

(9.13)

Fix a point eis0 ∈ T. By the definition of u (9.11) there is a linear (null) holomorphic disc D ζ → f (eis0 ) + ζ θs0 ∈ ω such that  2π 

v f (eis0 ) + eit θs0

0

 dt ε ε ≤ u( f (eis0 )) + ≤ u( ˜ f (eis0 )) + . 2π 5 5

Since v is upper semicontinuous and u˜ is continuous, there is an arc I  ⊂ T around the point eis0 such that for every arc I ⊂ I  we have that  2π 0

   ds dt   ds |I| ε ≤ u˜ f (eis ) + . v f (eis ) + eit θs0 2π 2π 2π 2π 4 s∈I I

(9.14)

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9 Minimal Hulls, Null Hulls, and Currents

Repeating this construction at other points of the circle T, we find finitely many triples of arcs eisi ∈ Ii ⊂ Ii ⊂ Ii ⊂ T and (null) vectors θi (i = 1, . . . , l) such that

• li=1 Ii = T, • the set li=1 ζ ∈Ii ( f (ζ ) + Dθi ) is compact and contained in ω, 



• I i ∩ I k = ∅ if i = k, • the estimate (9.14) holds on I = Ii for every i = 1, . . . , l, and • the set E = T \ li=1 Ii has arbitrarily small measure (to be specified later). Choose a smooth function χ : T → [0, 1] with support in i=1 Ii . Let g : T × D → ω be defined by

l

g(ζ , ξ ) = f (ζ ) + χ(ζ )ξ θi ,

l

 i=1 Ii

such that χ ≡ 1 on

ζ ∈ Ii , ξ ∈ D, i = 1, . . . , l.

(Note that g is well defined by the choice of χ.) If |E| is small enough then, since v is upper semicontinuous and u˜ is continuous, it follows that  2π  0

 2π   ds dt  dt ε  2π  it  dt 3ε u˜ f (eit ) u f (e ) ≤ + ≤ + , 2π 2π 2π 2 2π 4 0 0

v g(eis , eit )

where the last inequality holds in view of (9.13). We have used the estimate (9.14) 1, . . . , l, together with the fact that the error caused by the on I = Ii for every i = integral over E = T \ li=1 Ii is as small as desired provided |E| is small enough. By the approximate solution to the Riemann–Hilbert problem for the classes of discs D0 (ω) ⊂ D(ω) (see Lemma 6.1.1 and (6.4) for the holomorphic case and Lemma 6.2.1 for the holomorphic null case) there exists an f˜ ∈ D(B, p) satisfying  2π  0

 2π   dt  dt ≤ + ε. v f˜(eit ) u f (eit ) 2π 2π 0

This completes the proof for holomorphic discs and holomorphic null discs. The case of conformal minimal discs follows immediately by noting that every conformal minimal disc D → ω ⊂ Rn is the real part of a holomorphic null disc D → Tω = ω + iRn , and vice versa (see Theorem 2.3.4). Furthermore, linear conformal minimal discs in ω are real parts of linear holomorphic null discs in Tω . Hence, it suffices to extend the function v : ω → R ∪ {−∞} to the function v : ω × iRn → R ∪ {−∞} which is independent of the imaginary component (this extension is null plurisubharmonic by Lemma 8.2.3), apply the already proven result for holomorphic null discs in the tube Tω , and pass down to ω to obtain the desired result for conformal minimal discs in ω.   The next lemma is essentially [74, Proposition II.1] and [117, Proposition 2.9]. Lemma 9.2.12. Let ω, F (ω), and D0 (ω) ⊂ D(ω) be as above, and let v → T (v) denote the transformation defined by (9.11). Assume that φ : ω → R ∪ {−∞} is an upper semicontinuous function. Set u1 = φ and u j = T (u j−1 ) for j = 2, 3, . . .. Then the functions u j are upper semicontinuous and decrease pointwise to the largest function uφ ∈ F (ω) bounded above by φ .

9.2 Minimal Hulls and Null Hulls

389

Proof. We consider the case when ω ⊂ Cn and F (ω) is the space of plurisubharmonic or null plurisubharmonic functions. The remaining case when ω ⊂ Rn and F (ω) = MPsh(ω) follows from the null plurisubharmonic case by the argument given at the end of the proof of Lemma 9.2.11. By Lemma 9.2.11 the functions u j are upper semicontinuous, and since the sequence u j is decreasing, the limit function uφ is also upper semicontinuous. Pick z ∈ ω and θ ∈ Cn (resp. θ ∈ A∗ if F (ω) = NPsh(ω)) such that z + D θ ⊂ ω. By the monotone convergence theorem we have that  2π

uφ (z) = lim u j (z) ≤ lim j→∞

j→∞ 0

 2π    dt  dt = . u j−1 z + eit θ uφ z + eit θ 2π 2π 0

This shows that uφ ∈ F (ω). It remains to prove that uφ is the biggest function in F (ω) dominated by φ . Choose v ∈ F (ω) with v ≤ φ . It suffices to show that v ≤ u j for every j ∈ N. Suppose that v ≤ u j for some j ∈ N; this holds for j = 1 since u1 = φ . For every point z ∈ ω and vector θ ∈ Cn (resp. θ ∈ A∗ ) such that z + D θ ⊂ ω we then have v(z) ≤

 2π 

v z + eit θ

0

 2π   dt  dt ≤ . u j z + eit θ 2π 2π 0

Taking the infimum over all vectors θ we get v(z) ≤ u j+1 (z) by the definition of  u j+1 . This concludes the induction step and hence completes the proof.  Proof of Theorem 9.2.9. We consider the case when ω is a domain in Cn and F (ω) is the space of (null) plurisubharmonic functions. The remaining case when ω ⊂ Rn and F (ω) = MPsh(ω) follows from the null plurisubharmonic case by the argument given at the end of the proof of Lemma 9.2.11. Lemma 9.2.12 furnishes a decreasing sequence of upper semicontinuous functions u j on ω, with u1 = φ , converging pointwise to the largest function uφ ∈ F (ω) with uφ ≤ φ . It remains to show that uφ is given by the formula (9.9): uφ (z) =

inf

 2π 

f ∈D(ω,z) 0

 dt , 2π

φ f (eit )

z ∈ ω.

Denote the right-hand side of the above equation by u(z). ˜ Since uφ ∈ F (ω) and uφ ≤ φ , the following holds for any f ∈ D(ω, z): uφ (z) ≤

 2π 0

 2π    dt  dt uφ f (eit ) φ f (eit ) ≤ . 2π 2π 0

Taking the infimum over all f ∈ D(ω, z) gives uφ ≤ u˜ on ω. To prove the converse inequality, fix a point z ∈ ω and a number ε > 0. Since the sequence u j (z) decreases to uφ (z) as n → ∞, there is a j0 ∈ N such that uφ (z) ≤ u j0 (z) < uφ (z) + ε/2.

(9.15)

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9 Minimal Hulls, Null Hulls, and Currents

By the definition of u j0 −1 there exists a (linear) disc f j0 −1 ∈ D(ω, z) such that  2π 0

  dt ε < u j0 (z) + u j0 −1 f j0 −1 (eit ) . 2π 2 j0

Applying (9.12) with v = u j0 −1 , u = u j0 −2 , and ε replaced by ε/2 j0 gives a disc f j0 −2 ∈ D(ω, z) such that  2π 0

 2π    dt  dt ε < + u j0 −2 f j0 −2 (eit ) u j0 −1 f j0 −1 (eit ) . 2π 2π 2 j0 0

Continuing by a downward induction on j and applying (9.12) at every step we obtain a disc f = f1 ∈ D(ω, z) such that  2π  0

 dt < u j0 (z) + ε/2 < uφ (z) + ε, 2π

φ f (eit )

where we used that u1 = φ . (The last inequality holds by (9.15).) This shows that ˜ ≤ uφ (z), and hence u(z) ˜ = uφ (z).   u(z) ˜ < uφ (z) + ε. Letting ε → 0 gives u(z) Remark 9.2.13. An inspection of the proof shows that Theorem 9.2.9 holds for any class F of regular upper semicontinuous functions on a real or complex manifold ω, and classes of discs D0 (ω) ⊂ D(ω), satisfying the following conditions: • For every u ∈ F and f ∈ D(ω) the function u ◦ f is subharmonic on D. • If u : ω → R ∪ {−∞} is an upper semicontinuous function such that u(z) ≤

 2π 0

u( f (eit ))

dt 2π

for all z ∈ ω and f ∈ D0 (ω, z),

(cf. (9.8)), then u ∈ F . • The Riemann–Hilbert problem associated to any central disc f ∈ D(ω, z) and a continuous family of boundary discs gζ ∈ D0 (ω, f (ζ )) (ζ ∈ T) has an approximate solution f˜ ∈ D(ω, z). (This condition can often be weakened by considering more special families gζ as was the case in our proof.) F is also compact, If these conditions hold and K is a compact set in ω whose hull K F by using sequences of discs in D(ω) whose then Theorem 9.2.1 applies to K F .   images are contained in an open relatively compact subset of ω containing K There seem to be rather few known examples of minimal hulls described by minimal surfaces with boundaries in the given compact set K. The simplest one is when K is a closed Jordan curve in R3 whose projection to R2 bounds a convex domain D ⊂ R2 . In this case, the Plateau problem for K admits a unique solution which is a graph over D with boundary equal to K, and this graph is the minimal hull of K. The classical Dirichlet problem for the minimal graph equation, consisting of finding a solution of (2.100) in a bounded convex domain of R2 and taking on assigned continuous values on its boundary, was solved by T. Rad´o [295] in 1930; H. Jenkins and J. Serrin gave an alternative proof in the paper [200] from 1966.

9.3 Null Hulls of Curves

391

Here is a more general example in this direction. Example 9.2.14. Let D be a compact convex domain in R2 and let a, b : bD → R be continuous functions with a ≤ b. Consider the cylinder over bD given by   K = (x1 , x2 , x3 ) ∈ R3 : (x1 , x2 ) ∈ bD, a(x1 , x2 ) ≤ x3 ≤ b(x1 , x2 ) . (9.16) Let Γa ,Γb ⊂ R3 denote the unique minimal graphs over D with boundary values determined by a and b, respectively (see the references above). The maximum principle for minimal graphs (see Theorem 2.7.1) implies that Γa lies below Γb . M of the cylinder K (9.16) is the compact Proposition 9.2.15. The minimal hull K region L ⊂ D × R between Γa and Γb . In particular, a minimal graph in R3 over a compact convex domain in R2 is minimally convex. Proof. Clearly, L is filled by minimal graphs over D whose boundary values are M . It remains to see continuous functions c : bD → R with a ≤ c ≤ b, so L ⊂ K M . We may assume by approximation that bD, a and b are smooth, so that L = K the graphs Γa and Γb are smooth over D. Choose a pair of constants c1 , c2 ∈ R with c1 < a ≤ b < c2 on bD. Since the minimal hull is contained in the convex hull and D M ⊂ D×[c1 , c2 ]. For each t ∈ R we let Mt = Γb +(0, 0,t), and is convex, we see that K we define a smooth function u : D × R → R by u|Mt ≡ h(t) for each t ∈ R, where h is a strictly increasing and strictly convex function with h(0) = 0. A calculation using the minimal graph equation (2.100) shows that u is minimal plurisubharmonic on D × [c1 , c2 ] provided that the second derivative of h is sufficiently large. Since u ≤ 0 M . The analogous argument on K, we infer that no point in D × R above Γb lies in K M .  with Γa shows that no point in D × R below Γa belongs to K  It is well known (see G. Stolzenberg [325], H. Alexander [47], and E. L. Stout [329]) that every rectifiable embedded arc in Cn is polynomially convex. Problem 9.2.16. Is every smooth embedded arc in Rn (n ≥ 3) minimally convex?

9.3 Null Hulls of Curves The problem of determining the polynomial hull of a given compact set K in Cn for n > 1 is very difficult and in general impossible; we refer to E. L. Stout [329] for a survey of this subject. However, the situation is well understood when K is a union of curves or, more generally, a subset of a compact connected set of finite linear  \ K is either empty or a closed complex analytic measure: in such case, the set K n subvariety of C \ K with boundary in K. If Γ is a closed Jordan curve of finite linear measure, its polynomial hull is a nontrivial complex curve with boundary Γ if and only if Γ α = 0 for every entire holomorphic 1-form α on Cn . After the seminal work of J. Wermer in 1958 [342], major results on this topic were proved by G. Stolzenberg [326, 327], H. Alexander [47], and others.

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9 Minimal Hulls, Null Hulls, and Currents

As a consequence of these classical results, we have the following description of null hulls of rectifiable curves which was obtained in [117, Theorem 3.7]. Recall N denotes the null hull of a compact set K. that K Theorem 9.3.1. If Γ is a compact set in Cn (n ≥ 3) which is contained in a connected compact set of finite linear measure, then ΓN \ Γ is either empty or a (possibly singular) holomorphic null curve with boundary in Γ . Proof. Let Γ denote the polynomial hull of Γ . According to H. Alexander [47], the set V = Γ \ Γ is either empty or a closed bounded one-dimensional complex subvariety of Cn \ Γ with V \ V ⊂ Γ . Assume that the latter condition holds since otherwise there is nothing to prove. Let V = j V j be a decomposition of V into its irreducible components. Denote by A the union of all components V j which are null curves. Then, A is a bounded complex null curve, possibly reducible and with singularities, and A \ A ⊂ Γ . Since the restriction of every null plurisubharmonic function on Cn to A is a subharmonic function on A (see Proposition 8.2.7 (iii)), the maximum principle for subharmonic functions implies that A ⊂ ΓN . It remains to prove that ΓN = Γ ∪ A. To this end, we must show that for any point p ∈ V \ A there exists a function φ ∈ NPsh(Cn ) such that φ (p) > maxΓ φ . Let B denote the union of all irreducible component Vi of V which are not contained in A (i.e., which are not null curves). Then, B is a bounded 1-dimensional complex subvariety of Cn \ Γ and B \ B ⊂ Γ . Let C(B) denote the union of the singular locus Bsing and the set of points z ∈ Breg such that the tangent line Tz B ⊂ A is a null line. Then, C(B) is a closed discrete subset of B which clusters only on Γ . (To see that C(B) cannot cluster on a singular point z ∈ Bsing , choose a local irreducible component B j of B at z and parameterize Bi locally near z by a nonconstant holomorphic map f : D → Bi with f (0) = z. Clearly, the set {ζ ∈ D : f  (ζ ) ∈ A} is either all of D or discrete in D; the first case is impossible by the definition of B.) It follows that the set K := Γ ∪ A ∪C(B) ⊂ Γ (9.17) is compact. Fix a point p ∈ V \ A. Then p ∈ B \ A, and either p ∈ / K or p is an isolated point of K. Choose a pair of bounded open subsets U0 ,U1 ⊂ Cn such that K \ {p} ⊂ U0 ,

p ∈ U1 ,

and U 0 ∩U 1 = ∅.

Pick a smooth function h : Cn → [0, 1] such that h = 0 on U0 (in particular, h = 0 on Γ ) and h = 1 on U1 . Choose ε > 0 small enough such that the function ˜ = h(z) + ε|z − p|2 , h(z)

z ∈ Cn

satisfies maxΓ h˜ < 1. Note that h˜ is strongly plurisubharmonic on U := U0 ∪ U1 (since h is locally constant there and z → |z − p|2 is strongly plurisubharmonic on ˜ = 1. Pick a smooth function χ : Cn → [0, 1] which vanishes on an Cn ) and h(p) open neighbourhood of K (9.17) and is positive on Cn \ U. Since V is a closed complex curve in Cn \ Γ , there is a plurisubharmonic function ψ ≥ 0 on an open

9.3 Null Hulls of Curves

393

neighbourhood of V in Cn that vanishes to the second order on V and satisfies Lψ (z; θ ) > 0 for all z ∈ Vreg and θ ∈ Tz Cn \ TzV.

(9.18)

A function with these properties can be found of the form ψ = ∑k | fk |2 , where { fk } are holomorphic defining functions for V on a Stein neighbourhood of V in Cn . We claim that for C > 0 chosen big enough the function v = h˜ +Cχψ is null plurisubharmonic on an open neighbourhood of Γ ∪V = Γ in Cn . (Although ψ is only defined near V , the multiplier χ vanishes near Γ , and hence we can extend the product χψ as zero on a neighbourhood of Γ .) To see this, note first that v = h˜ on an open neighbourhood D  U = U0 ∪ U1 of K on which χ vanishes, so v is strongly plurisubharmonic there. Suppose now that z ∈ V \ D. By the definition of K it follows that z is a regular point of the complex one-dimensional subvariety B ⊂ V of Cn \ Γ and the tangent line TzV is not a null line. It follows from (9.18) that Lψ (z; θ ) > 0 for every θ ∈ A∗ . Since ψ vanishes to the second order on V , we also have that Lχψ (z; θ ) = χ(z)Lψ (z; θ ) ≥ 0. If z ∈ U then the above estimate and the fact that h˜ is strongly plurisubharmonic on U give Lv (z; θ ) = Lh˜ (z; θ ) + Lχψ (z; θ ) ≥ Lh˜ (z; θ ) > 0. On the other hand, on the compact set B \ U we have that χ > 0, so we can ensure by choosing the constant C > 0 big enough that Lv (z; θ ) > 0 for every z ∈ B \U and θ ∈ A∗ , thereby proving the claim. ˜ Note that v(p) = h(p) = 1 and maxΓ v = maxΓ h˜ < 1. Proposition 8.2.7 (iv) furnishes a null plurisubharmonic function φ ∈ NPsh(Cn ) which agrees with v near Γ. Then, 1 = φ (p) > maxΓ φ and hence p ∈ / ΓN . This completes the proof.   In the course of proof of Theorem 9.3.1 we have actually shown the following. Lemma 9.3.2. Let V be a smooth locally closed complex curve in Cn (n ≥ 3) whose tangent line TzV is not a null line for any z ∈ V . Then, for every h ∈ C 2 (V ) there exists an open neighbourhood Ω ⊂ Cn of V and a strongly null plurisubharmonic function v on Ω such that v|V = h. Theorem 9.3.1 also implies the following corollary. Corollary 9.3.3. Assume that Γ is a rectifiable Jordan curve in Cn (n ≥ 3) which contains an embedded arc Λ of class C 1 . If there is a point p ∈ Λ such that the tangent line TpΛ does not belong to the null quadric A, then ΓN = Γ . Proof. By a theorem of H. Alexander [47], either Γ is polynomially convex (in which case ΓN = Γ by (9.4)) or else V = Γ \ Γ is a connected complex curve with boundary bV = Γ . Since the arc Λ ⊂ bV is of class C 1 , at most points of Λ the union V ∪Λ is a local C 1 surface with boundary (see E. M. Chirka [88, Sect. 19.1]).

394

9 Minimal Hulls, Null Hulls, and Currents

If TpΛ does not belong to A for some p ∈ Λ , there is a nearby point q ∈ Λ such that TqΛ does not belong to A either and V ∪Λ is a local C 1 surface with boundary at q. Hence, V is not a null curve and the conclusion follows from Theorem 9.3.1.  

9.4 Rectifiable Sets, Varifolds, and Currents The theory of minimal submanifolds can be, and often must be treated in the more general setting of varifolds and currents. The basic reason for this is lack of compactness for ordinary submanifolds of a Riemannian manifold when applying variational methods, as these methods tend to introduce singularities. Therefore, one is led to consider generalized submanifolds, especially in the study of boundary value problems such as the Plateau problem. Regularity results and structure theorems for minimal varifolds and currents play a major role in this theory. Varifolds and currents are studied in the framework of geometric measure theory. Fundamental references are the books by H. Federer [124], F. Morgan [265], and L. Simon [319]; for minimal currents, see also H. B. Lawson [216]. Here we briefly discuss these notions, referring to the cited sources for the details. We use currents in the following two sections when describing the minimal hull of a compact set in Rn by Green currents. The basic generalization of an m-dimensional submanifold in Rn is an mdimensional rectifiable set. Let H m denote the m-dimensional Hausdorff measure on Rn . (For m = n, H n agrees with the Lebesgue measure L n on Rn .) A measurable subset M ⊂ Rn is called an m-dimensional rectifiable set, or m-rectifiable for short, if H m (M) < ∞ and H m almost all of M is contained in the union of the images of countably many Lipschitz maps from Rm to Rn . (See H. Federer [124, 3.2.14] or F. Morgan [265, 3.10].) In view of Rademacher’s theorem (see [124, 3.1.6]) one can take these Lipschitz maps to be C 1 diffeomorphisms f j on compact domains with disjoint images whose union coincides with M almost everywhere with respect to H m , and the Lipschitz constants of f j and f j−1 can be taken near 1. Rectifiable sets are the generalized manifolds of geometric measure theory. A more general notion is that of a varifold, another measure theoretic generalization of submanifolds. Varifolds were first introduced by L. C. Young (1951) under the name generalized surfaces. F. J. Almgren slightly modified the definition in his 1965 notes and introduced the name varifold to emphasize that these objects are substitutes for manifolds in problems of the calculus of variations. The modern approach to varifolds was developed by W. K. Allard in the paper [48] from 1972. The most important case is a rectifiable m-varifold. A basic example is a rectifiable set in Rn of dimension m ≤ n with finite H m volume. A general m-dimensional rectifiable varifold is a formal linear combination ∑ j r j A j of mdimensional rectifiable sets A j with positive real coefficients r j > 0 (multiplicities) and with finite H m volume counting multiplicities. By allowing only integer coefficients we get integral varifolds. One can define the m-dimensional mass measure of a rectifiable varifold and introduce minimal varifolds in the usual way.

9.4 Rectifiable Sets, Varifolds, and Currents

395

The oriented version of this theory is provided by currents. These are linear functionals on smooth differential forms, named by analogy with electrical currents, with a sort of direction as well as magnitude at every point. Currents were introduced by de Rham [104] as a generalization of distributions, the latter being linear functionals on spaces of test functions. The main motivation comes from viewing an oriented m-dimensional rectifiable set M ⊂ Rn (in particular, a submanifold) as a functional, [M], acting on a differential form φ by integrating it over the set: 

[M](φ ) = M

φ dH m .

(9.19)

This functional is called the current of integration over M. An m-dimensional current on Rn is a continuous linear functional on the space m D (Rn ) of smooth differential m-forms of compact support with the usual test function topology from the theory of distributions. Such a current can be viewed as a differential form T = ∑ TI dxI of degree k = n − m with distribution coefficients TI , where the sum is over multiindices I = (i1 , . . . , ik ) with 1 ≤ i1 ≤ · · · ≤ ik ≤ n and dxI = dxi1 ∧ · · · ∧ dxik . The value of T on a test form φ = adx j1 ∧ · · · ∧ dx jm = adxJ equals cTI (a), where c ∈ {0, ±1} is determined by dxI ∧ dxJ = cdx1 ∧ · · · ∧ dxn . A similar definition applies on any smooth manifold; however, by embedding the manifold into Rn , it suffices to consider the latter case. Furthermore, any smooth Riemannian manifold (M, g) admits an isometric embedding into a flat Euclidean space (Rn , ds2 ) by a theorem of J. Nash [269], so it suffices to treat metric questions on submanifolds of Euclidean spaces. This is no longer true for complex manifolds, but here we shall only consider currents on Euclidean spaces. The space Dm (Rn ) of m-dimensional currents on Rn (the dual space of D m (Rn )) carries the weak-star topology, also called the weak topology, determined by the condition that a net of currents T j ∈ Dm (Rn ) converges to a current T ∈ Dm (Rn ) if and only if T j (φ ) → T (φ ) for every φ ∈ D m (Rn ). Although the natural topologies on these spaces are not locally countable, we shall see that in most cases of interest (and certainly in all cases of interest to us) it suffices to consider sequences of currents as opposed to more general nets. The support of T ∈ Dm (Rn ) is the smallest closed subset supp T of Rn such that T (φ ) = 0 for every φ ∈ D m (Rn ) with supp(φ ) ∩ supp T = ∅. The space of m-dimensional currents on Rn of compact support is denoted Em (Rn ). A current T ∈ Em (Rn ) acts on any smooth m-form on Rn by T (φ ) = T (χφ ), where χ ≥ 0 is a smooth function of compact support which equals 1 on a neighbourhood of supp T . For currents on Cn (or, more generally, on complex manifolds) we have a further notion of bidimension, corresponding to bidegree of differential forms. Denote by D p,q (Cn ) ⊂ D p+q (Cn ) the space of smooth compactly supported (p, q)-forms (see (1.62)) for some p, q ∈ Z+ , p + q ≤ 2n. Its dual D p,q (Cn ) is the space of currents of bidimension (p, q) (and bidegree (n− p, n−q)) on Cn . A closed complex submanifold M ⊂ Cn of dimension m (possibly with boundary) determines a current of integration [M] ∈ Dm,m (Cn ) by (9.19). Similarly we define the space E p,q (Cn ) of

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9 Minimal Hulls, Null Hulls, and Currents

compactly supported currents of bidimension (p, q) as the dual of E p,q (Cn ), the space of smooth (p, q)-forms on Cn . Operations on currents are dual to those on forms. In particular, the exterior derivative of an m-dimensional current T ∈ Dm (Rn ) is the (m − 1)-dimensional current dT ∈ Dm−1 (Rn ) defined by φ ∈ D m−1 (Rn ).

dT (φ ) = T (dφ ),

(9.20)

Example 9.4.1. If M ⊂ Rn is a smooth, compact, oriented submanifold with coherently oriented boundary bM then, by Stokes’s theorem (see Theorem 1.3.1), (9.21)

d[M] = [bM].

Likewise, the current of integration over a compact manifold M without boundary is a closed current, d[M] = 0. Remark 9.4.2. In some texts, the current dT (9.20) is denoted ∂ T and called the boundary of T . This is motivated by the formula (9.21). However, in general one cannot think of dT as any kind of geometric boundary. For example, if h is a smooth function on a compact manifold M, then the current T = h[M] defined by 

h[M](φ ) =

(9.22)

hφ M

(the current of integration on M with the density function h) satisfies 

d(h[M])(φ ) =



h dφ = M

M

d(hφ ) −

 M

dh ∧ φ =

 bM

hφ −

 M

dh ∧ φ ,

so its support is typically all of M.   Recall that |v| and |φ | denote the mass and comass norm of m-vectors v ∈ Λm Rn and m-covectors φ ∈ Λm (Rn )∗ , respectively (see (1.5)). The mass M(T ) ∈ [0, ∞] of a current T ∈ Dm (Rn ) is defined by M(T ) = sup{T (φ ) : φ ∈ D m (Rn ), sup |φ (x)| ≤ 1}.

(9.23)

x∈Rn

The current of integration over an m-dimensional rectifiable set M ⊂ Rn has mass 

M([M]) = M

dH m = H m (M),

the m-dimensional Hausdorff volume of M. Similarly, the mass of the current h[M] given by (9.22) equals  M(h[M]) = M

|h| dH m .

Most currents have infinite mass; this holds in particular for all currents whose value on a test function involves derivatives of the latter.

9.4 Rectifiable Sets, Varifolds, and Currents

397

Rectifiable currents, which were introduced by H. Federer and W. H. Fleming [125] in 1960, provide a natural generalization of currents of integration from (9.19). These are currents T ∈ Em (Rn ) associated with integration on compact oriented rectifiable sets of dimension m with integer multiplicities and finite total mass counting multiplicities. (See H. Federer [124, 4.1.24–28] or F. Morgan [265, 4.3] for a precise description and note that their terminology differs from Allard’s [48] where rectifiable varifolds may have noninteger multiplicities.) We define Rm (Rn ) = {rectifiable m-currents on Rn }. The space of integral currents is defined by Im (Rn ) = {T ∈ Rm (Rn ) : dT ∈ Rm−1 (Rn )}. The larger class of normal currents allows for real multiplicities and smoothing. We now recall a representation theorem for currents of finite mass (see H. Federer [124, Sections 4.1.5–4.1.7]). Associated to any m-dimensional current T on Rn of finite mass there are a positive Radon measure T  on Rn of finite mass and a T -measurable frame of (not necessarily simple) m-vectors T(x) (x ∈ Rn ), with |T(x)| = 1 (in the mass norm (1.5)) for T  almost all x, such that 

T (φ ) =

Rn

φ , T d T ,

φ ∈ D m (Rn ).

(9.24)

Here, φ , T denotes the value of the m-covector φ on the m-vector T, a T (x)| ≤ 1 the measurable function on Rn . For any m-form φ ∈ D m (Rn ) with supx |φ  inequality φ , T ≤ 1 holds T  almost everywhere, so |T (φ )| ≤ Rn d T  and hence M(T ) ≤ Rn d T . By a suitable choice of φ we can ensure that φ , T is close to 1 except on a set of arbitrarily small T  measure, which shows that the mass of the current T given by (9.24) equals 

M(T ) =

Rn

d T  = T (Rn ) < ∞.

(9.25)

Any current of the form (9.24) is said to be representable by integration; hence, a current T ∈ Dm (Rn ) is representable by integration if and only if it has finite mass. Clearly, such a current acts on all compactly supported m-forms with continuous coefficients, and on all continuous m-forms if supp T is compact. Example 9.4.3. For the current of integration T = [M] (9.19) over a compact oriented m-dimensional submanifold M ⊂ Rn one has T  = H m |M , T(x) is the unit simple m-vector spanning the oriented tangent plane Tx M for each x ∈ M, and M(T ) = H m (M) is the volume of M. The same holds for a current of integration over a compact oriented m-dimensional rectifiable set M ⊂ Rn of finite total measure; such M is a differentiable manifold outside a set of T  measure zero.   Given a proper smooth map f : Rn → Rk , we associate to a current T ∈ Dm (Rn ) the image current f∗ T ∈ Dm (Rk ) (also called the pushforward of T by f ) by

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9 Minimal Hulls, Null Hulls, and Currents

( f∗ T )(φ ) = T ( f ∗ φ ),

φ ∈ D m (Rk ).

Here f ∗ φ is the pullback of φ by f (see (1.26)), and properness of f ensures that f ∗ φ has compact support. If T has compact support then f∗ T is well defined for any smooth map f since the support condition on the form becomes irrelevant. In particular, if T ∈ Em (Rn ) is a current of compact support and φt is the flow of a smooth vector field V on Rn (recall that the flow is well defined for small enough t on any given compact subset of Rn ), then Tt = (φt )∗ T ∈ Em (Rn ) is a family of currents depending smoothly on t, in the sense that for any α ∈ E m (Rn ) the map t → Tt (φ ) = T (φt∗ α) is smooth. This leads to the following definition of a minimal current (see R. Harvey and H. B. Lawson [185, Definition 4.8]), which generalizes the notion of a minimal surface. Definition 9.4.4. A rectifiable current T ∈ Rm (Rn ) is minimal or stationary if for every smooth vector field E on Rn vanishing on supp(dT ) we have that  d  M((φt )∗ T ) = 0, dt t=0 where φt denotes the flow of E. Formulas for the variation of mass of a rectifiable current can be found for example in H. B. Lawson and J. Simons [217, Theorem 1] and H. Federer [124, 5.1.8]. For the current of integration T = [M] over a compact submanifold M ⊂ Rn with boundary bM we have (see e.g. Lawson [216, Theorem 4, p. 7])   d  M((φt )∗ T ) = − E, HdH m , dt t=0 M

(9.26)

where E is the infinitesimal generator of φt and H is the mean curvature vector field of M, defined as the trace of the second fundamental form of M. For surfaces this agrees with our formula (2.38), except for the factor of 2 which is due to our definition of H as one half of the trace of the second fundamental form. Another variational formula for a current T of finite mass and compact support is given by F. R. Harvey and H. B. Lawson [185, Theorem 4.10]. It pertains to the case when the variational vector field is the gradient E = ∇u of a smooth function along supp T such that ∇u = 0 on a neighbourhood of the support of dT :   d  M((φt )∗ T ) = trT Hessu · dT . dt t=0 Rn

(9.27)

It follows in particular that if T ∈ R p (Rn ) is a minimal rectifiable p-current and u is a p-plurisubharmonic function near supp T which vanishes on a neighbourhood of supp(dT ), then trT Hessu = 0 on supp T (see [185, Corollary 4.11]). Remark 9.4.5. In Sections 9.6–9.7 we shall consider currents of the form T = h[M] (see (9.22)) given by integration over a compact connected minimal surface M ⊂ Rn with boundary endowed with a density function h ≥ 0. One may ask whether the

9.5 Hulls Defined by Minimal Rectifiable Currents

399

notion of minimality applies to such currents. We have seen in Remark 9.4.2 that supp(dh) = M implies supp(dT ) = M. Any such current is minimal according to Definition 9.4.4 since no deformations of supp(T ) are allowed. This is clearly not a satisfactory answer, especially if M is not a minimal submanifold. However, if M is minimal then the current T = h[M] is also minimal with respect to normal variations fixed on the boundary bM. Indeed, the first variation of the mass measure of T under a normal deformation φt of M fixing bM is easily seen to be   hdVt = h 1 − 2tE· H + O(t 2 ) dV (see (2.44)), where dVt is the volume element on φt (M), dV = dV0 on M, and H is the mean curvature vector field of M. If M is minimal then H ≡ 0 and we obtain  d  M((φt )∗ T ) = 0 for all normal variations. dt t=0 However, this fails for tangential variations. Indeed, if φ : M → M is a diffeomorphism whose restriction to bM is the identity, then for any test form α we have φ∗ (h[M])(α) = (h[M])(φ ∗ α) = 

= M

 M

hφ ∗ α

(φ −1 )∗ (hφ ∗ α) =



M

(h ◦ φ −1 )α = (h ◦ φ −1 )[M](α),

where the middle equality holds by the change of variables formula. Thus, φ∗ (h[M]) = (h ◦ φ −1 )[M],



M(φ∗ (h[M])) =

M

h ◦ φ −1 dH m .

This shows that the volume of the current h[M] can be changed in an almost arbitrary way by tangential deformations, unless h is constant, which corresponds to the case supp(dT ) = bM. This explains why the standard definition of minimal currents is restricted to rectifiable currents with locally constant multiplicities.  

9.5 Hulls Defined by Minimal Rectifiable Currents The following notion of a minimal p-current hull of a compact set in Rn was introduced by F. R. Harvey and H. B. Lawson [185, Definition 4.13]. Definition 9.5.1. The minimal p-current hull of a compact set K ⊂ Rn is p,min = K

!

supp T,

the union over rectifiable minimal p-currents T ∈ R p (Rn ) with supp(dT ) ⊂ K. The following result is due to Harvey and Lawson [185, Theorem 4.12].

400

9 Minimal Hulls, Null Hulls, and Currents

Theorem 9.5.2. If K is a compact set in Rn and T ∈ R p (Rn ) is a rectifiable minimal p , the p-hull of K (see Definition 8.1.9). current with supp(dT ) ⊂ K, then supp T ⊂ K In particular, we have that p,min ⊂ K p . K Proof. Let T ∈ R p (Rn ) be as in the theorem. If the conclusion fails, there is a point p . Proposition 8.1.12 furnishes a smooth function u ∈ Psh p (Rn ) which x ∈ supp T \ K vanishes on a neighborhood of K and is strongly p-plurisubharmonic at x. Therefore, that trT Hessu ≥ 0 on supp T since trT Hessu > 0 on a neighbourhood of x. Note also  u ∈ Psh p (Rn ) and T is minimal. It follows that Rn trT Hessu · dT  > 0. The vector field E = ∇u vanishes on a neighborhood of supp(dT ) ⊂ K. Let φt denote the flow of E. The variational formula (9.27) then shows that dtd M((φt )∗ T )t=0 > 0 and hence the current T is not minimal in contradiction to the assumption.   p,min approximate K p ? Is there Problem 9.5.3. How closely does the current hull K n  p ? a compact set K ⊂ R for some 2 ≤ p < n such that K p,min is a proper subset of K An affirmative answer to Problem 9.1.1 would clearly provide an affirmative answer to Problem 9.5.3. We are not aware of explicit results on this topic in the  literature. The answer to the analogous question concerning the polynomial hull K of a compact set K ⊂ Cn is negative in general, as shown by the following example. Example 9.5.4. We have already mentioned Stolzenberg’s example [324] of a  \ K is nonempty, but there is no nonconstant compact set K ⊂ C2 such that K  from the disc. In his example, the set K is the limit of holomorphic map D → K 2 boundaries of complex curves contained in the bidisc D which project properly onto the first disc and whose areas tend to infinity. It follows that there is no positive rectifiable (1, 1)-current T ∈ R2 (C2 ) (i.e., one whose frame field T consists of complex directions) such that supp(dT ) ⊂ K and supp T contains a point z ∈ C2 \ K.  cannot be described by such currents. Indeed, the same argument as In particular, K above, using plurisubharmonic functions on C2 , would show that for any such T we  Since T is d-closed on C2 \ K, the structure theorem of J. King have supp T ⊂ K. [206] shows that T is represented on C2 \ K by a holomorphic 1-chain, i.e., a linear \K combination of complex curves with constant multiplicities. It follows that K contains the image of a nonconstant holomorphic disc, a contradiction.  

9.6 Green Currents In this section we recall some results on Green currents that will be used in the characterizations of hulls provided in the following section. We follow the exposition in [117, Sect. 5]. We begin by recalling the notion of positivity of forms and currents on complex domains. Positive forms and currents play an important role in complex geometry

9.6 Green Currents

401

and have been studied extensively. We refer the reader to the monographs by J.-P. Demailly [105] and P. Griffiths and J. Harris [167] for more on this subject. A (1, 1)-form α on a domain Ω ⊂ Cn is said to be positive if α(p), ν ∧ Jν ≥ 0 for every p ∈ Ω and tangent vector ν ∈ Tp Cn ,

(9.28)

and strongly positive if strict inequality holds for every ν ∈ Tp Cn \ {0}. Here, J denotes the standard complex structure operator on T Cn . Equivalently, −iα(p), v ∧ v ¯ ≥ 0 for any (1, 0)-tangent vector v ∈ Tp1,0 Cn . If h is a Hermitian metric on T Cn then the fundamental form ω = −ℑh is a strongly positive (1, 1)-form (see (1.103)); conversely, a strongly positive (1, 1)-form is the fundamental form of a Hermitian metric determined by (1.104). Positive (1, 1)forms also arise as curvature forms of ample (positive) line bundles. A current T of bidimension (1, 1) is said to be positive if T (α) ≥ 0 for every positive (1, 1)-form α of compact support. The notion of positivity extends to (p, p)-forms and currents. An important example of a positive (p, p)-current is the current of integration, [M], over a p-dimensional complex submanifold M and, more generally, a current of the form T = h[M] where h ≥ 0 is a nonnegative function on such M (see (9.19) and (9.22)). Let z = x + iy be the coordinate on C ∼ = R2 . The Green current, G, on the closed unit disc D is defined on any 2-form α = adx ∧ dy with a ∈ C (D) by G(α) = −

1 2π



log |z|· α = −

D



1 2π

log |z|· a(z)dx ∧ dy.

(9.29)

D

Note that G = − log |· |· [D], where [D] is the current of integration over D (cf. (9.22)). Since − log | · | ≥ 0 on D, G is a positive current of bidimension (1, 1) whose density function z → − log |z| is locally integrable at the origin. Recall (cf. (1.67)) that for any function u of class C 2 on a domain in C we have that dd c u = 2i∂ ∂ u = Δ u · dx ∧ dy. The classical Green’s formula 1 u(0) = 2π

 2π 0

1 u(e ) dt + 2π



log |z|· Δ u(z) dx ∧ dy,

it

(9.30)

D

which holds for any function u ∈ C 2 (D), shows that (dd c G)(u) = G(dd c u) =

1 2π

 2π 0

u(eit )dt − u(0).

This means that dd c G = σ − δ 0

(9.31)

where σ denotes the normalized Lebesgue measure on the circle bD = T and δ 0 is the evaluation (point mass) at the origin 0 ∈ C. If u is subharmonic then

402

9 Minimal Hulls, Null Hulls, and Currents

G(Δ u · dx ∧ dy) = dd c G(u) =



u dσ − u(0) ≥ 0.

T

Let x = (x1 , . . . , xn ) be coordinates on Rn . Given a map f = ( f1 , . . . , fn ) : D → Rn of class C 1 , we denote by f∗ G the 2-dimensional current on Rn given on any 2-form α = ∑ni, j=1 ai j dxi ∧ dx j with continuous coefficients by 1 ( f∗ G)(α) = G( f α) = − 2π ∗



log | · |· f ∗ α.

(9.32)

D

Clearly, supp( f∗ G) ⊂ f (D). We call f∗ G the Green current supported by f . We shall be mainly interested in the case when f is a conformal harmonic immersion. Assume now that f : D → Cn is a C 2 map that is holomorphic on D. It is immediate that f∗ G is then a positive (1, 1)-current. Furthermore, the pullback f ∗ commutes with the operators ∂ and ∂ , and hence also with the conjugate differential d c = i(∂ − ∂ ). It follows that for any u ∈ C 2 (Cn ) we have that dd c ( f∗ G)(u) = ( f∗ G)(dd c u) = G( f ∗ dd c u) = G(dd c (u ◦ f )) 

=

(u ◦ f )dσ − (u ◦ f )(0),

T

where the last equality follows from (9.31). This shows that dd c ( f∗ G) = f∗ σ − δ f (0) for any f ∈ A 2 (D, Cn ).

(9.33)

The following lemma from [117, Lemma 5.1] gives an explicit upper bound for the mass (see (9.23)) of the Green current supported by a conformal minimal disc f : D → Rn in terms of the L2 -norm of f on the circle T = bD. Lemma 9.6.1. If f = ( f1 , . . . , fn ) : D → Rn is a conformal harmonic immersion of class C 2 (D), then the mass of the Green current f∗ G satisfies the inequality   1 2 2 | f | dσ − | f (0)| . (9.34) M( f∗ G) ≤ 4 T If f is injective outside of a closed set of measure zero in D, or if f : D → Cn is a holomorphic disc, then equality holds in (9.34). It was proved beforehand by E. F. Wold [347, Lemma 2.2] that for any holomorphic disc f : D → Cn the mass M( f∗ G) is bounded above in terms of the dimension n and the sup-norm supz∈D | f (z)| of the map f . However, for what follows it is crucial to have an estimate in terms of the L2 (T) norm. Our proof also applies in this case since a nonconstant holomorphic map is a conformal harmonic immersion outside a discrete set of points. Proof. We shall use the coordinate z = x + iy on D and will denote the partial derivatives of f = ( f1 , . . . , fn ) : D → Rn by fx and fy . Write

9.6 Green Currents

403 n n   2 2 . |∇ f |2 = ∑ |∇ fi |2 = ∑ fi,x + fi,y

n

| f |2 = ∑ fi2 , i=1

i=1

i=1

We consider fx and fy as vector fields along the image of f . Since f is conformal, these vector fields are orthogonal and satisfy | fx | = | fy | (see Lemma 1.8.4 (b)). Recall the notation (8.7): fx ∧ fy fx ∧ fy = . | fx |· | fy | | f x |2

T=

Given a 2-form α on Rn , we have that f ∗ α = α ◦ f , fx ∧ fy  dx ∧ dy = α ◦ f , T | fx |2 dx ∧ dy. The definition of the Green current T = f∗ G (9.32) and the above formula imply T (α) = −

1 2π



log |z| · α ◦ f , T· | fx |2 dx ∧ dy.

D

From this and the definition of the mass of a current (see (9.23)) it follows that M(T ) ≤ −

1 2π



log |z|· | fx |2 dx ∧ dy.

(9.35)

D

We now take into account that the map f is also harmonic. Given a C 2 function u on a neighbourhood of f (D), the formula (8.8) says that dd c (u ◦ f )(z) = tr T f (z) (Hessu ) · | fx (z)|2 dx ∧ dy,

z ∈ D.

(9.36)

Applying this identity with the function u(x1 , . . . , xn ) = ∑nj=1 x2j for which trT (Hessu ) ≡ 4 and u ◦ f = | f |2 , we obtain     1 1 1 log |z|· | fx |2 dx ∧ dy = − log |z|· dd c | f |2 = | f |2 dσ − | f (0)|2 , − 2π D 8π D 4 T where the last equation holds by Green’s formula (9.30). This proves (9.34). It remains to explain why equality holds under the conditions stated in the lemma. The argument leading to the formula (9.35) actually shows that if U ⊂ Rn is an open set such that f is injective on f −1 (U) ⊂ D, then T (U) = −

1 2π

 f −1 (U)

log |z|· | fx |2 dx ∧ dy.

(9.37)

The possible loss of mass, leading to a strict inequality in (9.35) and hence in (9.34), may be caused by the cancellation of parts of the immersed surface f (D) (considered as a current) due to the reversal of the orientation of the frame field T (8.7). (This happens for example by immersing the disc conformally onto a M¨obius band in R3 .) If f is injective outside a closed set E ⊂ D of measure zero, then for

404

9 Minimal Hulls, Null Hulls, and Currents

any open set V ⊃ E we can find a 2-form α such that |α, T| ≤ 1 on f (D) and α, T = 1 on f (D \ V ). By shrinking V down to E we see that the inequality in (9.35) becomes an equality. On the other hand, the total mass of the disc f (D), counted with multiplicities and with the density induced by − log |· | ≥ 0 on D, always equals the expression on the right-hand side of (9.34). In other words, the mass of f∗ G considered as a varifold (neglecting the orientation) is given by the equality in (9.34). Since a holomorphic disc f : D → Cn carries a canonical orientation induced by  the complex structure, there is no cancellation of mass in the current f∗ G. 

9.7 Currents Characterizing Minimal Hulls and Null Hulls We now show how Theorem 9.2.1 leads to characterizations of null hulls and minimal hulls by limits of Green currents supported by holomorphic null discs and conformal minimal discs, respectively. These results were obtained in [117, Sect. 6] in dimension n = 3 and in [15, Sect. 5] in arbitrary dimension n ≥ 3. The notion of positive forms and currents was discussed in the previous section. We shall need the following variant of this notion. Definition 9.7.1. A real (1, 1)-form α on a domain Ω ⊂ Cn (n ≥ 3) is null positive if for every point p ∈ Ω and null vector ν ∈ A∗ (2.54) we have that α(p), ν ∧ Jν ≥ 0. (We identify ν with a tangent vector in Tp Cn .) A (1, 1)-current T on Cn is null positive if T (α) ≥ 0 for every null positive (1, 1)form α of compact support. Note that a C 2 function u on a domain in Cn (n ≥ 3) is null plurisubharmonic (see Definition 8.2.2) if and only if the (1,1)-form dd c u is null positive. Let G denote the Green current (9.29). If f : D → Cn is a holomorphic null disc, then T = f∗ G is a null positive current. Indeed, if α is a null positive (1, 1)-form on oriented orthonormal frame field along f , then α, T ≥ 0 Cn and T is a positively  and hence T (α) = α, T d T  ≥ 0 by (9.24). Theorem 9.7.2. Let K be a compact set in Cn for some n ≥ 3. A point p ∈ Cn belongs N (9.2.2) of K if and only if there is a null positive (1, 1)-current to the null hull K n T on C of compact support satisfying dd c T = μ − δ p , where μ is a probability measure on K and δ p is the point mass at p. The support of any such current T is N of K, and we have that contained in the null hull K u(p) ≤



u dμ K

for all u ∈ NPsh(Cn ).

(9.38)

N then there is a current with these properties of the form Furthermore, if p ∈ K T = lim j→∞ ( f j )∗ G, where f j is a sequence of holomorphic null discs as in Theorem 9.2.6 and G is the Green current (9.29) on the disc D.

9.7 Currents Characterizing Minimal Hulls and Null Hulls

405

The proof of this theorem is similar to the one given for the polynomial hull by E. F. Wold [347, Theorem 2.3]. The original construction of currents characterizing polynomial hulls is due to J. Duval and N. Sibony [119]. Their currents are limits of currents of integration on bordered complex curves, with weights given by Green functions with finitely many poles. In Wold’s construction, and in the one presented here, one uses single-pole Green functions supported by (null) holomorphic discs. Proof. If a current T satisfies the conclusion of the theorem for a given point p, then for every u ∈ NPsh(Cn ) ∩ C 2 (Cn ) we have that 0 ≤ T (dd c u) =

 K

u dμ − u(p) ≤ max u − u(p). K

For an arbitrary function u ∈ NPsh(Cn ) the same holds by smoothing (see N and (9.38) holds. Proposition 8.2.7). Hence, p ∈ K For the converse implication, we follow the argument by E. F. Wold [347], providing more complete details from the functional analytic viewpoint. Choose a ball Ω ⊂ Cn containing K. Let f j ∈ N(D, Ω ) be a sequence of holomorphic null discs with f j (0) = p furnished by Theorem 9.2.6. The sequence of Green currents T j = ( f j )∗ (G) has uniformly bounded masses by Lemma 9.6.1. Each T j is a continuous linear functional on the separable Banach space of differential forms on Cn with bounded continuous coefficients, endowed with the sup-norm. A bounded set of functionals is metrizable (see J. B. Conway [99, V.5.1]), and hence the weak (in fact, the weak*) compactness of a closed bounded set of currents coincides with the sequential weak compactness. Thus, a subsequence of {T j } j∈N converges weakly to a null positive (1, 1)-current T of compact support and finite mass. By (9.33) we have that dd c T j = ( f j )∗ σ − δ p for all j. Condition (9.6) implies that the supports of the probability measures σ j = ( f j )∗ σ converge to K; passing to a subsequence gives a probability measure μ = lim j→∞ σ j on K. It follows that dd c T = lim dd c T j = lim σ j − δ p = μ − δ p . j→∞

j→∞

The inequality (9.38) holds by what has been said at the beginning of the proof. N . This is a special case of part (a) in the It remains to show that supp T ⊂ K following proposition. Part (b) will be used in the proof of Theorem 9.7.4. We consider Rn as the standard real subspace of Cn and denote by π : Cn → Rn M denotes the minimal hull (8.10). the projection π(x + iy) = x. Recall that K Proposition 9.7.3. Let T be a null positive (1, 1)-current on Cn for some n ≥ 3. (a) If T has compact support and satisfies dd c T ≤ 0 on Cn \ K for some compact N . set K ⊂ Cn , then supp T ⊂ K (b) If T has bounded mass, π(supp T ) ⊂ Rn is a bounded subset of Rn , and dd c T ≤ 0 M × iRn . on Cn \ (K × iRn ) for some compact set K ⊂ Rn , then supp T ⊂ K N . Proposition 9.2.5 furnishes a nonnegative Proof of (a). Fix a point q ∈ Cn \ K n smooth function u ∈ NPsh(C ) which is strongly null plurisubharmonic on a

406

9 Minimal Hulls, Null Hulls, and Currents

N . Since the neighbourhood U ⊂ Cn of q and vanishes on a neighbourhood of K n c support of u is contained in C \ K where dd T ≤ 0, we have that T (dd c u) = (dd c T )(u) ≤ 0. (We are using that T has compact support, so it may be applied to forms with arbitrary supports.) As T is null positive on Cn , we also have T (dd c u) ≥ 0; hence T (dd c u) = 0. Since u is strongly null plurisubharmonic on N . U, it follows that T has no mass on U. This proves that supp T ⊂ K Proof of (b). Write TU = π −1 (U) = U × iRn for any U ⊂ Rn . Choose a ball B ⊂ Rn M ⊂ B. Since T has bounded mass, it can be applied to any such that π(supp T ) ∪ K 2-form with bounded continuous coefficients on the tube TB . In particular, for any function u ∈ C 2 (Rn ) the current T can be applied to the (1, 1)-form dd c (u ◦ π). M . Proposition 8.1.12 furnishes a smooth nonnegative function Fix a point q ∈ /K n u ∈ MPsh(R ) that is minimal strongly plurisubharmonic on a neighbourhood M . The function U ⊂ Rn of q and vanishes on a neighbourhood V ⊂ Rn of K n u˜ = u ◦ π on C is then null plurisubharmonic (see Lemma 8.2.3), it is strongly null plurisubharmonic on the tube TU , and it vanishes on TV . Since the support of u˜ is ˜ = (dd c T )(u) ˜ ≤ 0. contained in Cn \ TK where dd c T is negative, we have T (dd c u) c ˜ ≥ 0; hence T (dd c u) ˜ = 0. Since u˜ is As T is null positive, we also have T (dd u) strongly null plurisubharmonic on the tube TU , we conclude as before that T has no M × iRn . mass there. This proves that supp T ⊂ K This completes the proof of Proposition 9.7.3 and hence of Theorem 9.7.2.   In the remainder of the section we obtain several characterizations of the minimal hull of a compact set in Rn by currents. Recall that π : Cn → Rn denotes the real part projection π(x + iy) = x. Theorem 9.7.4 (Characterization of minimal hulls by currents). Let K be a M compact set in Rn for some n ≥ 3. A point p ∈ Rn belongs to the minimal hull K n (8.10) if and only if there exists a null positive current T on C of finite mass such that π(supp T ) ⊂ Rn is a bounded set and dd c T = μ − δ p , where μ is a probability M × iRn . measure on the tube TK = K × iRn . For any such T we have supp T ⊂ K Before proving this theorem, we show how it implies the following characterization of the minimal hull by minimal Jensen measures. Corollary 9.7.5. Let K be a compact set in Rn (n ≥ 3). A point p ∈ Rn belongs to M if and only if there is a probability measure ν on K such that the minimal hull K u(p) ≤



u dν K

for all u ∈ MPsh(Rn ).

(9.39)

M . Any such measure ν is called a minimal Jensen measure for the point p ∈ K Proof. If ν is a probability measure on K such that (9.39) holds, then for every M . u ∈ MPsh(Rn ) we have that u(p) ≤ K u dν ≤ maxK u, and hence p ∈ K  Conversely, assume that p ∈ KM . Let T and μ be as in Theorem 9.7.4. Then,

9.7 Currents Characterizing Minimal Hulls and Null Hulls

u(p) ≤

 TK

(u ◦ π) dμ ≤ max u K

407

for all u ∈ MPsh(Rn ).

(9.40)

Indeed, if u ∈ MPsh(Rn ) ∩ C 2 (Rn ) then u˜ := u ◦ π ∈ NPsh(Cn ) ∩ C 2 (Cn ) by Lemma 8.2.3, and u˜ is bounded on the tube TB = B × iRn for every bounded set B ⊂ Rn . Since T is null positive and dd c T = μ − δ p , we have that ˜ = 0 ≤ T (dd c u)

 TK

u˜ dμ − u(p),

thereby proving (9.40). The projection ν = π∗ μ is then a probability measure on K  satisfying u(p) ≤ K u dν. The same inequality follows for every u ∈ MPsh(Rn ) by approximating it with smooth ones; see Proposition 8.1.6.   Proof of Theorem 9.7.4. If a current T and a measure μ satisfy the conclusion of the M . theorem, then (9.40) holds and hence p ∈ K M . Theorem 9.2.1 furnishes Let us now prove the converse. Fix a point p ∈ K a bounded sequence of conformal minimal discs f j ∈ M(D, Rn ) with f j (0) = p satisfying (9.2). We may assume that each f j is smooth on D. Let g j be the harmonic conjugate of f j on D with g j (p) = 0. Then, Fj = f j + ig j ∈ N(D, Cn ) is a holomorphic null disc. Let Θ j = ( f j )∗ G and T j = (Fj )∗ G be the associated Green currents on Rn and Cn , respectively. Then, T j is null positive and π∗ T j = Θ j for every j = 1, 2, . . .. By Lemma 9.6.1 we have that 

4M(T j ) =

|Fj |2 dσ − |p|2 =

T

 T

| f j |2 dσ +



|g j |2 dσ − |p|2 .

T

Since the conjugate function operator is bounded on the Hilbert space L2 (T) (see [156, Theorem 3.1, p. 111]) and the sequence f j is uniformly bounded, we see that M(T j ) ≤ C < ∞ for some constant C and for all j ∈ N. We may assume by passing to a subsequence that the sequence T j converges weakly to a null positive (1, 1)-current T with finite mass (but not necessarily compact support since the harmonic conjugates g j of f j need not be uniformly bounded), and Θ j converges to a 2-dimensional current Θ on Rn . From π∗ T j = Θ j for all j ∈ N we infer that π∗ T = Θ . By (9.33) we have that dd c T j = (Fj )∗ σ − δ p for all j ∈ N. Note that π∗ (Fj )∗ σ = ( f j )∗ σ . Condition (9.2) implies that the supports of the probability measures ( f j )∗ σ converge to K, and hence the supports of the measures (Fj )∗ σ converge to the tube TK . By passing to a subsequence we obtain a probability measure μ = lim j→∞ (Fj )∗ σ supported on TK . It follows that dd c T = lim dd c T j = lim (Fj )∗ σ − δ p = μ − δ p . j→∞

j→∞

M × iRn .   Finally, Proposition 9.7.3 (b) implies that supp T ⊂ K Remark 9.7.6. As pointed out in Problem 9.5.3, it is not clear whether every point p of a compact set K ⊂ Rn lies in the support of a minimal rectifiable in the p-hull K

408

9 Minimal Hulls, Null Hulls, and Currents

p-current T with supp dT ⊂ K. On the other hand, Theorem 9.7.4 fully explains the minimal hull (2-hull) by Green currents supported by conformal minimal discs.   We wish to compare the minimal hull of a compact set K ⊂ Rn with the null hull of the tube TK = K × iRn ⊂ Cn over K. The latter set is unbounded, and the standard definition of its polynomial hull (and, by analogy, of its null hull) is by exhaustion with compact sets. Let Br ⊂ Rn denote the closed ball of radius r centred at the origin. Then, TK = r>0 TK,r where TK,r = K × iBr , and we set 3K = T

!

2 T K,r ,

 (T K )N =

r>0

!

 (T K,r )N .

(9.41)

r>0

It is easily seen that these hulls are independent of the choice of exhaustion. Clearly,  3 (T K )N ⊂ TK . From (9.7) we also get that  M × iRn ⊂ Co(K) × iRn . (T K )N ⊂ K We do not know whether the first inclusion is strict in general. On the other hand, Theorem 9.7.4 motivates the following definition of the current null hull of TK . Definition 9.7.7. Let K be a compact set in Rn and TK = K × iRn ⊂ Cn be the tube  over K. The current null hull of TK , denoted (T K )N∗ , is the union of supports of all null positive (1, 1)-currents T on Cn with finite mass such that π(supp T ) ⊂ Rn is a bounded set and dd c T ≤ 0 on Cn \ TK .   Theorem 9.7.2 shows that (T K )N ⊂ (TK )N∗ . Now, Theorem 9.7.4 implies the following result which extends the classical relationship between conformal minimal discs and holomorphic null discs to the corresponding hulls. Corollary 9.7.8. If K is a compact set in Rn and TK = K × iRn ⊂ Cn , then  M × iRn . (T K )N ∗ = K

(9.42)

Question 9.7.9. Let T be a current as in Theorem 9.7.4 with dd c T = μ − δ p , where μ is a probability measure on TK and p ∈ Rn . Is the point p contained in the support of the projected current Θ = π∗ T on Rn ? M is the union of supports If the above question has an affirmative answer, then K of currents of the form Θ = lim j→∞ ( f j )∗ G, where f j is a bounded sequence of conformal minimal discs as in Theorem 9.2.1 whose boundaries converge to K in measure. However, the problem is that cancellation of mass may occur in Θ as explained in the proof of Lemma 9.6.1. This can be circumvented by considering the Green currents ( f j )∗ G as bounded linear functionals on the separable Banach space Q(Rn ) consisting of all real symmetric quadratic forms n

h=

∑

i, j=1

hi j (x)dxi ⊗ dx j ,

hi j = h ji

9.7 Currents Characterizing Minimal Hulls and Null Hulls

409

with continuous coefficients and finite sup-norm h =

n

∑

sup |hi j (x)| < ∞.

i, j=1 x∈Rn

Assume that T is 2-dimensional current of finite mass on Rn (hence representable by integration, see (9.24)), and let T and T  denote its frame field and mass measure, respectively. Then, T defines a bounded linear functional on Q(Rn ) by 

T (h) =

Rn

trT h · d T ,

h ∈ Q(Rn ),

(9.43)

where trT h denotes the trace of the restriction of h to the 2-plane span T. Since trT h is independent of the orientation determined by T, every compact surface M ⊂ Rn (also nonorientable) defines a bounded linear functional on Q(Rn ). More generally, one can use rectifiable surfaces of finite area, that is, countable unions of images of Lipschitz maps f : D → Rn (see Section 9.4). Note that T (Hessu ) ≥ 0 for every u ∈ C 2 (Rn ) ∩ MPsh(Rn ) since trT Hessu ≥ 0 for every 2-frame field T. We now prove the following characterization of the minimal hull in Rn . Theorem 9.7.10. Let K be a compact set in Rn for some n ≥ 3. A point p ∈ Rn M of K if and only there exist a continuous real linear belongs to the minimal hull K n functional T on Q(R ) with support in a compact subset of Rn and a probability measure μ on K such that 

T (Hessu ) =

K

u dμ − u(p) for all u ∈ C 2 (Rn ),

(9.44)

and T (Hessu ) ≥ 0 for every minimal plurisubharmonic function u of class C 2 on M . Rn . The support of every such functional T is contained in K Note that μ in (9.44) is a minimal Jensen measure for the point p; see Corollary 9.7.5. For n = 3 the above result is [117, Corollary 6.10]. Proof. If such T and μ exist, then for every u ∈ C 2 (Rn ) ∩ MPsh(Rn ) we have 0 ≤ T (Hessu ) =

 K

udμ − u(p) ≤ max u − u(p), K

M . To prove the converse, we consider linear functionals (9.43) and hence p ∈ K defined by minimal discs. Suppose that f : D → Rn is a smooth conformal minimal immersion. Denote by T the unit 2-frame (8.7) along f determined by the partial derivatives fx , fy . In view of (9.37) the associated functional T f = f∗ G (9.43) on the Banach space Q(Rn ) takes the form T f (h) = −

1 2π



log |z| (trT h ◦ f ) | fx |2 · dx ∧ dy,

h ∈ Q(Rn ).

D

In particular, when h = Hessu with u ∈ C 2 (Rn ), the formula (8.8) gives

(9.45)

410

9 Minimal Hulls, Null Hulls, and Currents

T f (Hessu ) = −

1 2π

 D

log |z|· dd c (u ◦ f ) =

 2π 

 dt − u( f (0)), 2π

u f (eit )

0

(9.46)

where the second equality holds by Green’s formula (9.30). M . Let f j : D → Rn Assume now that K is a compact set in Rn and p ∈ K be a bounded sequence of conformal minimal immersions furnished by Theorem 9.2.1, with f j (0) = p for all j ∈ N. The associated linear functionals T j = T f j on Q(Rn ) given by (9.45) then form a bounded sequence in the dual space Q(Rn )∗ . By the same argument as in the proof of Theorem 9.7.2 (using Lemma 9.6.1), we see that a subsequence of {T j } j∈N converges weakly to a bounded linear functional T ∈ Q(Rn )∗ of compact support. Similarly, we may assume by passing to a subsequence that the sequence of probability measures ( f j )∗ σ on Rn converges weakly to a probability measure μ supported on K. Since every T j satisfies (9.46), we get in the limit the identity (9.44). If u ∈ C 2 (Rn ) is minimal plurisubharmonic then T j (Hessu ) ≥ 0 for every j, and hence T (Hessu ) ≥ 0. Proposition 9.7.3 (b) then M , thereby completing the proof.   shows that supp T ⊂ K Remark 9.7.11. N. Sibony [317] found nonnegative directed currents of bidimension Γ of a compact set K ⊂ Cn in any directed system (1, 1) describing the Γ -hull K determined by a closed, fibrewise conical subset Γ of the tangent bundle T Cn . The Γ is defined by the maximum principle (see (9.1)) in terms of the family hull K FΓ of Γ -plurisubharmonic functions on Cn , i.e., functions whose Levi form at any point z ∈ Cn is nonnegative in directions belonging to the fibre Γz of Γ at z. Sibony’s characterization holds even if there are no Γ -directed holomorphic discs; in particular, the currents constructed in his paper need not be limits of directed Green currents. The classical case of the polynomial hull (with Γ = T Cn ) is due to J. Duval and N. Sibony [119]; see also E. F. Wold [347]. The null hull also falls within this framework; in this case the fibre Γz ⊂ T Cn ∼ = Cn of Γ over any point z ∈ Cn is the null quadric (2.54), Γ -plurisubharmonic functions are null plurisubharmonic functions, and Γ -discs are null discs. It seems an interesting problem to decide in which systems directed by a complex analytic subvariety Γ ⊂ T Cn with conical fibres it is possible to describe the hull Γ by sequences of Γ -directed holomorphic discs Fj : D → Cn whose boundaries K converge to K in measure (cf. (9.6)). For the polynomial hull, this holds by Poletsky [291, 292] and Bu–Schachermayer [74]. (For generalizations to plurisubharmonic hulls in complex manifolds, see [214, 215, 301, 302]; for complex spaces, see [116].) For the null hull, this holds by Theorem 9.7.2. These seem to be the only cases studied so far. One expects that such a description holds in any holomorphic directed system in which every Riemann–Hilbert boundary value problem for directed discs admits approximate solutions (cf. Remark 9.2.13). The latter result is also known to hold in the case of Legendrian holomorphic discs in the standard holomorphic contact structure on C2n+1 for any n ∈ N; see [31, Sect. 3]. The corresponding theory of contact hulls has not been investigated yet.  

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Index

(0, 1)-form, 31 (1, 0)-form, 31 (p, q)-form, 31 1-cocycle, 26 D f (p), 4 H 0 (M, E), 64 k (M), 14 HdR H1 (M, Z), 7 Hρ (x), 19 J-positive form, 41 KM , 55 N-normal curve, 85 T M, 9 T ∗ M, 12 A (K), 5 A r (K, X), 74 A∞r (S | A, A∗ ), 166 AD (S), 202 Aut(M), 8 Bn (a, r), 2 BnR (a, r), 2 CMI(M, Rn ), 133 CMIr (M, Rn ), 133, 134 CMI∞ (S | E, Rn ), 203 CMIfull (M, Rn ), 134 CMIrfull (M, Rn ), 134 CMInf (M, Rn ), 134 CMIrnf (M, Rn ), 134 C (C, u), 207 Co, 113 Crit( f ), 16 C (D), 4 C (K), 5 C ω (D), 4 C r atlas, 8 C r structure, 8 C r (D), 4

C r (K), 4 C r,α (D), 4

D(a, r), 2 Diff r (M), 8 Div(S), 201 D m (Rn ), 395 Dm (Rn ), 395 E (M), 13 E k (M), 13 Em (Rn ), 395 Fluxx , 99 F -convex manifold, 380 F -convex set, 380 GCMIrfull (S, Rn ), 135 GCMIrnf (S, Rn ), 135 GNCrfull (S, Cn ), 135 GNCrnf (S, Cn ), 135 Γ (M, p0 ), 306 Γqd (M, p0 ), 306 Hn+ , 5 Hessρ , 19 I-invariant conformal minimal immersion, 103 Im (Rn ), 397 Jac( f ), 4 Jac C , 8 Δ , 38 Lρ , 32 MPsh(D), 340 M (M), 55 M (S, S∗ ), 202 M (S, A∗ ), 202 M (S | E), 201 M (S | E, S∗ ), 202 M (S | E, A∗ ), 202 M 1 (S), 201 M 1 (S | E), 201 M 1 (M), 55

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Alarcón et al., Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69056-4

425

426 M∗1 (M), 55 M∗ (M), 55 NC(M, Cn ), 133 NCr (M, Cn ), 133 NCfull (M, Cn ), 134 NCrfull (M, Cn ), 134 NCnf (M, Cn ), 134 NCrnf (M, Cn ), 134 OD (K, u), 206 OD (S), 202 Ω (M), 55 Ω (S), 201 O (D), 5 O (K), 5 O (K, X), 74 O (M)-convex set, 66 O (M, X), 74 O (X)-convex hull, 23 Onf (M, A∗ ), 185 O(K), 5, 68 O(K, X), 74 Psh(M), 33 Psh p (D), 340 Q(Rn ), 408 Rm (Rn ), 397 S2 (D), 207 TC(x), 109, 193 ϒ [r], 229 ϒm [r], 229 χ(M), 7 ∂I f, 4 div, 38 Sn−1 ∗ , 202 ∇, 19 π1 (M), 7 supp T , 395 trL , 337 ΛkV , 2 M,D , 343 K M , 343 K p,D , 343 K p , 343 K f  Z, 17 f ∗ α, 13 f∗ v, 12 k-vector, 2 p-convex compact set, 343 p-convex domain, 337 p-convex hull, 343 p-harmonic function, 343 p-plurisubharmonic function, 337, 340 p-positive quadratic form, 337 inverse line bundle, 27

Index absolute neighbourhood retract, 77 adjunction space, 20 admissible set, 69 affine algebraic variety, 23 against set, 229 algebraic Gauss map, 111 algebraically elliptic manifold, 81 algebraically subelliptic manifold, 81 almost complex structure, 50 angle, 46 anti-self-dual manifold, 323 arc length, 36 area of a domain, 37 associated minimal surfaces, 101 atlas, 5 attaching set, 20 Baire space, 17 basic Oka principle, 76 basic transversality theorem, 17 Beltrami equation, 51 bidimension, 395 biholomorphism, 8 Bishop–Mergelyan theorem, 68 bordered conformal surface, 59 bordered Riemann surface, 59 boundary of a current, 396 boundary of a manifold, 5 bounded exhaustion hulls, 168 branched minimal surface, 102 branched null curve, 102 branching number, 56 Bryant correspondence, 324 Calabi-Yau problem, 295, 299 Calabi-Yau property, 321 calibration, 45 canonical bundle, 55 canonical class, 62 canonical divisor, 62 Carleman admissible set, 168 Carleman set, 168 Cartan pair, 281 Cartan–Oka–Weil theorem, 73 Cartan pair, 79 catenal curve, 116 catenoid, 101, 116 Cauchy estimates, 217 Cauchy–Riemann equation, 54 Cauchy–Riemann system, 27 Chen–Gackstatter surfaces, 125 Chern number, 63 Chern–Osserman Theorem, 193 closed domain, 2

Index closed form, 14 coherent orientation, 9, 15 comass norm, 3 compact bordered conformal surface, 59 compact bordered Riemann surface, 59 compact domain, 2 complete Riemannian manifold, 36, 306 complete vector field, 12 complex atlas, 8 complex contact manifold, 322 complex contact structure, 322 complex Gauss map, 107 complex manifold, 8 complex normal bundle, 30 complex structure operator, 28 complex vector bundle, 27 complexified cotangent bundle, 30 complexified tangent bundle, 28 conformal atlas, 49, 59 conformal Calabi-Yau for surfaces, 299 conformal Calabi-Yau problem, 295 conformal immersion, 46 conformal structure, 46, 49 conformal surface, 49 conformally equivalent metrics, 46 conjugate differential, 32 conjugate minimal surfaces, 101 connected sum, 6 convex hull, 113 convex hull of an affine variety, 151 convex hull property, 114 convex integration lemma, 153 core, 20, 79, 136 Costa’s surface, 126 cotangent bundle, 12 cotangent space, 12 critical point, 16 critical total curvature, 199 critical value, 16 current, 102, 394, 395 current null hull, 408 current of finite mass, 397 current of integration, 395 current representable by integration, 397 curvature of a curve, 84 curvature vector, 84 cusp point, 219 de Rham cohomology group, 14 decomposable conformal minimal immersion, 106 degree of a divisor, 61 degree of a map, 56 degree of projective variety, 25

427 desingularization, 152 diffeomorphism, 8, 12 difference map, 144 differential 1-form, 12 differential k-form, 13 Dirichlet integral, 98 disc formula, 385 disc functional, 385 divergence, 38 divergence theorem, 38 divergent path, 36, 306 divisor, 60 divisor of poles, 202 divisor of zeros, 202 domain, 2 domain of holomorphy, 24 dominating spray, 79 effective divisor, 61 Einstein manifold, 323 elliptic manifold, 80 embedding, 12 end, 6 Enneper’s surface, 122 equivalent atlases, 8 Euclidean metric, 34 Euler number, 7, 63, 218, 229 everywhere full path, 234 exact form, 14 exhaustion function, 6 exterior derivative, 13 exterior power, 2 exterior product, 2 finite sublinear multigraph, 376 finite total curvature, 109 first Chern class, 27 first fundamental form, 35 first homology group, 7 first variation of area, 92 first variational formula, 92 flat conformal minimal immersion, 106 flat map, 106 flexible manifold, 77 flow of a vector field, 11 flow-spray, 80 flux, 99 Fubini–Study form, 42 Fubini–Study metric, 35 full conformal minimal immersion, 106 full generalized conformal minimal immersion, 134 full generalized null curve, 135 full map, 106

428 full path, 234 fundamental domain, 11 fundamental form, 40 fundamental group, 7 Gauss map, 87, 88, 105 Gaussian curvature, 86 Gaussian image, 200, 250 general position, 249 generalized conformal minimal immersion, 134 generalized conformal minimal immersion of finite total curvature, 164 generalized conformal minimal immersion of locally finite total curvature, 170 generalized null curve, 135 generic property, 17 genus, 7 genus formula, 57 gradient, 19, 38 Green current, 379, 401, 402 halfspace theorem, 176, 177 handlebody, 16, 20 harmonic conjugate, 55 harmonic function, 38 Hausdorff measure, 340, 394 helicatenoid, 101 helicoid, 101, 117, 118 Henneberg’s surface, 127 Hermitian metric, 39 Hessian, 19, 339 hitting condition, 354 H¨older continuous function, 4 hole, 6 holomorphic 1-form, 55 holomorphic disc, 266 holomorphic hull, 66, 168 holomorphic line bundle, 63 holomorphic map, 8 holomorphic null curve, 100 holomorphic spray of maps, 79 holomorphic tangent bundle, 28 holomorphic vector bundle, 26 holomorphic vector field, 29 homogeneous polynomial, 24 homotopy equivalence, 77 Huber’s Theorem, 111 Hurwitz Theorem, 214 hyperbolic Riemann surface, 60 image current, 397 immersion, 12 integral current, 397

Index integral curve, 11 intrinsic diameter, 306 isometric immersion, 34 isometry, 34 isoperimetric inequality, 115 isothermal coordinate, 48 isotopy of immersions, 185 Jacobian determinant, 4 Jacobian matrix, 4 jet, 18 jet transversality theorem, 18 Jorge–Meeks formula, 196, 197 K¨ahler form, 41 K¨ahlerian manifold, 41 K¨ahler manifold, 41 K¨ahler metric, 41, 48 K¨ahler potential, 41 Kobayashi hyperbolicity, 60 Laplace operator, 32 Laplacian, 19, 38 Laplacian operator, 54 Legendrian curve, 322 length, 36 Levi form, 32 line bundle, 26 linearly equivalent divisors, 61 Liouville manifold, 60 local chart, 5 local diffeomorphism, 12 localization theorem, 68 locally Euclidean space, 5 locally nilpotent derivation, 82 L´opez Klein bottle, 130 manifold of class C r , 8 mass of a current, 396 maximal atlas, 8 maximum principle, 113 mean curvature flow, 382 mean curvature vector, 87 mean-convex domain, 338, 346 mean-convex hull, 382 Meeks M¨obius strip, 128 Mergelyan property, 69, 74 Mergelyan theorem, 74 Milnor link, 186 minimal current, 398 minimal current hull, 399 minimal graph equation, 114 minimal hull, 114, 343 minimal Jensen measure, 406

Index minimally convex compact set, 343 minimally convex domain, 338, 343 minimally convex end, 355 minimal plurisubharmonic function, 338, 340 minimal surface, 92 minimal varifold, 394 Morse critical point, 19 Morse function, 20 Morse index, 19 Morse normal form, 19 Morse–Sard lemma, 17 multiplier, 234 negative crossing, 219 nice critical point, 363 nondegenerate conformal minimal immersion, 106 nondegenerate map, 106 nonflat generalized conformal minimal immersion, 134 nonflat generalized null curve, 135 nonflat map, 106 nonorientable surface, 6 normal curvature, 85 normal exhaustion, 67 normal space, 84 normal variation, 94 normal vector, 84 normal vector field, 84 null hull, 383 null plurisubharmonic function, 351 null positive current, 404 null positive form, 404 null pseudoconvex domain, 354, 359 null quadric, 47, 98 null strongly plurisubharmonic function, 351 null vector, 47 null-positive Green current, 379 Oka–Grauert principle, 77 Oka–Grauert theorem, 65 Oka manifold, 75 open Riemann surface, 23 open surface, 5 orientable manifold, 8 orientable surface, 6 orientation, 8 oriented atlas, 8 pair of pants, 22 parabolic Riemann surface, 60 parametric h-principle, 76 parametric Oka principle, 76 period dominating multiplier, 235, 236

429 period dominating spray, 136, 207 period map, 136, 234 Perron’s method, 373 Picard group, 27 piecewise regular curve, 219 pluricomplex Laplacian, 32 pluriharmonic function, 32 plurisubharmonic hull, 381 Poincar´e metric, 91, 252 Poisson functional, 385 Poletsky sequence, 381 polynomial hull, 66, 381 polynomially convex set, 66 positive crossing, 219 positive current, 401 positive form, 401 primitive immersion, 221 primitive pair, 226 principal curvatures, 86 principal divisor, 61 principal normal vector of the curve, 84 projective manifold, 24 projective space, 24 proper map, 5 pullback, 13 pullback metric, 34 puncture, 59 pure dimension, 25 pushforward, 397 quasiconformal map, 51 quasidivergent path, 306 real vector bundle, 11 rectifiable current, 97 rectifiable set, 394 rectifiable varifold, 394 regular curve, 219 regular function, 380 regular homotopy, 185 regular neighbourhood, 69 regular point, 16 regular upper semicontinuous function, 340 regular value, 16 residue, 55 Riemann–Hilbert boundary value problem, 265 Riemann–Hilbert deformation, 266 Riemann–Hurwitz formula, 56 Riemannian manifold, 33 Riemannian metric, 33 Riemann–Koebe uniformization theorem, 53 Riemann–Roch theorem, 62 Riemann’s minimal examples, 123

430 Riemann surface, 8, 52 Runge set, 66 Scherk’s surfaces, 120 Schwarz reflection principle, 102, 230 Schwarz surfaces, 124 second variational formula, 96 section, 26 self-dual manifold, 323 separating hyperplane theorem, 152 Serre duality, 64 Serre fibration, 190 set of first category, 17 set of second category, 17 shape operator, 88 simple k-vector, 3 simple double point, 144 singular points, 219 smooth map, 8 solution of the Dirichlet problem, 373 spherical metric, 35 spinorial representation, 187 stable minimal surface, 97 standard handle, 20 standard volume form, 14 stationary current, 398 Stein compact, 74 Stein manifold, 23 stereographic projection, 36 Stokes’s theorem, 15 strictly stable minimal surface, 97 strongly minimally convex domain, 347 strongly p-convex domain, 347 strongly plurisubharmonic function, 32 strongly p-plurisubharmonic function, 342 sub-mean value property, 340 subharmonic function, 340 sublinear asymptotic growth, 196 submanifold, 12 submersion, 12 submersive family of maps, 18

Index subsolution of the Dirichlet problem, 374 superminimal surface, 323 supersolution of the Dirichlet problem, 374 support of current, 395 symmetry, 230 tangent bundle, 9 tangent covector, 12 tangent space, 9 tangent vector, 9 three points problem, 201 time-t map, 11 topological manifold, 5 topological surface, 5 total curvature, 109 transition map, 8 transverse map, 17 turning number, 220 twistor bundle, 324 unit normal vector field, 84 upper semicontinuous function, 339 variation, 92 variational vector field, 92 varifold, 102, 394 vector bundle atlas, 10 vector bundle chart, 10 vector field, 11 Vitushkin property, 69, 74 volume, 37 volume element, 37 volume form, 14 weak homotopy equivalence, 76 weak topology, 395 wedge product, 2 Weierstrass data, 101 Weingarten map, 88 winding number, 219 Wirtinger’s inequality, 44