Geometry of Holomorphic Mappings (Frontiers in Mathematics) 3031371488, 9783031371486

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Frontiers in Mathematics

Sergey Pinchuk Rasul Shafikov Alexandre Sukhov

Geometry of Holomorphic Mappings

Frontiers in Mathematics Advisory Editors William Y. C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Sprößig, TU Bergakademie Freiberg, Freiberg, Germany

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Sergey Pinchuk • Rasul Shafikov • Alexandre Sukhov

Geometry of Holomorphic Mappings

Sergey Pinchuk Department of Mathematics Indiana University Bloomington Bloomington, IN, USA

Rasul Shafikov Department of Mathematics University of Western Ontario London, ON, Canada

Alexandre Sukhov Department of Mathematics University of Lille Villeneuve d’Ascq, France

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-031-37148-6 ISBN 978-3-031-37149-3 (eBook) https://doi.org/10.1007/978-3-031-37149-3 Mathematics Subject Classification: 32-02, 32H40, 32H02, 32D15, 32T15 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Introduction

The subject of this book is holomorphic mappings between domains in .Cn . The theory of holomorphic (conformal) mappings is a cornerstone in the classical geometric function theory of one complex variable, and it is an important tool for applications in many areas of mathematics. Such applications to a great extent are based on the results in the following two directions: the fundamental Riemann Mapping theorem, which establishes conformal equivalence of simply connected domains in .C, and the Schwarz reflection principle together with further major contributions by Carathéodory, Kellogg and others, which provide a comprehensive description of boundary behaviour of conformal mappings. The goal of this book is to present some substantial results concerning holomorphic mappings in several complex variables. In contrast to the case .n = 1, for .n > 1 the dominating feature of holomorphic mappings is their rigidity which may appear in various forms. For instance, in dimension bigger than 1, different domains are not in general biholomorphically equivalent. But when they are, it may lead to certain unusual phenomena. The first results of such kind were obtained in the beginning of the twentieth century by H. Poincaré who proved that (a) the unit ball .B2 and the bidisk .2 in .C2 are not biholomorphically equivalent; (b) if p is a point in the sphere .S = bB2 and .fp is the germ of a biholomorphic map at p which maps the germ of S to S, then .fp extends to a biholomorphic map .f : B2 → B2 . Subsequent investigation of holomorphic mappings was largely focused on boundary regularity of biholomorphic and proper holomorphic mappings, the problem of holomorphic equivalence of real hypersurfaces and various rigidity phenomena in .Cn . The investigation of boundary regularity of biholomorphic and proper holomorphic mappings is one of the central themes of the book. A general goal in this context is to show that any biholomorphic map between domains in .Cn with .C ω -, or m ω s .C -smooth boundaries extends as a .C -, or .C -smooth map between their closures for some .s = s(m). This problem remains open in full generality despite numerous partial positive results, and there are no known counterexamples. Research in this direction led to creation of many new methods that were successfully applied to other problems in SCV and that remain to be of independent interest. One of these methods is an analytic approach based on regularity properties of the Bergman projection and subelliptic estimates of the .∂-Neumann problem. There are many excellent monographs that cover this subject and so it is only natural for us to concentrate on alternative, more geometric methods. This v

vi

Introduction

geometric approach forms the core of the book and includes invariant metrics, holomorphic correspondences, the Scaling method and the Reflection principle. Some minimal background material is given in Chap. 1 with a few more technical proofs deferred to the Appendix. In Chap. 2, we present a number of striking results in several complex variables that illustrate the importance of boundary regularity in the development of other aspects of the theory of holomorphic mappings. In Chap. 3, we establish the estimates for the invariant metrics that are used to prove continuous extension of holomorphic mappings between strictly pseudoconvex domains. In Chaps. 4 and 5, we discuss smooth and holomorphic extension of biholomorphic and proper holomorphic maps between smoothly bounded strictly pseudoconvex domains. This requires some additional tools, such as the Scaling method. A local version of the extension is proved in the language of holomorphic extension of CR mappings. It should be noted that the results obtained in these chapters, like all the results proved in the book, are independent of the properties of the Bergman spaces or the .∂-Neumann problem methods mentioned above. The Scaling method proved to be a powerful tool in analysing domains with noncompact automorphism groups, and this is the subject of Chap. 6. Another important topic of the book is the biholomorphic equivalence problem of real analytic hypersurfaces in .Cn and the domains that they bound. For biholomorphisms with nice boundary behaviour, the equivalence of domains translates to the equivalence of their boundaries. In 1974, Chern and Moser in different terms obtained necessary and sufficient conditions for local equivalence of two real analytic hypersurfaces with nondegenerate Levi form. This theory is briefly discussed in Chap. 7. The construction of local invariants makes it possible in certain cases to decide whether the germs of boundaries of two domains are locally equivalent. While it may not be enough to conclude that the domains are themselves equivalent, the result by Poincaré above shows that in the case of the unit balls, a local equivalence of the boundaries extends to a global biholomorphism between the balls. In Chap. 8, such rigidity phenomenon of analytic continuation of germs of mappings along hypersurfaces is extended to real analytic strictly pseudoconvex hypersurfaces. This allows us to claim that local equivalence of germs of the boundaries of certain domains implies the global equivalence of these domains. Various modifications of the Reflection principle are discussed and used in the book for analytic continuation of mappings. The most advanced results of this kind were obtained by using the technique of Segre varieties, these are introduced in Chap. 9. In Chap. 10 we discuss holomorphic correspondences—multiple-valued maps that naturally appear in the problems of analytic continuation. We also prove a result concerning critical sets of holomorphic mappings. Chapter 11 contains one of the central results of the book: a proper holomorphic map .f : D → D  between bounded domains in .Cn with real analytic boundaries that extends continuously to .D necessarily extends holomorphically to a neighbourhood of .D. Finally, in Chap. 12, we prove that in dimension .n = 2, the map f extends holomorphically without the assumption of continuous extension. These results do not assume any pseudoconvexity, and the proofs heavily rely on the geometric

Introduction

vii

properties of the Segre varieties associated with the boundaries of the domains, which shows the power of this approach. In a short monograph is it impossible to mention all the results pertinent to the subject. Our choice of the material for the book is heavily influenced by our personal taste in the subject and the wish to keep the book largely self-contained and readable. The third author is partially supported by Labex CEMPI and the Institute of Mathematics of the Ufa Research Centre. Bloomington, IN, USA London, ON, Canada Villeneuve d’Ascq, France

Sergey Pinchuk Rasul Shafikov Alexandre Sukhov

Contents

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Classes of Functions and Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Smooth and Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 CR Manifolds and CR Functions in Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 CR Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 CR Functions and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Totally Real Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Real Analytic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Pseudoconvex Domains, Defining and Exhaustion Functions, the Hopf Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 5 5 6 8 8 10

2

Why Boundary Regularity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Schwarz Reflection Principle for Strictly Pseudoconvex Domains . . 2.2 The Poincaré-Alexander Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Wong-Rosay Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 21

3

Continuous Extension of Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Kobayashi Distance and Kobayashi-Royden Pseudometric . . . . . . . . . . . . . . . 3.2 Boundary Estimates for the Kobayashi-Royden Metric . . . . . . . . . . . . . . . . . . . 3.3 Other Invariant Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Continuous Extension of Holomorphic Mappings. . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 30 31 35

4

Boundary Smoothness of Holomorphic Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Scaling Method and Continuous Extension of the Lift . . . . . . . . . . . . . . . 4.1.1 The Scaling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Continuous Extension of the Lift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Boundary Regularity of Discs Near a Totally Real Manifold . . . . . . . . . . . . .

37 37 38 39 41

11

ix

x

Contents

4.3 4.4

Gluing Complex Discs and the Proof of Theorem 4.1. . . . . . . . . . . . . . . . . . . . . Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 46

5

Proper Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Holomorphic Correspondences and Algebroid Functions. . . . . . . . . . . . . . . . . 5.2 Proper Holomorphic Mappings of Strictly Pseudoconvex Domains . . . . . . 5.3 Local Invertibility of CR Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Comments and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 52 56

6

Uniformization of Domains with Large Automorphism Groups . . . . . . . . . . . . . 6.1 Existence of Parabolic Subgroups of Automorphisms . . . . . . . . . . . . . . . . . . . . 6.2 Geometry Near a Parabolic Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Parabolic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 66 70 73

7

Local Equivalence of Real Analytic Hypersurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Complexification, Segre Varieties, and Differential Equations . . . . . . . . . . . 7.2 Equivalence Problem I: Moser’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Local Equivalence II: The Cartan-Chern Approach . . . . . . . . . . . . . . . . . . . . . . . 7.4 Fefferman’s Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 79 82 83

8

Geometry of Real Hypersurfaces: Analytic Continuation . . . . . . . . . . . . . . . . . . . . 85 8.1 Extension of Germs of Holomorphic Mappings, I: The Spherical Case . . 87 8.1.1 The Reflection Principle Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.1.2 Extension Across Generic Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.1.3 Holomorphic Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.1.4 Analytic Continuation Along Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.2.1 Continuous Extension to M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2.2 Analytic Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.3 From Local to Global Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8.4 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

9

Segre Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Geometry of Segre Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Compact Real Analytic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Analytic Continuation Using Segre Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Holomorphic Mappings of Real Algebraic Hypersurfaces . . . . . . . . . . . . . . . . 9.5 Comments and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 121 123 128 131

Contents

xi

10

Holomorphic Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Proper Holomorphic Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Invariance of Segre Varieties for Correspondences . . . . . . . . . . . . . . . . . . . . . . . . 10.3 From Multiple-Valued to Single-Valued Extension. . . . . . . . . . . . . . . . . . . . . . . . 10.4 Proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Critical Sets of Holomorphic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 135 138 141 147 153

11

Extension of Proper Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Examples and General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Extending the Map to a Dense Subset of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Correspondences that Extend the Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Extension as an Analytic Set Implies Holomorphic Extension . . . . . . . . . . . 11.5 Pairs of Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Extension Along Segre Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Final Steps in the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Comments and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 157 159 162 164 166 170 172

12

Extension in .C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Possible Extension as a Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Structure of the Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Extension Across a Dense Subset of the Boundary. . . . . . . . . . . . . . . . . . . . . . . . 12.4 Extension Across Strictly Pseudoconvex Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Proof of Theorem 12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Boundary Regularity: Some Historic Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 178 182 183 185 189

A

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Edge-of-the-Wedge Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 PSH Defining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Hörmander-Wermer Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Separate Algebraicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 195 197 199

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

1

Preliminaries

In this chapter we review some standard definitions and results in complex analysis and lay out the technical framework for the core material of the book. We assume that the reader is familiar with the classical monographs on several complex variables such as Gunning and Rossi [74], Hörmander [79], Range [119], or Shabat [129]. The reader who is well-versed in the subject may directly proceed to the next chapter.

1.1

Classes of Functions and Domains

In this section we fix some basic notation and recall elementary properties of holomorphic and plurisubharmonic functions.

1.1.1

Smooth and Holomorphic Functions

We denote by .z = (z1 , . . . , zn ) the standard complex coordinates in .Cn . We often use the (vector) notation .z = x + iy for the real and imaginary parts. We identify .Cn with .R2n via Cn  (z1 , . . . , zn ) ∼ = (x1 , y1 , . . . , xn , yn ) ∈ R2n .

.

By .|z| we denote the Euclidean norm of z, i.e., |z| =

.

 x12 + y12 + · · · + xn2 + yn2 ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_1

1

2

1 Preliminaries

 and by .z, w = j zj wj the standard Hermitian inner product in .Cn . Note that its real part .Rez, w defines the standard Euclidean scalar product of z and w viewed as vectors in .R2n . A domain . in .Cn is defined to be a nonempty connected open subset of .Cn . By .b we denote the boundary of .. The unit ball in .Cn is denoted by .Bn = {z ∈ Cn : |z| < 1}, while for .n = 1 we use the notation .D := B1 for the unit disc in .C. The ball .p + rBn of radius .r > 0 centred at a point .p ∈ Cn may be also denoted by .Bn (p, r) or simply .B(p, r) if the dimension is clear from the context. Another basic example of a domain in .Cn is the unit polydisc .Dn , or more generally, Dn (p, r) := p + rDn , p ∈ Cn , r = (r1 , . . . , rn ), rj > 0.

.

Finally, H = {z ∈ Cn : 2 Re zn + |z1 |2 + . . . + |zn−1 |2 < 0}

.

(1.1)

is an unbounded realization of the unit ball .Bn , it is biholomorphic to .Bn via Cayley’s transform  (z1 , . . . , zn ) →

.

2zn−1 zn − i 2z1 , ,..., zn + i zn + i zn + i

 .

(1.2)

We denote by .O(1 ) the space of holomorphic functions on a domain .1 . If .2 is another domain in .CN , we use the notation .O(1 , 2 ) to denote the space of holomorphic mappings from .1 to .2 . For a positive integer k, .C k () denotes the space of .C k -smooth complex-valued functions on .. Also .C k () denotes the class of .C k -smooth functions whose partial derivatives up to order k extend as continuous functions to .; this space becomes a Banach space if it is equipped with the standard norm ||f ||C k () =

.

k 

sup |D (j ) f (z)|,

|j |=0 z∈

where, as usual, .D (j ) denotes the partial derivative of order .j = (j1 , . . . , jn ) with .|j | = j1 + . . . jn , and .D (0) f = f . If .s > 0 is a real noninteger and k is its integer part, by .C s () we denote the space of functions of class .C k () such that their partial derivatives of order k are (globally) Höldercontinuous on . with the exponent .α := s − k; these derivatives automatically satisfy the global Hölder condition on ., so the notation .C s () for the same space of functions is also appropriate. These are Banach spaces equipped with the standard norm

1.1 Classes of Functions and Domains

3

|f (z) − f (w)| . |z − w|α z,w∈

||f ||C s () = ||f ||C k () + sup

.

Let . be a domain in .Cn . A biholomorphic map .f :  →  is called an automorphism of .. The set of all automorphisms form a group, which we denote by .Aut(), with the group operation being composition of maps. The group .Aut() is typically equipped with the topology of uniform convergence on compacts in .. The automorphism group carries important information about the domain, for example, it is well-known that the automorphism group of the unit ball .Bn , .n > 1, is different from that of the polydisc .Dn , and this implies that these domains are not biholomorphically equivalent. The following result is due to H. Cartan, see, e.g., [100]. Theorem 1.1 Let . ⊂ Cn be a bounded domain, and let .{fj } ⊂ Aut() be a sequence of automorphisms of .. Suppose that .{fj } converges uniformly on compact subsets of . to a holomorphic map .f :  → Cn . Then the following are equivalent. (i) .f ∈ Aut(). (ii) .f () ⊂ b. (iii) There exists .p ∈  such that the determinant of the Jacobian matrix, .det Df (p), is nonzero. Consider a domain . ⊂ Cn and a map .f :  → CN , not necessarily holomorphic or smooth. Let . be a subset of the boundary .b. The cluster set .C (f, ) of f on . is defined to be the set of all limit points of the sequences .{f (zk )} in .CN , where .{zk } is any sequence in . converging to a point in .. The cluster set .C (f, ) is empty if .lim |f (z)| = +∞ when .  z → . In what follows we usually omit . and simply write .C(f, ). A map .f : 1 → 2 is called proper if for every compact subset .K ⊂ 2 its pull-back −1 (K) is a compact subset in . . Note that a map .f :  →  between two domains .f 1 1 2 is proper if and only if the cluster set .C1 (f, b1 ) does not intersect .2 . For bounded domains in .Cn one can state this property in the equivalent form: .C1 (f, b1 ) ⊂ b2 . A continuous function .f :  → R is called an exhaustion function for . if the sublevel sets .f −1 ((−∞, c)) are relatively compact in . for all .c ∈ R. A bounded exhaustion function for . is a continuous proper function .f :  → (a, b) for some .a, b ∈ R.

1.1.2

Plurisubharmonic Functions

An upper semicontinuous function .u : D → R ∪ {−∞}, .u ≡

−∞, from a domain D ⊂ Rm , .m > 1, is called subharmonic if the following Mean value inequality holds for all .x ∈ D:

.

4

1 Preliminaries



1 .u(x) ≤ σm−1 r m−1

u(y)dσ (y), S(x,r)

where .r > 0 is so small that the sphere .S(x, r) centred at x of radius r, is contained in D, .σm−1 is the volume of the unit .(m−1)-sphere, and .dσ (y) is the volume measure on .S(x, r). A .C 2 -smooth function u is subharmonic iff .u ≥ 0 on D. If .f : Cn → C is a holomorphic function, then .Re f and .Im f are pluriharmonic functions, i.e., harmonic functions on n 2n such that their restrictions to any complex line remain to be harmonic. These .C = R are in particular subharmonic functions. A subset .E ⊂ D is called a polar set, if for any .p ∈ E there exists a neighbourhood V of p and a subharmonic function v on V such that .E ∩ V = v −1 (−∞). If E is a closed polar set in a domain .D ⊂ Rm , and u is subharmonic on .D \ E and is bounded above, then the function defined by u(x) = lim sup u(y)

(1.3)

.

E y→x

on E is, in fact, subharmonic. Thus, polar sets are removable singularities for subharmonic functions. The same result holds for pluriharmonic functions, see, e.g., [87, Thm 2.7.1, p. 53]. One of the applications of this is Rado’s theorem: if f is a continuous function on a domain . ⊂ Cn and holomorphic on . \ f −1 (0), then f is holomorphic on .. This can be seen by applying the removable singularity result to .Re f and .Im f and the polar set −1 (0) = {log |f | = −∞}. .f Let . ⊂ Cn be a domain. An upper semicontinuous function .u :  → R ∪ {−∞} is called plurisubharmonic if the restriction of u to any complex line in . is a subharmonic function or is identically .−∞. A .C 2 -smooth u is plurisubharmonic iff its complex Hessian given by Lu (p) =

.

 ∂ 2u (p) ∂zj ∂zk

(1.4)

j,k

is a positive-semidefinite form for all .p ∈ . A .C 2 -smooth function u is called strictly plurisubharmonic if the above form is positive-definite. We denote by .P SH () the class of plurisubharmonic functions on .. It is straightforward to verify that if f is a holomorphic function on ., then .|f |2 and .log |f | are plurisubharmonic on .. More generally, if .f :  →  is holomorphic, and .u ∈ P SH ( ), then .u ◦ f is plurisubharmonic on .. If . ⊂ Cn is a bounded domain and u is a plurisubharmonic function on . then for all .z ∈ 

 u(z) < sup

.

p∈b

lim sup u(w) . w→p

1.2 CR Manifolds and CR Functions in Cn

5

This Maximum principle also holds on relatively compact complex analytic sets of positive dimension. Pluripolar sets are defined similarly to polar sets, i.e., locally these are defined as the locus where a plurisubharmonic function takes the value .−∞. From the removable singularity result stated above for subharmonic functions one can deduce the following result. Theorem 1.2 Let . ⊂ Cn and .E = {z ∈  : v(z) = −∞} for some plurisubharmonic function v on .. If .u ∈ P SH ( \ E) is bounded above, then the extension of u defined on E by (1.3) is plurisubharmonic on .. In particular, complex analytic sets, being pluripolar, are removable singularities for bounded plurisubharmonic functions. Let . ⊂ Cn be a domain and .p ∈ b. A function .φ ∈ P SH () ∩ C() is called a peak function for . at p if .φ(z) < φ(p) for all .z ∈  \ {p}. A function .φ is called a local peak function if it is a peak function for the domain . ∩ U , where U is some neighbourhood of p. Holomorphic peak functions are defined in the same way. A point .p ∈ b is said to have a PSH-barrier property if there exist a neighbourhood U of p and constants .α ∈ (0, 1], .β > 1, and .M > 0 such that for any point .ζ ∈ U ∩ b there exists a function .φζ (z) continuous on . ∩ U and plurisubharmonic in . ∩ U that satisfies the condition .

− M|z − ζ |α ≤ φζ (z) ≤ −|z − ζ |β ,

(1.5)

for any .z ∈ ∩U . Thus, .φζ is a local plurisubharmonic peak function of algebraic growth.

1.2

CR Manifolds and CR Functions in Cn

In this section we recall some basic facts about CR manifolds and CR functions. We will use this throughout the book.

1.2.1

CR Manifolds

For .s ∈ [1, ∞] (resp. .s = ω) a closed real submanifold E of a domain . ⊂ Cn is of class s .C (resp. real analytic) if for every point .p ∈ E there exists an open neighbourhood U of p and a map .ρ = (ρ1 , . . . , ρd ) : U −→ Rd of maximal rank .d < 2n such that .ρj are of class .C s (U ) (resp. real analytic in U ), and E ∩ U = {z ∈ U : ρj (z) = 0, j = 1, . . . , d}.

.

(1.6)

6

1 Preliminaries

The functions .ρj are then called local defining functions of E. The positive integer d is the real codimension of E. In the fundamental special case .d = 1 we obtain the class of real hypersurfaces. Let  ∇z ρj (p) =

.

 ∂ρj ∂ρj (p) . (p), . . . , ∂zn ∂z1

Then the condition that .ρ has rank d simply means that the vectors .∇z ρj , .j = 1, . . . , d, are .R-linearly independent. The (real) tangent space .Tp E to E at .p ∈ E can be defined by the equations

 Tp E = w ∈ Tp Cn ∼ = Cn : Re w, ∇z ρj (p) = 0, j = 1, . . . , d .

.

Let J denote the standard complex structure of .Cn given by the multiplication by i, in other words, J acts on a vector .w ∈ Cn as .J w = iw. For every .p ∈ E the holomorphic (or complex) tangent space .Hp E := Tp E ∩ J (Tp E) is the maximal complex subspace of the tangent space .Tp E of E at p. Clearly,



Hp E = w ∈ Cn : w, ∇z ρj (p) = 0, j = 1, . . . , d .

.

(1.7)

The complex dimension of .Hp E is called the CR dimension of E at p and is denoted by CR- dimp E. A manifold E is called a (an embedded) Cauchy-Riemann or simply a CR manifold if its CR dimension is independent of .p ∈ E. A real submanifold .E of real codimension d in .Cn is called generic (or generating) if the complex span of .Tp E coincides with all of .Cn for all .p ∈ E. Note that every generic manifold of real codimension .d < n is a CR manifold of CR dimension .n−d. If a function .ρ = (ρ1 , . . . , ρd ) satisfies .

∂ρ1 ∧ . . . ∧ ∂ρd = 0,

.

(1.8)

or, equivalently, ∂ρ1 ∧ . . . ∧ ∂ρd = 0,

.

then the vectors .∇z ρj , .j = 1, . . . , d, are .C-linearly independent, and therefore .E = {ρ = 0} is a generic submanifold of .Cn .

1.2.2

CR Functions and Mappings

Let E be a generic manifold of real codimension d in .Cn , .d < n, defined by (1.6). Locally, consider tangent vector fields .Xj , .j = 1, . . . , n − d, on E (of type (1,0)) which form a basis in the space of local sections of the holomorphic tangent bundle H E.

1.2 CR Manifolds and CR Functions in Cn

7

A .C 1 -smooth function .f : E → C is called a CR function if it satisfies the first order PDE system on E X j f = 0, j = 1, . . . , n − d.

.

(1.9)

These are the tangential Cauchy-Riemann equations. If E is defined by (1.6) and .f˜ is any 1 n .C -smooth extension of f to a neighbourhood of E in the ambient .C , then the function f is CR iff ∂ f˜ ∧ ∂ρ1 ∧ · · · ∧ ∂ρd = 0 on E.

.

In particular, if W is a wedge in .Cn defined by W = {z ∈ U : ρj (z) < 0, j = 1, . . . , d}

.

with the edge E, and f is a holomorphic function in W that extends smoothly to E, then the restriction of f to E is a CR function. By Stokes’ formula the Eq. (1.9) can be rewritten in the equivalent form .[E](f ∂φ) = 0 for every test form .φ on E of bidegree .(n, n − d); over E, i.e., a linear functional on the space here .[E] denotes the current of integration  of test forms acting as .[E](ω) = E ω, where .ω is a form of degree .= dim E. In this weak formulation the notion of a CR function can be extended to the class of continuous or locally integrable functions on E. If E is a hypersurface (.d = 1) given by a defining function .ρ with .∂ρ/∂zn = 0, then the tangential Cauchy-Riemann operators can be written in the form Xj =

.

∂ρ ∂ ∂ρ ∂ − , j = 1, . . . , n − 1. ∂zn ∂zj ∂zj ∂zn

(1.10)

In general, not every CR function on a real hypersurface E extends to a one-sided neighbourhood of E (or to a wedge in higher codimension). However, locally a CR function can be represented as a “jump” of two holomorphic functions: if f is a CR function in a neighbourhood U of a point .p ∈ E ⊂ Cn , and E divides U into two connected components, say, .U + and .U − , then there exist functions .F ± ∈ O(U ± ) such that .f = F + −F − (understood in the proper sense if f is not continuous), see Chirka [35]. + , the envelope of holomorphy of the domain .U + , then the function .F + extends If .p ∈ U holomorphically to a neighbourhood of p, and we conclude that any CR function f extends holomorphically to a one-sided neighbourhood of E near p. This is the case, for example, if the Levi form of E (see Sect. 1.4) has at least one nonzero eigenvalue (the Lewy extension theorem). The most general result in this direction is due to Trépreau [143]. Theorem 1.3 If a real hypersurface .E ⊂ Cn does not contain any germs of complex hypersurfaces at a point .p ∈ E, then there exists a one-sided neighbourhood of p, say, − ⊂ Cn , such that all CR functions near p extend holomorphically to .U − . .U

8

1 Preliminaries

Finally, if .E1 and .E2 are smooth generic submanifolds in .Cn and .Cm respectively, with .codimE1 < n, then a continuous map .f : E1 → E2 is called a CR map, if it can be written in the form .f = (f1 , . . . , fm ), where .f : E1 → C are CR functions and .f (E1 ) ⊂ E2 . As an example, a biholomorphic map .f : 1 → 2 between smoothly bounded domains in n .C that extends continuously to .1 defines a CR map .f |b1 : b1 → b2 .

1.3

Totally Real Manifolds

Of special importance are the so-called totally real manifolds, i.e., submanifolds E for which .Hp E = {0} at every .p ∈ E. Clearly, the dimension of a totally real submanifold in n .C does not exceed n. A totally real manifold is generic if and only if its real dimension is equal to n, i.e., maximal possible. Such manifolds are called maximally totally real, the simplest such example is the real subspace Rn = {z = x + iy ∈ Cn : y = 0}.

.

The simplest example of a compact maximally totally real manifold is the torus S 1 × S 1 ⊂ C2 .

.

1.3.1

Real Analytic Case

An open cone in .Rn is an open set C such that .y ∈ C implies .ty ∈ C for all .t > 0. Given a cone C, let .V = C ∩ B(0, 1), and let E be a domain in .Rn . We define W + = E + iV , W − = E − iV .

.

(1.11)

The sets .W + and .W − are called the wedges with the edge E. In general, the set .W + ∪ W − ∪ E does not contain an open neighbourhood of E in .Cn . The following is a simple version of the classical edge-of-the-wedge theorem. Theorem 1.4 For E, .W + and .W − as above, there exists an open neighbourhood . of E in .Cn with the following property: Every continuous function f on .W + ∪ W − ∪ E, which is holomorphic on .W + ∪ W − , extends holomorphically to .. The proof is given in Sect. A.1 of the Appendix. Different proofs can also be found, for example, in [155] and [124]. Further generalizations also exist: one can take E to be a

1.3 Totally Real Manifolds

9

maximally totally real submanifold, and replace continuity of f on E with convergence on E in the sense of distributions, see Pinchuk [106]. For our purposes, however, the above formulation will be sufficient. Real analytic totally real submanifolds can be locally viewed as subsets of .Rn . In fact, the following holds. Proposition 1.5 Let E be a real analytic totally real submanifold of dimension n in .Cn . For every point .p ∈ E there exists an open neighbourhood . in .Cn and a map . :  → Cn , which is a biholomorphism onto its image, such that . (p) = 0 and . (E ∩ ) = Rn ∩ (). Proof The proof is an application of complexification of real analytic functions. After a linear holomorphic change of coordinates we may assume that in a neighbourhood of the origin the set E is the graph of .y = φ(x), where .φ is a real analytic map from a neighbourhood of the origin in .Rn , .φ(0) = 0 and .dφ(0) = 0. We complexify this equation by replacing .z = x−iy with an independent complex variable w. Then, in a neighbourhood of the origin in .Cnz × Cnw , we obtain a holomorphic equation z−w −φ . 2i



z+w 2

 = 0.

By the Implicit function theorem this equation can be resolved for w to obtain w = z + ψ(z),

.

where .ψ is holomorphic near the origin, .ψ(0) = 0, and .dψ(0) = 0. By substituting w with .z in the above equation we see that E is defined by the equation .z = z + ψ(z). Let . (z) = 2z + ψ(z). By construction, . is a biholomorphic map in a neighbourhood of the origin in .Cn . Further, if .z ∈ E, then .

Im (z) = Im(z + (z + ψ(z)) = Im(z + z) = 0.

This means that E is mapped by . onto a neighbourhood of the origin in .Rn .

 

The above proposition simplifies many aspects of complex analysis near real analytic totally real submanifolds of maximal dimension. As an example, let E be a totally real submanifold defined by E = {p ∈ Cn : ρj (p) = 0, j = 1, . . . , n}.

.

(1.12)

10

1 Preliminaries

Since .{ρj = 0} and .{ρk = 0} intersect transversely for .j = k, the set W = {p ∈ Cn : ρj < 0, j = 1, . . . , n}

(1.13)

.

contains a (linear) wedge with the edge E as defined in (1.11). The converse to this statement is also true, and so W in (1.13) is also called a wedge with the edge E. The next result gives a simple multidimensional version of the classical Schwarz reflection principle. Proposition 1.6 Let .E1 and .E2 be real analytic totally real manifolds of dimension n and N in .Cn and .CN respectively, .n, N > 1. Suppose that W is a wedge with the edge N is a map holomorphic on W , continuous on .W ∪ E and such that .E1 , .f : W → C 1 .f (E1 ) ⊂ E2 . Then f extends holomorphically to a neighbourhood of .E1 . Proof By Proposition 1.5 one can assume that .E1 and .E2 coincide with open pieces of n N respectively. The map .f ∗ (z) = f (z) is holomorphic in a wedge with the edge .R and .R n n .R that is opposite to W and coincides with f on .R . We now may apply the edge-of-the  wedge theorem (Theorem 1.4), and this proves the result.

1.3.2

Smooth Case

Let M be a totally real manifold of class .C 2 in .Cn . By .dist(M, z) we denote the standard Euclidean distance from a point .z ∈ Cn to a set M. The following well-known result is often useful, see Wells [160] and Hörmander and Wermer [80]. Lemma 1.7 There exists a neighbourhood . of M in .Cn such that the function φ(z) = [dist(M, z)]2

.

is strictly plurisubharmonic in .. The lemma clearly holds if .M = Rnx . Indeed, in this case .φ(z) = plurisubharmonic.



yj2 , which is strictly

Proof Let .k = dim M, and .p ∈ M be an arbitrary point. Since M is totally real, after a biholomorphic change of coordinates in .Cn we may assume that .p = 0 and the tangent space to M at .p = 0 is given by T0 M = {(z1 , . . . , zn ) ∈ Cn : y1 = · · · = yk = zk+1 = · · · = zn = 0},

.

1.4 Pseudoconvex Domains, Defining and Exhaustion Functions, the. . .

11

where .zj = xj + iyj , .j = 1, . . . , n. Let .δ(z) be the normal distance from a point z in a neighbourhood of the origin in .Cn to .T0 (M). It is easy to see that .dist(M, z) = δ(z) + O(|z|2 ), and moreover, δ 2 (z) =

k 

.

j =1

yj2 +

n 

|zj |2 ,

j =k+1

which has a positive-definite complex Hessian in .Cn , see (1.4). Then .φ(z) = δ 2 (z) + O(|z|3 ), and so it also has a positive-definite complex Hessian in some neighbourhood of the origin. By taking the union of sufficiently small neighbourhoods of all points in M we   obtain the required neighbourhood . of M. We also need the following lemma due to Hörmander and Wermer [80]. Lemma 1.8 Let .M ⊂ Cn be a maximally totally real manifold of class .C k , .k ≥ 1, .k ∈ R, with .0 ∈ M and .T0 M = Rn . For any function .f ∈ C k (M) there exists an extension .f˜ in a neighbourhood of 0, which is of class .C k on M and infinitely differentiable outside M and which has the property that the coefficients of .∂ f˜ vanish on M to order .k − 1 in a neighbourhood of the origin. The proof is given in Sect. A.3 of the Appendix.

1.4

Pseudoconvex Domains, Defining and Exhaustion Functions, the Hopf Lemma

Let . be a bounded domain in .Cn . Suppose that its boundary .b is a (compact) real hypersurface of class .C s in .Cn , .s ≥ 1. Then there exists a .C s -smooth real function .ρ in a neighbourhood U of the closure . such that . = {ρ < 0} and .dρ|b = 0. We call such a function .ρ a global defining function. If .s ≥ 2 we may consider the Levi form of .ρ: Lρ (p, w) =

n 

.

j,k=1

∂ 2ρ (p)wj w k , w ∈ Hp b, p ∈ b. ∂zj ∂zk

(1.14)

If .σ = α · ρ is another defining function, then .α(p) = 0, and one can easily check that Lρ (p, w) = α(p)Lρ (p, w), for w ∈ Hp ∂.

.

(1.15)

12

1 Preliminaries

In particular, this means that the signature of the Levi-form is independent of the choice of the defining function. Another important property of the Levi form is that it is invariant under biholomorphic change of coordinates: if .f : U  → U is a biholomorphic map and n .U ⊂ C is a neighbourhood of .b, then Lρ◦f (z, w) = Lρ (f (z), Df (z)w),

.

(1.16)

where .Df (z) is the differential of f . For a .C 2 -smooth defining function .ρ the real hypersurface .M = {ρ = 0} is called Levi-nondegenerate, if the Levi form of .ρ is nondegenerate at every point of M. In a neighbourhood of a Levi-nondegenerate point there exists a local change of coordinates such that in the new coordinates .0 ∈ M and near 0 the hypersurface M is defined by the function ρ(z) = 2xn + Lρ (0, z) + o(|z|2 ).

.

(1.17)

In particular, if the Levi-form is positive-definite, the defining function can be chosen to be ρ(z) = 2xn +

n−1 

.

|zj |2 + o(|z|2 ).

(1.18)

j =1

We refer to (1.17) and (1.18) as a normal form of the defining function. A bounded domain . with .C 2 -smooth boundary is called pseudoconvex (resp. strictly pseudoconvex) if .Lρ (p, w) ≥ 0 (resp. .> 0) for all .p ∈ b and every .w ∈ Hp ∂ (resp. for every nonzero .w ∈ Hp ∂). For domains with .C 2 -smooth boundary this definition of pseudoconvexity coincides with the general notion of pseudoconvexity in the sense of Grauert-Oka: . is pseudoconvex if and only if it can be exhausted by a sequence of strictly pseudoconvex domains, or, equivalently, iff . admits a strictly plurisubharmonic exhaustion function. A pseudoconvex domain . ⊂ Cn is a domain of holomorphy, i.e., .O() contains a function that does not extend analytically to any larger domain. If . is not pseudoconvex, then all functions in .O() extend across the nonpseudoconvex part of the boundary. Let . be a real hypersurface of class .C 2 in .Cn . One can view every holomorphic tangent space .Hp  as an element of the (complex) Grassmannian .G(n − 1, n) of complex hyperplanes in .Cn , which can be identified with the complex projective space .CPn−1 . Then the holomorphic tangent bundle .H () can be viewed as a real submanifold of dimension n .2n − 1 of the complex manifold .C × G(n − 1, n) of complex dimension .2n − 1. We call it the projectivization of the holomorphic tangent bundle and denote by .PH (). The following statement is due to Webster [159].

1.4 Pseudoconvex Domains, Defining and Exhaustion Functions, the. . .

13

Lemma 1.9 Let . be strictly pseudoconvex. Then .PH () is a maximally totally real manifold in .Cn × G(n − 1, n). Proof Clearly the problem is local. Any strictly pseudoconvex hypersurface is a local deformation of the sphere, and since being totally real is an open condition, it suffices to verify the statement for the sphere, or, more conveniently, for its realization .H given by (1.1). Using (1.7) we see that for any .z ∈ H, Hz H = {w ∈ Cn : w1 z1 + · · · + wn−1 zn−1 + wn = 0}.

.

Any such complex linear subspace can be uniquely identified with its complex normal vector (z1 , . . . , zn−1 , 1),

.

and so in the affine chart .(z, ζ ) ∈ Cn × Cn−1 of .Cn × G(n − 1, n) the set .PH () can be defined by .2n − 1 real equations zn + z n +

n−1 

.

zj zj = 0; ζk = zk , k = 1, . . . , n − 1.

j =1

Direct computation shows that the condition (1.8) is satisfied by the above system, and thus .PH () is generic. Finally, it is totally real because of the dimension considerations.   Every strictly pseudoconvex domain . admits a global plurisubharmonic defining function which is strictly plurisubharmonic on a neighbourhood of .b. Indeed, if .ρ is an arbitrary defining function of ., consider the function .ρ˜ = ρ + Kρ 2 , for .K > 0. By a direct computation one can show that the constant K can be chosen large enough so that .ρ is strictly pseudoconvex in a neighbourhood of .b. Then, for a small .ε > 0, the function .max(ρ, ˜ −ε) is a pseudoconvex defining function of . which is strictly pseudoconvex on .b. From this argument it also follows that if a domain . is strictly pseudoconvex near a point .p ∈ ∂, then there exists a local biholomorphic change of coordinates near p that transforms . into a strictly geometrically convex domain. This property does not hold in general for weakly pseudoconvex domains (i.e., pseudoconvex domains which are not strictly pseudoconvex). Indeed, Kohn and Nirenberg [90] and Diederich and Fornæss [43] constructed explicit examples of smoothly bounded pseudoconvex domains that do not admit even local plurisubharmonic defining functions. In particular, the famous “worm” domains of Diederich and Fornæss do not admit plurisubharmonic defining functions. However, Diederich and Fornæss [44] proved that pseudoconvex domains admit bounded plurisubharmonic exhaustion functions.

14

1 Preliminaries

Theorem 1.10 Let . ⊂ Cn be a bounded pseudoconvex domain with .C 2 -smooth boundary. Then there exist a defining function .r(z) of . and .0 < η0 < 1 such that for any .0 < η ≤ η0 the function ρ(z) = −(−r(z))η

(1.19)

.

is a bounded strictly plurisubharmonic exhaustion function on .. The proof is given in Sect. A.2 of the Appendix. In many applications it is important to connect the rate of growth of a defining function of a domain with the distance to the boundary of the domain. An instrumental role here is played by the classical Hopf lemma, which we formulate below. Lemma 1.11 Let .φ be a negative plurisubharmonic function in a bounded domain . ⊂ Cn with .C 2 -smooth boundary. Then there exists a constant .M > 0 such that |φ(z)| > Mdist(z, b) for all z ∈ .

.

Proof We first prove the lemma for the unit disc .D ⊂ C and a subharmonic .φ. Clearly, it suffices to establish the estimate away from any compact subset of .D, say, on .{1/2 < |z| < 1}. Let C = (2 log(1/2))−1 max φ(z) > 0.

.

|z|=1/2

(1.20)

˜ ˜ = φ(z) − C log |z| is subharmonic with .lim|z|→1 φ(z) ≤ 0, and Then the function .φ(z) .

˜ max φ(z) = max φ(z) − C log(1/2) =

|z|=1/2

|z|=1/2

1 max φ(z) < 0. 2 |z|=1/2

˜ < 0 for .1/2 < |z| < 1. It follows that for z close Thus, by the Maximum principle, .φ(z) to the unit circle, |φ(z)| ≥ −C log |z| ≥

.

C C (1 − |z|) = dist(z, bD). 2 2

For the case of a general domain . ⊂ Cn , observe that since .b is .C 2 -smooth, there exists .δ > 0 with the property that if .z ∈  and .dist(z, b) = ε ≤ δ, then the closed ball .B(z, ε) intersects .b at exactly one point .zb . Let .U = {z ∈  : dist(z, b) < δ/2}. For .z ∈ U , let .Lz be a unique complex line that passes through z and .zb . On .Lz ∩  we choose a point .zc such that .dist(zc , b) = δ. Then .D(zc , δ) = B(zc , δ) ∩ Lz is a complex disc in . that contains the point z and whose closure touches .b at one point .zb . We may apply the one-dimensional case proved above to .φ|D(zc ,δ) . Further, the estimates

1.4 Pseudoconvex Domains, Defining and Exhaustion Functions, the. . .

15

can be made uniform if in (1.20) the constant C is chosen using the maximum of .φ on {z ∈  : dist(z, b) = δ}. And this completes the proof.  

.

The following variation of the Hopf lemma plays an important role in many questions concerning boundary regularity of holomorphic maps. It originally appeared in Pinchuk and Tsyganov [114]. Proposition 1.12 Let . be a bounded domain with .C 2 -smooth boundary in .Cn and let K be a relatively compact open subset of .. For every constant .L > 0 there exists a constant .C = C(K, L) > 0 with the following property: if a function .u ∈ P SH () is such that .u(z) < 0 for every .z ∈  and .u(z) ≤ −L for all .z ∈ K, then |u(z)| ≥ Cdist(z, ∂) for all z ∈ .

.

Proof Let . (K, G) be the class of nonpositive plurisubharmonic functions .ψ on . satisfying .ψ|K ≤ −1. Define ω∗ (z, K, ) = limw→z {ψ(w) : ψ ∈ (K, G)}.

.

This is known as the .P-measure of the set K relative to .. It is known (see, e.g., [126]) that .ω∗ is a plurisubharmonic function which is either everywhere nonzero or vanishes identically. The latter happens iff the set K is pluripolar in .. By assumption, K has nonempty interior and so .ω∗ is a negative plurisubharmonic function. By Lemma 1.11, there exists .M > 0 such that |ω∗ (z, K, )| ≥ Mdist(z, b), z ∈ ,

.

from which it follows that |φ(z)| ≥ L|ω∗ (z, K, G)| ≥ LMdist(z, b).

.

 

2

Why Boundary Regularity?

In this chapter we present some classical results in several complex variables that relate boundary smoothness of domains in .Cn , .n > 1, with the geometric properties of holomorphic maps between these domains. The results motivate and determine the general subject of the book and outline the methods that will be developed. Some of them generalize their one-dimensional analogs, while other results describe higher-dimensional phenomena that distinguish several complex variables from complex analysis in dimension one. Our main goal is to explain how the boundary behaviour of holomorphic mappings is related to fundamental problems in complex analysis and geometry.

2.1

The Schwarz Reflection Principle for Strictly Pseudoconvex Domains

The classical Schwarz reflection principle in one variable provides holomorphic extension of a biholomorphic map between two domains in .C with real analytic boundaries. The boundary of each domain is a real analytic curve in .C. There are two ways to generalize this to higher dimensions. One is to view the curve as a 1-dimensional totally real manifold in .C. This approach leads to Proposition 1.6. The other way is to view the curve as a boundary of a domain, and so in higher dimensions it should be replaced with a real hypersurface in n .C . This point of view leads to far-reaching generalizations discussed in this book. One of them is the following multi-dimensional Reflection principle, obtained independently by Pinchuk [107] and Lewy [95]. Theorem 2.1 Let .f : 1 → 2 be a biholomorphic mapping between two strictly pseudoconvex domains with real analytic boundaries. Assume that f is of class .C 1 on .1 . Then f extends holomorphically to a neighbourhood of the closure .1 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_2

17

18

2 Why Boundary Regularity?

Proof This proof is due to Webster [159]. Consider the space .Cn × CPn−1 . For a point n .(z, p) in this space we identify p with a complex hypersurface in .C that passes through z. Recall that .PH (b1 ) denotes the projectivization of the holomorphic tangent bundle of .b1 as discussed in Sect. 1.4. By Lemma 1.9, .PH (b1 ) is a real analytic maximally totally real submanifold of .Cn × CPn−1 . Further, it is straight forward that in .Cn × CPn−1 the set n−1 contains a wedge with the edge .PH (b ). The same applies to . . .1 × CP 1 2 Let .F : 1 × CPn−1 → 2 × CPn−1 be the lift of the map f defined by F (z, p) = (f (z), Df (z) p).

.

Since f is of class .C 1 on .b1 , the map F extends continuously to .b1 × PH (b1 ), and since f is holomorphic in .1 , we have F (b1 × PH (b1 )) ⊂ b2 × PH (b2 ).

.

Now by Proposition 1.6, F extends holomorphically to a neighbourhood of .b1 × PH (b1 ) in .Cn × CPn−1 , which gives holomorphic extension of f to a neighbourhood   of .1 . The above theorem illustrates how the regularity of the holomorphic map on the boundary along with the regularity of the boundary itself leads to an analytic continuation of the map. This phenomenon can be generalized to a wider class of domains. We will return to this throughout the course of the book.

2.2

The Poincaré-Alexander Theorem

The following result is due to Poincaré [117] in dimension two and to Alexander [2] for general n. Theorem 2.2 Let .S 2n−1 be the real unit sphere in .Cn , .n > 1, and .U1 , .U2 be connected neighbourhoods of two points .p1 and .p2 of .S 2n−1 . Suppose that .f : U1 → U2 is a biholomorphic mapping such that f (S 2n−1 ∩ U1 ) ⊂ S 2n−1 ∩ U2 .

.

(2.1)

Then f extends to an automorphism of the unit ball .Bn . There are several ways to prove this result. The proof presented here is geometric in nature and has many important generalizations.

2.2 The Poincaré-Alexander Theorem

19

Proof The unit sphere is defined by the equation S 2n−1 = {z ∈ Cn : z, z =

n 

.

zj zj = 1}.

j =1

We consider its complexification by replacing the variable .z with an independent variable w. For a fixed .w ∈ Cn we consider the complex hyperplane

.

Qw = {z ∈ Cn : z, w = 1}.

.

It is called the Segre variety of the point w associated with the sphere. Segre varieties can be defined in a similar way for arbitrary real analytic hypersurfaces; their importance is due to the fact that the family of Segre varieties associated with a real analytic hypersurface is invariant under biholomorphic mappings. This will be discussed in detail in Chap. 9. Invariance of Segre varieties can be viewed as another multi-dimensional analog of the Schwarz reflection principle in one variable. Let us illustrate this for the map f as in the theorem. If .z ∈ U1 ∩ S 2n−1 , then z, z = 1 and f (z), f (z) = 1.

.

In the complexified space consider the complex hyperplane P =

.

⎧ ⎨ ⎩

(z, w) ∈ C2n :

 j

⎫ ⎬ zj w j = 1 . ⎭

Define .f ∗ (z) = f (z), and consider the holomorphic function h(z, w) := f, f ∗ − 1 =

n 

.

fj (z)fj∗ (w) − 1.

j =1

Then .h(z, w) vanishes on the totally real submanifold .{(z, w) ∈ Cn × Cn : z, z = 1, w = z}, which is a set of uniqueness for P . Hence, h vanishes identically on P . By the Weierstrass theorem, there exists a holomorphic function .g(z, w) such that (f (z), f (w) − 1) = g(z, w)(z, w − 1).

.

This means that f (Qw ∩ U ) ⊂ Qf (w) .

.

(2.2)

20

2 Why Boundary Regularity?

Note that when .n = 1, this is exactly the Schwarz reflection with respect to the unit circle in .C. To continue with the proof of the theorem, consider first the case .n = 2. When 3 2 .w ∈ U1 \ B2 is close enough to .S , the intersection of the complex line .Qw with .B is contained in .U1 . This intersection is some complex disc in .Qw . The map f takes this disc biholomorphically to the disc .Qf (w) ∩ B2 by the invariance property (2.2). Hence, the restriction of f to .Qw is a linear-fractional map by the standard result in one complex variable. We now may isolate two one-parameter families of .Qw (which are complex lines!) such that after a complex linear change of coordinates they become parallel to the coordinate axes in .C2 . What we showed is that the map .f (z1 , z2 ) satisfies the property that the restriction of f to .{z1 = const} or .{z2 = const} is linear-fractional. Now we apply the following classical result: if a function f is rational in each variable, then f is a rational function, see Theorem A.7 of the Appendix. Thus, f is the restriction of a rational map .F : C2 → C2 . The case of an arbitrary dimension follows by an induction on n. If . is the set of poles of the map F , then by the uniqueness theorem, F (S 2n−1 \ ) ⊂ S 2n−1 .

.

(2.3)

It follows that . ∩ Bn = ∅. Indeed, by (2.3) . may intersect .S 2n−1 only along the indeterminacy locus of F (i.e., the intersection of zeros and poles of F ). However, if . enters .Bn , then . ∩ S 2n−1 is too big to be contained in the indeterminacy locus. From this it follows that F is holomorphic in .Bn . Further, from (2.2), which by analyticity holds for F and all .w ∈ Bn , the map F is injective, and by (2.3) maps .Bn to itself. Hence, F is the   automorphism of .Bn . The theorem is false when .n = 1, and this shows remarkable rigidity of holomorphic mappings in several variables. A substantial part of the book is devoted to the extension of this phenomenon to a wider class of domains. In Theorem 2.2 the assumption on f can also be relaxed, for example, we may merely assume that f is holomorphic in .U1 ∩ Bn and extends as a .C 1 -smooth map to .S 2n−1 ∩ U1 so that (2.1) holds. Indeed, in this setting we may apply Theorem 2.1 to conclude that f is holomorphic in a neighbourhood of .S 2n−1 ∩ U1 . But what is the minimal initial boundary regularity assumption on the map f that guarantees the same conclusion? Classical Carathéodory theorem states that a conformal map .f : D1 → D2 between two domains in .C bounded by simple Jordan curves extends to a homeomorphism .f˜ : D 1 → D 2 . Further, by Kellogg’s theorem if .bD1 and .bD2 are of class .C m , .m ≥ 1, then f extends to .D1 as a diffeomorphism of class .C m−0 . What are the analogs of these results in higher dimensions? Another purely higher dimensional phenomenon concerns proper holomorphic mappings. The result, due to Pinchuk [105], is local, but for simplicity we formulate the global version.

2.3 The Wong-Rosay Theorem

21

Theorem 2.3 Let .f : 1 → 2 be a proper holomorphic mapping of class .C 2 (1 ) between strictly pseudoconvex domains in .Cn , .n > 1. Then the Jacobian determinant of f , .det Df , does not vanish on .1 . Proof Note that by the assumption the domains .j have .C 2 -smooth boundaries, .j = 1, 2. Let .ρj be a strictly plurisubharmonic defining function of .j (these exist, see Sect. 1.4). Then the function .u = ρ2 ◦ f is negative plurisubharmonic in .1 . Since .f ∈ C 2 (1 ), the function u extends smoothly to a neighbourhood of .1 . Applying the Hopf lemma (Lemma 1.11) to u, we see that .du = 0 on .b1 , and therefore, u is a .C 2 -smooth defining function of .1 . Since .1 is strictly pseudoconvex, we have Lu (z, w) > 0 for all z ∈ b1 , w ∈ Hz b1 , w = 0.

.

(2.4)

Suppose that .det Df (p) = 0 for some .p ∈ b1 . Then there exists a nonzero vector w ∈ Tp Cn , such that .Df (p)w = 0. Since .d(ρ2 ◦ f ) = 0, we conclude that .w ∈ Hp (b), and by the Eq. (1.16) we have

.

Lu (p, w) = Lρ2 (f (p), Df (p)w) = 0,

.

which contradicts (2.4). This argument proves that the complex hypersurface .{det Df = 0} does not intersect .b1 , which simply means that it is empty.   Finite Blaschke products in the unit disc of .C show that this theorem is false when n = 1. We will see that this purely higher dimensional phenomenon is closely related to Theorem 2.2. Theorem 2.3 is proved under the assumption of smoothness up to the boundary. Is it possible to weaken this assumption? For which classes of domains proper holomorphic mappings are in fact (locally) biholomorphic? We see again in the above theorems that boundary behaviour of holomorphic mappings is a key property that captures rigidity of holomorphic mappings in several variables.

.

2.3

The Wong-Rosay Theorem

The central result of the classical geometric function theory is the Riemann uniformization theorem. This does not admit a direct generalization to higher dimensions. We already observed that the unit ball .Bn , .n > 1, is not biholomorphic to the polydisc .Dn , and so the biholomorphic classification of domains (or even classes of domains) in .Cn is considerably more complicated than its topological counterpart. However, under certain additional assumptions biholomorphic equivalence to some model domains can be established. An example of that is the following result which was first proved by Wong [162] and Rosay [121].

22

2 Why Boundary Regularity?

Theorem 2.4 Let . be a bounded domain in .Cn and let .b be strictly pseudoconvex near a point .a ∈ b. Assume that there exists a point .z0 ∈ , and a sequence .(f k ) of holomorphic automorphisms of ., such that .f k (z0 ) → a as .k → ∞. Then . is biholomorphic to .Bn . Proof By Montel’s theorem, we may assume that the sequence .(f k ) converges uniformly on compact subsets of . to a holomorphic mapping f with .f (z0 ) = a. Let .ρ be a local strictly plurisubharmonic defining function of . near a. It follows by the Maximum principle that .ρ ◦ f ≡ a. Hence, .f ≡ a is a constant map. Without loss of generality assume that .a = 0, and that ρ(z) = Re zn + |z|2 + o(|z|2 )

.

near the origin. Set .a k = f k (z0 ) and let .bk ∈ b be the closest point to .a k . Note that k k k .a belong to the real inward normal of .b at .b . Let .L denote the composition of the k translation .b → 0 and a unitary transformation taking .Hbk b to the hyperplane .{wn = 0}. For every k, the map .g k := Lk ◦ f k sends the domain . biholomorphically onto some bounded domain .D k . Furthermore, .g k (z0 ) = (0, . . . , 0, −δk ) with .δk = |bk − a k | > 0. The defining function .ρk = ρ ◦ (Lk )−1 can be written in the from ρk = Re wn + ck |w|2 + o(|w|2 ),

.

where .ck → 1. Consider now the anisotropic dilations k : (w1 , . . . , wn−1 , wn ) →

.

w1 wn−1 wn √ ,..., √ , δk δk δk

,

and the maps F k := k ◦ g k .

.

˜ k = F k () is a domain defined by The image . φk = δ −1 ρk

.



δk w1 , . . . δk wn−1 , δk wn .

After the substitution we obtain φk (w) = Re wn + ck (|w1 |2 + . . . . + |wn−1 |2 + δk |wn |2 ) + δk o(|w|2 ).

.

(2.5)

2.3 The Wong-Rosay Theorem

23

It is easy to check that as .k → ∞ these functions converge uniformly on compact subsets of .Cn to the function .φ(w) = Re wn + |w1 |2 + . . . + |wn−1 |2 , the defining function of the domain .H, which, as discussed before, is biholomorphic to .Bn . We claim that .{F k } is a normal family. Indeed, if we write .F k = (F1k , . . . , Fnk ), it follows from (2.5) that .Fnk < 0; also one has .Fnk (z0 ) = −1 for all k. By Montel’s theorem, k k .{Fn } contains a subsequence (again denoted by .Fn for simplicity) converging uniformly on compact subsets of . to a holomorphic function .Fn . But then again from (2.5) we see that the other components of .F k are also uniformly bounded. This shows that .{F k } is a normal family, and so there is a subsequence that converges to a holomorphic map .F :  → H. Since the inverse maps .(F k )−1 are also uniformly bounded, we may assume that they converge to a holomorphic mapping . : H → . Finally, since .F (z0 ) = (0, . . . , 0, −1) and .(0, . . . , 0, −1) = z0 , it follows from Cartan’s theorem (Theorem 1.1) that .F :  →   H is a biholomorphism. The above method of dilation of coordinates (or the scaling method) will be used several times in this book. Theorem 2.4 provides uniformization by the unit ball of strictly pseudoconvex domains with large groups of automorphisms. Is it possible to solve the biholomorphic equivalence problem for some other classes of domains in several complex variables? In this chapter we raised a number of interesting questions concerning holomorphic mappings. The reader will find some answers in the next chapters. Some of them are definitive. Other ones are only partial and will require further investigation. It is the goal of this book to inspire further interest in this beautiful subject.

3

Continuous Extension of Holomorphic Mappings

The fundamental question concerning boundary regularity of holomorphic mappings is the following: suppose that .j , .j = 1, 2, are bounded domains in .Cn , .n > 1, with smooth boundaries. The exact required smoothness of .bj is part of the question, but generally we assume that they are at least of class .C 2 , while real analytic boundaries play a special role. Suppose that .f : 1 → 2 is a biholomorphic or proper holomorphic map. Does f extend to .b1 ? If such an extension can be proved, the regularity of f on the boundary will clearly depend on the regularity of the boundary itself. While the problem remains open in full generality, we will explore known results and the methods that they use. The first step in this journey is continuous extension up to the boundary, and the most effective tool for that is metrics that are invariant under biholomorphic maps. This is the subject of this chapter.

3.1

Kobayashi Distance and Kobayashi-Royden Pseudometric

For .ζj ∈ D set   ζ1 − ζ2 .δ(ζ1 , ζ2 ) =  1 − ζ ζ

1 2

  . 

Recall that the Poincaré distance on the unit disc .D in .C is defined by ρ(ζ1 , ζ2 ) =

.

1 1 + δ(ζ1 , ζ2 ) ln . 2 1 − δ(ζ1 , ζ2 )

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_3

25

26

3 Continuous Extension of Holomorphic Mappings

In the infinitesimal form it is given by |v| , 1 − |ζ |2

FD (ζ, v) =

.

(3.1)

so that the length of a curve .γ : [0, 1] → D can be computed as 

1

length(γ ) =

.

0

|γ  (t)| dt . 1 − |γ (t)|2

The Poincaré metric is invariant under conformal maps of the unit disc. The disc equipped with this metric is one of the primary models for hyperbolic geometry. Moving to higher dimensions, let . be a domain in .Cn . Given two points .p, q ∈ , a holomorphic chain in . from p to q is a finite family of holomorphic discs, i.e., holomorphic maps .hj : D → , and pairs of points .a j , .bj in .D, .j = 1, . . . , m, such that the following holds (1) .h1 (a 1 ) = p, .hm (bm ) = q; (2) .hj (bj ) = hj +1 (a j +1 ) for .j = 1, . . . , m − 1. The Kobayashi (pseudo) distance on . is the function .k :  ×  → R≥0 defined for any points .p, q ∈  by k (p, q) = inf

m 

.

ρ(a j , bj ),

j =1

where the infimum is taken over all (finite) holomorphic chains from p to q. In general, .k does not separate points and therefore it is only a pseudometric. Its fundamental property is that it is nonincreasing under holomorphic mappings: if .f : 1 → 2 is a holomorphic map between two domains in .Cn and .Cm respectively, then k2 (f (p), f (q)) ≤ k1 (p, q).

.

This is clear from the definition. In particular, if .f : 1 → 2 is a biholomorphic map, then f is an isometry of the Kobayashi distance function on .1 and .2 . Let . ⊂ Cn be a domain, .p ∈ , and .v ∈ Tp  be a tangent vector. The (infinitesimal) Kobayashi-Royden (pseudo) metric is defined by  F (p, v) = inf

.

 1 dh (0) = λv , : h(0) = p, dζ λ

(3.2)

3.2 Boundary Estimates for the Kobayashi-Royden Metric

27

where the infimum is taken over all holomorphic discs .h : D → . This metric is also contracting under holomorphic mappings, that is, if .f : 1 → 2 is a holomorphic map between two domains in .Cn and .Cm respectively, then F2 (f (p), Df (v)) ≤ F1 (p, v).

.

It follows from the Schwarz lemma in one variable that for the unit disc .D, .FD coincides with the infinitesimal Poincaré metric (3.1). From this it is easy to calculate that for the domain .B(0, r), .r > 0, and .v ∈ T0 B(0, r), FB(0,r) (0, v) =

.

|v| . r

(3.3)

Unlike any Riemannian metric, the Kobayashi-Royden metric in general is not smooth, but the following holds. Theorem 3.1 The function .F is upper semicontinuous on the tangent bundle .T  and  k (p, q) = inf

.

0

1

  dγ (t) dt, F γ (t), dt

where the infimum is taken over all smooth paths .γ : [0, 1] →  with .γ (0) = p and .γ (1) = q. A domain . is called hyperbolic if .k is a distance, in other words, if .k (p, q) > 0 whenever .p = q. It is immediate that any bounded domain in .Cn is hyperbolic. A hyperbolic domain is called complete if any ball of finite radius with respect to the Kobayashi distance is compactly contained in .. Any bounded strictly pseudoconvex domain in .Cn is complete, see Graham [71].

3.2

Boundary Estimates for the Kobayashi-Royden Metric

In applications it is important to establish the estimates for the Kobayashi-Royden metric from above and below, that is, to bound the magnitude of a vector by its Euclidean norm and the distance from the reference point to the boundary of the domain. Let . be a convex domain in .Cn . For any .p ∈  and .v ∈ Tp  denote by .L(p, v) the complex line passing through p in the direction of v and by .δ(p, v) the Euclidean distance from p to .L(p, v) ∩ b. The general result can be formulated as follows.

28

3 Continuous Extension of Holomorphic Mappings

Theorem 3.2 Let . be a convex domain in .Cn . For any .p ∈  and .v ∈ Tp  one has .

|v| |v| . ≤ F (p, v) ≤ δ(p, v) 2δ(p, v)

(3.4)

Note that in this theorem we do not assume that . is strictly convex. Proof The upper bound is straight forward. For .p ∈  let .r > 0 be such that the ball p + rBn is contained in .. Then from (3.3) it follows that

.

F (p, v) ≤ Fp+rBn (p, v) =

.

|v| , v ∈ Cn . r

(3.5)

Further, the estimate (3.5) can be rewritten in a more precise form F (p, v) ≤

.

|v| . δ(p, v)

(3.6)

Indeed, it suffices to replace the ball .p + dBn with the disc of maximal radius .δ(p, v) contained in .L(p, v) ∩ . For the lower bound, we first consider the domain  = {z ∈ Cn : Re z1 < 0}.

.

Let .p = (p1 , . . . , pn ) = (p1 ,  p) ∈ , .v = (v1 , . . . , vn ) = (v1 ,  v). Let 1 := {z1 ∈ C : Re z1 < 0}

.

be the left half-plane. Since . = 1 × Cn−1 and the Kobayashi distance .kCn−1 vanishes identically, we obtain that

k (p, v) = max k1 (p1 , v1 ), kCn−1 ( v, p) = k1 (p1 , v1 ),

.

and F (p, v) = F1 (p1 , v1 ) =

.

|v1 | . 2| Re p1 |

Observe that by similar triangles, .|v|/δ(p, v) = |v1 |/| Re p1 |. (Here .δ(p, v) is defined with respect to .). Therefore, F (p, v) =

.

|v| . 2δ(p, v)

3.2 Boundary Estimates for the Kobayashi-Royden Metric

29

Assume now that .L(p, v) is not contained in . (otherwise .δ(p, v) = ∞). Denote by q a point in .L(p, v) ∩ b which is the closest to p in the Euclidean metric, and denote by . the supporting half-space for . at q. By the observation above, we have .

|v| ≤ F (p, v) ≤ F (p, v), 2δ(p, v)

which gives the lower bound and completes the proof.



Theorem 3.2 can also be applied to strictly pseudoconvex domains. Proposition 3.3 Let . be a bounded domain in .Cn . Assume that the boundary .b is strictly pseudoconvex of class .C 2 near a point .z0 ∈ b. Then there exists a neighbourhood U of .z0 and a constant .C > 0 such that for every .p ∈ U ∩  and every .v ∈ Tp  one has .

C|v| |v| . ≤ F (p, v) ≤ dist(p, b) dist(p, b)1/2

(3.7)

Note that for vectors .v ∈ Tp  close to the normal direction of .b the proposition gives a better estimate with the exponent 1 for .dist(p, b). Proof After a biholomorphic change of coordinates in a neighbourhood U of p (see Sect. 1.4) one can assume that .p = 0 and . ∩ U is convex:  ∩ U = {z : Re zn + |z|2 + o(|z|2 ) < 0}.

.

In order to apply the above estimates we need to localize .F near p, that is, to compare it with .F∩U . Fix .0 < r < R small enough. Clearly, .F∩RBn (z, v) ≥ F (z, v) for any n .z ∈  ∩ rB and any tangent vector v. Since the boundary of . is strictly pseudoconvex, it follows that if a family of holomorphic discs .hk : D →  converges on compact subsets in .D to a holomorphic disc h that satisfies .h(0) ∈ b, then the limit disc is constant–a point of .b. From the definition of the Kobayashi-Royden metric we conclude that .

lim

(∩rBn ) z→0

F (z, v) → 1. F∩RBn (z, v)

(3.8)

Now we can conclude the proof. The second inequality is immediate. For the first inequality note that by elementary geometry there exists a constant .C  , which depends only on ., such that δ(p, v) ≤ C  (dist(p, b))1/2 , p ∈ , v ∈ Cn .

.

From this and Theorem 3.2 the required estimate follows.



30

3 Continuous Extension of Holomorphic Mappings

Note that any domain with a .C 2 -smooth boundary and with a .C 2 -defining function .ρ admits a global constant .C > 0 such that C −1 |ρ(z)| ≤ dist(z, b) ≤ C|ρ(z)|, z ∈ .

.

(3.9)

This allows one to replace .dist(z, b) with .|ρ(z)| in the estimates for the KobayashiRoyden metric.

3.3

Other Invariant Metrics

There are other useful metrics on complex manifolds that are invariant under biholomorphic maps. In this section we give two examples: the Carathéodory and the Sibony metrics. For two points .p, q ∈ , the Carathéodory (pseudo-)distance is defined to be c (p, q) =

.

sup

h∈O (,D)

ρ(h(p), h(q)).

The pseudo-distance .c (p, q) is a distance if, for example, . ⊂ Cn is bounded. One can recover the Carathódory distance from its semi-norm similarly to the case of the Kobayashi metric. The infinitesimal form of the metric can be computed to be FC (p, v) = sup {|Df (p)v| : f ∈ O(D, ), f (p) = 0} .

.

It is immediate that the Carathéodory distance is nonincreasing under holomorphic mappings, and it agrees with the Poincaré and Kobayashi metric in the unit disc .D. One can further show that c (p, q) ≤ k (p, q)

.

for any domain .. From this inequality one may expect that the estimates from below for the Carathéodory metric are generally more difficult to obtain than those for the Kobayashi metric; in fact, they are not always the same. However, the estimates of Theorem 3.2 and Proposition 3.3 also hold for the Carathéodory metric. But while the proof of the theorem remains the same, a localization of the Carathéodory metric required by the proposition is more subtle. One approach to this problem is based on holomorphic peak functions on strictly pseudoconvex domains. Let now . be a domain in .Cn , define ⎧ ⎪ ⎨

FS (p, v) = sup (∂∂(p)(v, v))1/2 ⎪ ⎩

.

⎞1/2 ⎫ ⎪ n ⎬ 2  ∂ u(p) ⎝ ⎠ = vj v k , ⎪ ∂zj ∂zk ⎭ ⎛

j,k=1

(3.10)

3.4 Continuous Extension of Holomorphic Mappings

31

where the supremum is taken over all plurisubharmonic functions on . with the following properties: .u(p) = 0, u is of class .C 2 near p, .log u is plurisubharmonic, and .0 ≤ u ≤ 1 on .. This an infinitesimal form of a (pseudo) metric that was introduced by Sibony [128]. Similarly to the Kobayashi and Carathéodory metric, one can show that the Sibony metric coincides with the Poincaré metric on .D and is nonincreasing under holomorphic maps. Further, one can prove that after a normalization we have the following inequality FC (p, v) ≤ FS (p, v) ≤ F (p, v).

.

3.4

Continuous Extension of Holomorphic Mappings

The following theorem is one of the first results on boundary behaviour of holomorphic mappings, see Margulis [96], Henkin [76], and Vormoor [156]. In this formulation it was obtained by Pinchuk [105]. Theorem 3.4 Let .f : 1 → 2 be a proper holomorphic mapping between two strictly pseudoconvex domains with .C 2 -smooth boundaries in .Cn . Then f extends to .1 as a 1/2-Hölder-continuous mapping. Proof Let .ρ and .ψ be strictly plurisubharmonic global defining functions of .1 and .2 respectively. The function .v(z) = ψ(f (z)) is negative plurisubharmonic on .1 . We have Cdist(z, b1 ) ≤ |v(z)| ≤ C −1 dist(f (z), b2 ).

.

(3.11)

Indeed, the first inequality follows from the Hopf Lemma applied to the function .v(z) on 1 , and the second inequality follows from (3.9) applied to the function .ψ. Let .f ⊂ 1 × 2 be the graph of f and .π : 1 × 2 → 2 be the natural projection. Since f is proper, .π : f → 2 is a (finite) branched analytic covering and its branch locus is a complex hypersurface . ⊂ 2 . The function

.

u(w) = sup{ρ(z) : f (z) = w}

.

is negative plurisubharmonic on .2 \ . Hence, by Theorem 1.2, it admits a unique extension as a plurisubharmonic function to .2 . By the construction of the function u, if .w = f (z), then .u(w) ≥ ρ(z), but since the functions on both sides are negative, we have .|ρ(z)| ≥ |u(f (z))|. Applying the Hopf Lemma to the function u on .2 , we obtain .|u(w)| ≥ Cdist(w, b2 ), which yields .Cdist(f (z), b2 ) ≤ |ρ(z)|. Applying (3.9) to .ρ then gives the following inequality Cdist(f (z), b2 ) ≤ |ρ(z)| ≤ C −1 dist(z, b1 ), z ∈ 1 .

.

(3.12)

32

3 Continuous Extension of Holomorphic Mappings

Combining (3.11) and (3.12) we deduce the following boundary distance preserving property C dist(z, b1 ) ≤ dist(f (z), b2 ) ≤ C −1 dist(z, b1 ).

.

(3.13)

From the nonincreasing property of the Kobayashi-Royden pseudometric and Proposition 3.3 we obtain .

C|Df (z)v| dist(f (z), ∂2 )−1/2 ≤ F2 (f (z), Df (z)v) ≤ F1 (z, v) ≤ C −1 |v| dist(z, ∂1 )−1 ,

for every point .z ∈ 1 and every tangent vector v. In view of (3.13) this yields the estimate ||Df (z)|| ≤ Cdist(z, ∂1 )−1/2

.

(3.14)

for the operator norm of the differential. The theorem now follows by the classical HardyLittlewood lemma discussed below.  Lemma 3.5 (Hardy-Littlewood) Suppose that .f : 1 → 2 is a smooth map between smoothly bounded domains in .Rn such that .||Df (z)|| ≤ Cdist(z, ∂1 )−1/2 . Then f extends as a .1/2-Hölder continuous map to .1 . Proof Let .z1 and .z2 be points in .1 and .l = |z1 − z2 |. One may assume that l is small enough and .zj are sufficiently close to .b1 . Let .pj ∈ b1 be a point closest to .zj . Then j j j j j .z belongs to the real inward normal .n to .b1 at .p . Fix a point .w which belongs to .n , j j j j j j j .|w − z | = l and .dist(w , b1 ) > dist(z , b1 ). Let .γ : [0, 1] → [z , w ] be an affine map with .γ j (0) = zj and .γ j (1) = w j . Then .dist(γ j (t), b1 ) ≥ tl, .t ∈ [0, 1], and by the assumption on Df we have     j j  . f (w ) − f (z ) ≤

1

C|dγ j (t)/dt|(tl)−1/2 dt ≤ Cl 1/2 .

0

Note that the length of the segment .[w 1 , w 2 ] does not exceed 3l, and .dist(w j , b1 ) ≤ l. A similar integration argument implies that .|f (w 1 ) − f (w 2 )| ≤ Cl 1/2 . Gathering these estimates together, we conclude that f is .1/2-Hölder continuous up to the boundary. This completes the proof.  The proof of Theorem 3.4 with minor modifications works for the case when the domain 1 is merely pseudoconvex: instead of the defining function .ρ one can use the bounded exhaustion function of Diederich and Fornæss (Theorem 1.10). The assumptions on f and .2 also can be weakened. The most difficult part of the proof is to derive estimates from .

3.4 Continuous Extension of Holomorphic Mappings

33

below for the Kobayashi-Royden metric on .2 . It was the idea of Diederich and Fornæss [47] to use plurisubharmonic peak functions of algebraic growth near a boundary point. Their method leads to the estimates for the Kobayashi-Royden metric for a wide classes of domains. In particular, the following holds Theorem 3.6 Let .1 ⊂ Cn be a bounded pseudoconvex domain with .C 2 -smooth boundary, and let .2 ⊂ Cn be a bounded pseudoconvex domain with real analytic boundary. Then a proper holomorphic mapping .f : 1 → 2 extends as a Hölder continuous map to .1 . The hard part of their proof provides a family of plurisubharmonic peak functions of polynomial growth near the boundary of .2 . There is a number of generalizations of this result in the literature, both global and local. One of them, that will be used later in the book, is the following local version. Theorem 3.7 Let .f : 1 → 2 be a holomorphic mapping between two bounded domains in .Cn and let the boundary .b1 be of class .C 2 and strictly pseudoconvex in a neighbourhood .1 ⊂ b1 of a point .z0 ∈ b1 . Assume that for each point .z ∈ 1 the cluster set .C(f, z) does not intersect .2 . Further assume that the cluster set .C(f, z0 ) contains a point .w 0 ∈ b2 such that .b2 is of class .C 2 and strictly pseudoconvex in a neighbourhood .2 of .w 0 . Then f extends as a .1/2-Hölder continuous map in a neighbourhood of .z0 in .1 . For the proof of Theorem 3.7 we first prove the following key lemma. Lemma 3.8 In the assumptions of Theorem 3.7 there exists .C > 0 such that C −1 dist(z, 1 ) ≤ dist(f (z), 2 ) ≤ Cdist(z, 1 )

.

(3.15)

for all .z ∈ 1 . Proof Let .ρ2 be a defining strictly plurisubharmonic function of .2 . Applying the Hopf Lemma to the negative plurisubharmonic function .ρ2 ◦ f defined on a one-sided neighbourhood of .z0 in .1 , we obtain the inequality on the left. The proof of the righthand inequality is more delicate because f is not necessarily proper. One can assume that .1 is strictly convex near .z0 . There exists a real sphere S touching 0 ε .1 from outside at .z . Denote by .S the translation of S by distance .ε > 0 (small enough) along the real inward normal direction to .1 at .z0 . Denote by .ε1 ⊂ 1 the domain whose boundary is formed by an open pieces of .1 and .S ε . We claim that for any .w ∈ 2 the intersection .A = f −1 (w) ∩ ε1 is at most finite (for any .ε > 0 small enough). Suppose that A is not empty. Since for .z ∈ 1 the cluster set .C(f, z) does not intersect .2 , the set A cannot have a limit point on .1 . Then the

34

3 Continuous Extension of Holomorphic Mappings

boundary .bA := A \ A is contained in .S ε , and so .A ∪ bA is a compact complex-analytic set with boundary on .S ε . We may apply the Maximum principle to the restriction to A of a translation of the function .|z|2 to conclude that A is a 0-dimensional. Let .a ∈ S ε be a limit point of A for some .ε. Then for .ε1 > ε the complex 0-dimensional analytic set A has an interior limit point .a ∈ ε11 . This yields a contradiction that proves the claim. Now fix .ε > 0. Let .ρ be a defining strictly plurisubharmonic function of .1 near .z0 . By strict pseudoconvexity, we can choose .ρ such that .ρ = −1 on .1 \ V , where V is a neighbourhood of .z0 . By choosing V sufficiently small, we may assume that .−1 ≤ ρ < 0 on .1 and .ρ = −1 on .S ε . Define the function .φ on .2 by letting φ(w) = sup{ρ(z) : f (z) = w}

.

for .w ∈ f (ε1 ), and .φ(w) = −1 for .w ∈ 2 \ f (ε1 ). The key step is to show that .φ is a continuous negative plurisubharmonic function on .2 . First we observe that since the map f is finite in .ε1 , for every point .z ∈ ε1 there exists a neighbourhood U such that .f : U → f (U ) is proper; in particular, f is open. Consider ε k .ω ∈ f ( ) and a sequence .ω , .k = 1, 2, .., converging to .ω. For .k = 0, 1, 2, . . . there 1 k −1 k exists .ζ ∈ f (ω ) with .ρ(ζ k ) = φ(ωk ). An arbitrary limit point a of the sequence ε k ε −1 (ω) and .ζ , .k = 1, 2, . . . belongs to .S or is an interior point of . . In any case, .a ∈ f 1 0 0 .ρ(a) ≤ ρ(ζ ) = φ(ω). Assume that .ρ(a) < ρ(ζ ). Then we may choose a neighbourhood U of .ζ 0 such that .f −1 (ω0 ) ∩ U = {ζ 0 } and the restriction .f : U → f (U ) is proper. By passing to a subsequence, we have .ζ k → a. For any sufficiently large k, there exists .bk ∈ f −1 (ωk ) ∩ U . By continuity, .ρ(ζ k ) < ρ(bk ), which contradicts the choice of .ζ k . Thus, .φ is continuous on .f (ε1 ). If .ω0 ∈ 2 is a boundary point for .f (ε1 ), then .φ(ω0 ) = −1. For k k ε .ω , .ζ and a defined as above, a similar argument shows that .a ∈ S and .ρ(a) = −1 = φ(a). This implies the continuity of .φ on .2 . It remains to verify that .φ satisfies the Mean value inequality (see Sec. 1.1.2). Choose ε 0 0 ∈ f −1 (ω0 ) ∩ ε and a neighbourhood U of .ζ 0 such that .ω ∈ f ( ). Also, choose .ζ 1 1 −1 (ω0 ) ∩ U = {ζ 0 }, .ρ(ζ 0 ) = φ(ζ 0 ) and .f : U → f (U ) is proper. The function .f ψ(w) = sup{ρ(z) : z ∈ f −1 (w) ∩ U }

.

is plurisubharmonic on U . It satisfies .ψ(ω0 ) = φ(ω0 ) and .ψ ≤ φ. Since .ψ satisfies the Mean value property at .ω0 , so does the function .φ. Finally, if .ω0 ∈ 2 is a boundary point for .f (ε1 ), we have a point .a ∈ S ε ∩ f −1 (ω0 ) with .ρ(a) = φ(ω0 ) = −1 and it suffices to repeat the above argument. Now, applying Proposition 1.12 to the function .φ in .2 , we conclude the proof of the lemma.  Under the assumptions of Theorem 3.7, let .U1 and .U2 be sufficiently small neighbourhoods of .z0 and .w 0 respectively. Using the inequality of Lemma 3.8 and arguing as in the proof of Theorem 3.4, we easily obtain the following.

3.5 Comments

35

Lemma 3.9 There exists a constant .C > 0 such that if .z ∈ 1 ∩ U1 and .f (z) ∈ 2 ∩ U2 , then .

 Df (z) ≤ Cdist(z, ∂1 )−1/2 .

To conclude the proof of the theorem it suffices to show that f extends continuously to .z0 . By assumption, there exists a sequence .(zj ) in .1 converging to .z0 and such that .f (zj ) converges to .w 0 . Arguing by contradiction suppose that f does not extend continuously to .z0 . Then one can find a sequence .(˜zj ) in .1 converging to .z0 such that dist(f (zj ), f (˜zj )) ≥ δ > 0,

.

and .f (˜zj ) tends to .w˜ 0 ∈ b2 ∩ U2 as .j → ∞. Here .δ > 0 is small enough so that the ball .(w 0 + δBn ) is contained in .U2 . For every couple of points .zj , .z˜ j consider a path j j j .γ constructed as in the proof of the Hardy-Littlewood lemma. Let .ζ be the point of .γ j j j j closest to .z and such that .|f (ζ ) − f (z )| = δ. Integrating along .γ , we obtain that |f (ζ j ) − f (zj )| ≤ Cdist(zj , z˜ j )1/2 −→ 0,

.

which is a contradiction. This proves the theorem.

3.5

Comments

The Kobayashi distance was introduced by Kobayashi [88], and its infinitesimal form (3.2) by Royden [123]. Estimates of the Kobayashi-Royden metric (3.4) appeared in Frankel [69], Graham [72], and Bedford and Pinchuk [17]. The key estimate from below (3.7) and its generalizations for a wider class of domains were obtained by several authors using various methods. One of them is analytic in nature and is based on the construction of families of barrier plurisubharmonic functions (with an algebraic growth near a boundary point). This uses the ideal theory of subelliptic multipliers developed by Kohn [89]. Such functions represent a useful tool for localization of the Kobayashi-Royden metric and its estimates from below. This idea goes back to the work of Diederich and Fornæss [47] and Sibony [128]. The key observation is that barrier plurisubharmonic functions in combination with the Maximum principle can be used to control the derivatives of holomorphic discs near a boundary point; this yields estimates on the metric. In various forms this approach was used by many other authors. In particular, this allows one to obtain estimates on pseudoconvex domains with real analytic boundary, or, more generally, on smoothly bounded pseudoconvex domains of finite type, see Cho [39]. Such estimates are similar to (3.7). Flexibility of plurisubharmonic functions makes this approach effective in many situations. Note that it is easy to construct barrier plurisubharmonic functions near strictly pseudoconvex boundary points. More general cases require subtle local analysis near a given boundary point.

36

3 Continuous Extension of Holomorphic Mappings

Another approach, more geometric, is based on the scaling argument presented in the proof of the Wong-Rosay theorem. The idea is to argue by contradiction. If the constants in the estimate (3.7) are not bounded above, then there exists a sequence of holomorphic discs converging to a boundary point with a strong distortion in the complex tangent and normal directions. Application of the scaling method (with respect to the centres of discs) yields a complex line in the limit domain .H, which gives a desired contradiction. This approach also allows one to obtain precise estimates of the Kobayashi-Royden metric on pseudoconvex domains of finite type in .C2 , see Berteloot [22]. The above-mentioned estimates of the Kobayashi-Royden metric naturally lead to generalizations of Theorem 3.4. For example, continuous extension of the map f can be established for a very important case when f is a proper holomorphic mapping from a smoothly bounded pseudoconvex domain onto a bounded pseudoconvex domain with real analytic boundary, see Diedrich-Fornæss [47]. A local version was proved by ForstneriˇcRosay [68], and Fornaess and Low [62]. The local version presented in Theorem 3.7 also admits several generalizations. The following result is due to Sukhov [139]. Theorem 3.10 Let .f : 1 → 2 be a holomorphic mapping between domains in .Cn , 2 0 .n > 1, not necessarily bounded. Assume that .b1 is .C -smooth near some point .z ∈ b1 with the PSH-barrier property and that there exists a neighbourhood U of .z0 such that 2 .C1 (f, z) is contained in .b2 for any .z ∈ U ∩ b1 . Assume also that .b2 is .C -smooth 0 0 0 near a point .w ∈ b2 with the PSH-barrier property and that .w ∈ C1 (f, z ). Then f extends to a mapping that is Hölder continuous in some neighbourhood of .z0 in .1 . The assumption of PSH-barrier property is satisfied, for example, when the boundaries are pseudoconvex, real analytic and do not contain complex curves near the source and the target points. More generally, this property holds if .b1 is smooth pseudoconvex in .C2 , or convex in .Cn , .n ≥ 2, and of finite type, as proved by Fornæss and Sibony [63].

Boundary Smoothness of Holomorphic Mappings

The central result of this chapter is the following Theorem 4.1 Suppose that .1 and .2 are strictly pseudoconvex domains with .C m smooth boundaries, .m > 2. Let .f : 1 → 2 be a biholomorphic map. Then f extends to .1 as a mapping of class .C m−1−0 . Recall that for a real noninteger .k > 0, .f ∈ C k−0 means that f has derivatives of order up to .[k] and the the derivatives of order [k] are Hölder continuous of order .k − [k]. If k is an integer, then .f ∈ C k−0 if f is of class .C k−ε for any arbitrarily small .ε > 0. To simplify some technical considerations, we will assume that .bj are .C ∞ -smooth, in which case the map f extends to .1 as a map of class .C ∞ . In this formulation the result is due to Ch. Fefferman [60]. Determining the precise smoothness of the extension has been one of the central questions in the subject and has shaped the development of the theory of holomorphic mappings for many decades. There are several different approaches to the proof of the above theorem. Our presentation follows the geometric ideas developed by Nirenberg et al. [103] and Pinchuk and Khasanov [111]. The proof is spread over Sects. 4.1 through 4.3.

4.1

The Scaling Method and Continuous Extension of the Lift

As a first step we show in this section that the lift of the map f into the projectivized complex tangent bundle over .1 extends continuously to the boundary. Our proof relies on Pinchuk’s scaling method [110].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_4

37

4

38

4.1.1

4 Boundary Smoothness of Holomorphic Mappings

The Scaling Method

We now outline a more general version, suitable for our purposes, of the scaling method that was introduced in the proof of Theorem 2.4. Let . be a domain with strictly pseudoconvex boundary of class .C 2 and a defining function .ρ near a point .w 0 ∈ b. By composing an affine map with a suitable quadratic transformation, for every .ω in a neighbourhood U of .w 0 in .Cn we construct a family of biholomorphic mappings .hω : Cn → Cn , continuously depending on .ω ∈ b ∩ U , so that the following conditions are satisfied: (i) .hω (ω) = 0. (ii) The defining function .ρω := ρ ◦ h−1 ω for the domain .hw () has the form ρω (w) = 2 Re wn + 2 Re Qω (w) + Hω (w) + Rω (w),

.

n n where .Hw (w) = μ,ν=1 hμ,ν (ω)wμ w ν , .Qω (w) = μ,ν=1 qμν (ω)wμ wν , and 2 .Rω (w) = o(|w| ). Furthermore, .Qω (w) = 0 and .Hω (w) = 0 when .wn = 0. (iii) Each mapping .hω sends the real normal line to .b at the point .ω to the real line .{w1 = . . . = wn−1 = Im wn = 0} which is normal to the boundary of the domain .hω (ω ) at the origin. In applications of this construction we may further assume that .w 0 = 0 and that .b is normalized near the origin, i.e., that .hw0 is the identity map. We set . w = (w1 , . . . , wn−1 ) so that .w = ( w, wn ). Consider a sequence of points .{w k } in ., .k = 1, 2, . . . , converging to a point .w 0 ∈ ∂. Denote by .ωk ∈ ∂ the point closest to .w k . Set .hk := hωk and .ρk := ρωk . Set .δk = dist(w k , b). Then .hk (q k ) = ( 0, −δk ), or more precisely, we replace .δk with the corresponding sequence keeping the same notation. Consider the sequence of dilations  : (w, wn ) →

.

k

w

wn √ , δk δk

 ,

and finally define the biholomorphic maps .D k := k ◦ hk . Note that this sequence of biholomorphic mappings is determined by . and the sequence .{w k }. We call the sequence −1 −1 k k k k .{D } the scaling of . along a sequence .{w }. Let . = D () = {δ k ρk ◦ dk < 0}, −1 where the factor .δk is introduced for additional normalization. It is easy to see that the sequence of functions .{δk−1 ρk ◦ dk−1 } converges uniformly on compact subsets of .Cn to the function φ(w) = 2 Re wn + | w|2 ,

.

(4.1)

which defines the domain .H given by (1.1). As a consequence, the sequence of domains {k } converges to .H with respect to the Hausdorff distance.

.

4.1 The Scaling Method and Continuous Extension of the Lift

39

In the proof of the Wong-Rosay theorem, given a sequence of biholomorphic mappings f k we considered the composition .D k ◦ f k . But the scaling construction may also be used to determine the properties of a single biholomorphism f between strictly pseudoconvex domains .j , .j = 1, 2, in .Cn . In this case this construction is used simultaneously in the source and in the target domains. Let .f : 1 → 2 be a biholomorphic mapping between two smoothly bounded strictly pseudoconvex domains. Recall that in this situation by Theorem 3.4 the map f extends as a .1/2-Hölder continuous map to .1 . Consider a sequence of points .(zk ), .k = 1, 2, . . . , that converges to some .z0 ∈ b1 . Set .w k = f (zk ), .k = 0, 1, 2, . . .. Let .(D1k ) and .(D2k ) be scaling sequences for .1 and .2 along the sequences .(zk ) and .(w k ) respectively. We may use the same boundary distance values .δk for both scalings since f up to equivalence preserves the boundary distance. As above we set .k1 = D1k (1 ) and .k2 = D2k (2 ). Consider the sequence of biholomorphic mappings .F k : k1 → k2 defined by .F k = D2k ◦ f ◦ (D1k )−1 . The sequences of domains .kj converge to .H, .j = 1, 2. As in the proof of the Wong-Rosay theorem, we obtain that .{F k } and .{(F k )−1 } are normal families that contain subsequences converging uniformly on compact subsets to F and .F −1 respectively. By construction, .F ( 0, −1) = ( 0, −1). Hence, by Cartan’s theorem (Theorem 1.1) it follows that .F : H → H is a biholomorphism.

.

4.1.2

Continuous Extension of the Lift

In the setting of Theorem 4.1 we consider the bundles of (tangential) complex hyperplanes over .1 and .2 . Every complex hyperplane p in .Cn passing through the origin can be identified by the equation  p = t = (t1 , . . . , tn ) ∈ Cn :

n 

.

 pk t k = 0 .

k=1

The n-tuple .[p1 , . . . , pn ] may be viewed as homogeneous coordinates for the plane p, and the set of all such hyperplanes is naturally identified with .CPn−1 . The mapping .fˆ : 1 × CPn−1 → 2 × CPn−1 , given by fˆ(z, p) = (f (z), Df (z)p),

.

is clearly holomorphic. Further, we consider the projectivizations of holomorphic tangent bundles over .bj , .

M = {(z, p) : z ∈ b1 , p = Hz (b1 )}, M˜ = {(w, q) : w ∈ b2 , q = Hw (b2 )}.

These are .(2n − 1)-dimensional submanifolds in .Cn × CPn−1 that are totally real by Lemma 1.9. The key step in the proof of Theorem 4.1 is the following.

40

4 Boundary Smoothness of Holomorphic Mappings

Lemma 4.2 The mapping .fˆ extends continuously to M along some wedge, and ˜ F (M) ⊂ M.

.

Proof By the Chirka–Lindelöf principle, F extends to M along some wedge with the edge M with .f (M) ⊂ M˜ provided that .fˆ extends along some admissible direction (see [34, 125]). By Theorem 3.4, f extends to .1 as a map of class .C 1/2 , and there exist constants .C1 , C2 > 0 such that C1 dist(z, b1 ) ≤ dist(f (z), b2 ) ≤ C2 dist(z, b1 ).

.

(4.2)

Since the statement is local it suffices to consider a small neighbourhood of a point (z0 , p0 ) ∈ M. Without loss of generality we will assume that .z0 = 0 = f (z0 ), and that the defining functions .ρ and .φ of the boundaries .b1 and .b2 respectively have the form

.

ρ(z) = 2xn + |z|2 + o(|z|2 ), φ(w) = 2un + |w|2 + o(|w|2 ),

.

in a neighbourhood of the origin. In particular, .H0 (b1 ) = H0 (b2 ) = {t ∈ Cn : tn = 0}, in other words, .p0 = (0, . . . , 0, 1). On the set Pn = {p = [p1 , p2 , . . . pn ] ∈ CPn−1 : pn = 0}

.

the coordinates .p1 , . . . , pn−1 of any plane .p = (p1 , ..., pn−1 , pn ) are uniquely determined by the condition .pn = 1. Therefore, .p ∈ Pn will be viewed both as a hyperplane in .Cn and a point in .Cn−1 with coordinates .(p1 , . . . , pn−1 ). Locally M is given by the system of equations ρ(z) = 0, pk = ∂zk ρ(z)/∂zn ρ(z), k = 1, . . . , n,

.

where .∂zj ρ =

∂ρ ∂zj

. Let

N = {(z, p) : ρ(z) < 0, pk = ∂zk ρ(z)/∂zn ρ(z), k = 1, . . . , n}.

.

Then N is a generic submanifold of .Cn × CPn−1 of real dimension 2n with boundary M. The map .fˆ is naturally defined on N and we show that .fˆ extends continuously to M along ˜ This is where the scaling method comes into play. N with values in .M. It suffices to verify that if a sequence of points .(zk , pk ) ∈ N converges to .(z0 , p0 ) ∈ M, then .fˆ(zk , pk ) converges to .(f (z0 ), q 0 ), where .q 0 = Hz0 (b2 ). We argue by contradiction: suppose on the contrary that there exists a sequence of vectors .a k ∈ pk ⊂

4.2 Boundary Regularity of Discs Near a Totally Real Manifold

41

Cn such that .a k → a 0 ∈ p0 , and such that the images .bk = Df (zk )a k ∈ Cn satisfy the estimate |bnk |/|bk | ≥ C0 > 0,

.

(4.3)

0 in view of .q10 = . . . = qn−1 = 0. Now we apply to the mapping f the scaling along the k sequence .(z ). Let .F : H → H be a biholomorphic mapping which is the limit of the scaling sequence k .(F ) (in the notation of the previous section). Using the boundary distance preserving property (4.2) and the definition of .(F k ), we easily obtain (passing to the limit) that for every .R > 0 there exists a constant .C = C(R) > 0 such that for .z ∈ H ∩ RBn one has

|φ(F (z))| ≤ C|φ(z)|.

.

(4.4)

This implies that F is bounded on every .H ∩ RBn , and therefore F is .1/2-Hölder continuous up to .bH. Indeed, assume that there exists a sequence of points .(ζ k ) in k .H, converging to a (finite) point .ζ ∈ bH such that .|F (ζ )| → ∞. Consider the line .l(t) = ( 0, −t) with .t > 0. Since F is a biholomorphism, we conclude that −1 (l(t)) = ζ . But this contradicts the estimate (4.4). .limt→+∞ F Furthermore, the condition (4.3) implies D f˜(0, 1)a 0 = (0, bn ),

.

(4.5)

with .bn = 0. Since .f˜ is a linear-fractional automorphism of .H, it takes complex lines to complex lines. In view of (4.5), the image of the line .l = {z = (0, −1) + a 0 t, t ∈ C} is a line .l˜ of the form .w = const. But this is impossible since the set .l ∩ H is bounded and as   we saw above its image must be bounded as well, but .l˜ ∩ H is not. Using Lemma 4.2 one can immediately conclude the proof of Theorem 4.1 in the case when the boundaries .bj are real analytic. Indeed, by Proposition 1.6 the lift F extends holomorphically past M. This means that the mapping f extends holomorphically to a neighbourhood of .1 . In the .C ∞ category this approach requires some additional technical tools that we introduce in the next two sections.

4.2

Boundary Regularity of Discs Near a Totally Real Manifold

In this section we prove the following result. Proposition 4.3 Let M be an n-dimensional totally real manifold of class .C ∞ in .Cn . Let n .f : D → C be a holomorphic mapping. Suppose that .γ ⊂ bD is an open arc, f is continuous on .D ∪ γ and .f (γ ) ⊂ M. Then .f ∈ C ∞ (D ∪ γ ).

42

4 Boundary Smoothness of Holomorphic Mappings

To prove the proposition we will need three lemmas below. We begin with the following C ∞ version of the Reflection principle. Denote by .D+ the upper half-disc, i.e.,

.

D+ = {ζ ∈ C : |ζ | < 1, Im ζ > 0}.

.

Lemma 4.4 Let .h : D+ → C be a .C ∞ -smooth function such that .∂h/∂ζ vanishes to order m, .m > 0 noninteger, as .ζ → (−1, 1), in other words, there exists a constant .C > 0 such that    ∂h  m  .  ∂ζ  ≤ C(Im ζ ) . Assume that h is continuous on .D + ∪ (−1, 1) and .h((−1, 1)) ⊂ R. Then h ∈ C m+1 (D+ ∪ (−1, 1)).

.

Proof We extend the function .g = ∂h/∂ζ to the whole disc .D by setting .g(ζ ) = g(ζ ), and denote the extended function by .G(ζ ). Since by the assumption g vanishes to order m on .(−1, 1), the function G is of class .C m (D). Fix a .C ∞ -smooth function .λ : R → R≥0 such that .λ ≡ 1 on .(−1/2, 1/2) and the support of .λ is contained in .(−1, 1). Consider the function .f (ζ ) = λ(|ζ |)G(ζ ). Then f is of class .C m (C) and its support is contained in .D. Recall that the Cauchy transform of f is defined as f (ω) 1 dω ∧ dω. .Tf (ζ ) = (4.6) 2π i D ω − ζ It is classical that .T : C m (D) → C m+1 (D) is a bounded linear operator when m is not an integer (and if m is an integer, .T : C m (D) → C m+α (D) is bounded for any .0 < α < 1). By the properties of the Cauchy transform we have .∂Tf/∂ζ = f . Furthermore, it is easy to check that in our case .Tf (ζ ) = Tf (ζ ), and therefore, Tf is real-valued on .(−1, 1). By construction, the function .h − Tf is holomorphic on a one-sided neighbourhood of the origin and is real-valued on the real axis. By the Schwarz reflection principle, it extends holomorphically to a neighbourhood U of the origin. Therefore, h is of class .C m+1 (D + ∩   U ). Since the desired statement is local, the proof of the lemma is complete. The next step toward the proof of Proposition 4.3 is the following. Lemma 4.5 In the assumptions of Proposition 4.3, there exists a constant .C > 0 such that dist(f (ζ ), M) ≤ Cdist(ζ, γ )1/2

.

for all .ζ ∈ D in a neighbourhood of .γ .

4.2 Boundary Regularity of Discs Near a Totally Real Manifold

43

Proof Let .φ be the square distance function to M. It is nonnegative, vanishes precisely on M and by Lemma 1.7 it is strictly plurisubharmonic in a neighbourhood of M. Consider the function .φ ◦ f . There exists a simply-connected smoothly bounded onesided neighbourhood U of .γ in .D such that .f (U ) is contained in the region where .φ is strictly plurisubharmonic. Let .h : D → U ∩ D be a conformal parametrization. Then the function .φ ◦ f ◦ h is nonnegative, subharmonic on .D, and continuous on .D. Therefore, by an estimate of the Poisson integral (see, e.g., [120]) it follows that there exists .C1 > 0 such that for all .ξ = reit with .r < 1, we have 1−r 1 .φ ◦ f ◦ h (re ) ≤ 1 + r 2π



2π

it

φ ◦ f ◦ h(eiθ )dθ ≤ C1 dist(ξ, bD).

(4.7)

0

Let .ζ = h(ξ ). Then, by the definition of the function .φ, by the estimate .dist(ξ, bD) ≤ C2 dist(ζ, bD) for some .C2 > 0, and (4.7), we have dist2 (f (ζ ), M) ≤ C1 dist(ξ, bD) ≤ Cdist(ζ, γ ).

.

From this the required estimate holds.

 

Finally, we need Lemma 4.6 Let M be an n-dimensional totally real manifold of class .C m in .Cn and let p be a point in M. Then, in a neighbourhood U of p in .Cn there exists a local .C m -smooth diffeomorphism . such that .(M ∩ U ) ⊂ Rn and .∂ vanishes to order .m − 1 on M. Proof Assume that .p = 0 and M is defined by the equation .y = φ(x) near the origin. Here φ is vector function of class .C m near the origin in .Rn , satisfying .φ(0) = 0, .dφ(0) = 0. It follows by Lemma 1.8 that the defining function can be extended to a neighbourhood of the origin in .Cn as functions of class .C m and such that its .∂-derivative vanishes on M to   order .m − 1. The remaining argument is the same as in the proof of Proposition 1.5.

.

Now we are able to conclude the proof of Proposition 4.3. After a conformal reparametrization, we may assume that f is defined on .D+ , continuous up to .(−1, 1) and .f ((−1, 1)) ⊂ M. (Thus, we replace .γ with .(−1, 1)). Let . be given by Lemma 4.6 near .p = f (0). Consider the function .h =  ◦f . In order to apply the Reflection principle from Lemma 4.4, we need to check that .∂h vanishes on .(−1, 1) to order .m − 1. Since the holomorphic function f is bounded, it follows by the Cauchy inequalities that (k) .|f (ζ )| ≤ Ck | Im ζ |−k , for any positive integer k. Given a positive integer m, consider m the differential operator .D m = ∂I J where .|I | + |J | = m. Since .∂ vanishes to infinite ∂z ∂z order on M, in view of Lemma 4.5 we have the estimate |(D m ∂)(f (ζ ))| ≤ Cdist(f (ζ ), M)s ≤ C| Im ζ |s/2 ,

.

for any positive integer s.

44

4 Boundary Smoothness of Holomorphic Mappings

Together with the Chain rule, these estimates imply that any derivative of .∂h/∂ζ tends to 0 as .Im ζ tends to 0. Hence, f is of class .C m (D ∪ γ ) for any m. This completes the proof.

4.3

Gluing Complex Discs and the Proof of Theorem 4.1

By Lemma 4.2 we know that the lift F of the biholomorphism f extends continuously to the projectivization M of the holomorphic tangent bundle of .b1 and that F takes M to the projectivization .M˜ of the holomorphic tangent bundle of .b2 . Locally, both of these are totally real manifolds of dimension .2n−1 in .C2n−1 . In order to apply the smooth reflection principle to F , we need to fill the wedge where F is defined by a family of holomorphic discs attached to M along a boundary arc. We now describe this construction. Consider a wedge-type domain .

W = {z ∈ Cm : φj (z) < 0, j = 1, . . . , m}

(4.8)

E = {z ∈ Cm : φj (z) = 0, j = 1, . . . , m}.

(4.9)

with the edge .

We assume that the defining functions .φj are of class .C 1+α with .α > 0, and that .∂φ1 ∧ . . . ∧ ∂φm = 0 in a neighbourhood of E. This implies that E is a generic manifold. Given .δ > 0 (assumed to be small enough) we also define a truncated wedge Wδ = {z ∈ Cm : φj − δ



.

φl < 0, j = 1, . . . , m} ⊂ W.

(4.10)

l =j

A complex or holomorphic disc is a holomorphic map .h : D → Cn which is at least continuous on the closed disc .D. Denote by .S + = {eiθ : θ ∈ [0, π ]} and .S − = {eiθ : θ ∈ (π, 2π )} the upper and the lower semicircles respectively. Proposition 4.7 Fix .δ > 0. There exists a smooth map .H : D × R2m → W , .H : (ζ, c, t) → h(c,t) (ζ ) with the following properties: (i) for every .(c, t) ∈ R2m the map .ζ → h(c,t) (ζ ) is holomorphic in .D and .h(c,t) (S + ) is contained in E. (ii) every disc .h(c,t) (D) is transverse to E. (iii) .Wδ ⊂ ∪(c,t) h(c,t) (D). This gluing discs argument can be quite helpful in many problems that involve totally real submanifolds. It was introduced in [104] and was subsequently used by many authors.

4.3 Gluing Complex Discs and the Proof of Theorem 4.1

45

Proof We outline the proof. Without loss of generality, we may assume that in a neighbourhood . of the origin a smooth totally real manifold E is defined by the equation .x = r(y), where a smooth vector function .r = (r1 , . . . , rn ) satisfies the conditions .rj (0) = 0, .drj (0) = 0. Fix a positive noninteger s and consider for a real function s .u ∈ C (bD) the Hilbert transform .T : u → T (u) given by 1 .T (u)(e ) = v.p. 2π iθ





θ −t u(e ) cot 2 −π π

it

 dt.

T (u) is uniquely defined by the conditions that the function .u + iT (u) is the trace of a function holomorphic on .D and that .T (u) vanishes at the origin. This is a singular linear integral operator which is bounded on the space .C s (bD) for any noninteger .s > 0. Fix a .C ∞ -smooth real function .ψj on .bD such that .ψj |S + = 0 and .ψj |S − < 0, .j = 1, . . . , n. Set .ψ = (ψ1 , . . . , ψn ). Consider the generalized Bishop equation .

u(ζ ) = r(u(ζ ), T (u)(ζ ) + c) + tψ(ζ ), ζ ∈ bD,

.

(4.11)

where .c ∈ Rm and .t = (t1 , . . . , tm ), .tj ≥ 0, are real parameters. It follows by the Implicit function theorem that this equation admits a unique solution .u(c, t) ∈ C s (bD) (for any noninteger .s > 0) depending smoothly on the parameters .(c, t). Consider now the complex discs .f (c, t)(ζ ) = P (u(c, t)(ζ ) + iT (u(c, t))(ζ )), where P denotes the Poisson operator of the harmonic extension to .D. The map .(c, t) → f (c, t)(0) (the centres of the discs) is smooth. Every disc is attached to E along the upper semicircle. It is easy to see that this family of discs fills out the wedge .Wδ (, E) when .δ > 0 and a neighbourhood . of the origin are chosen sufficiently small. Indeed, this is immediate when the function r vanishes identically (i.e., .E = iRm ), while the general case follows by a small perturbation argument.   Now we can complete the proof of Theorem 4.1. The map F is holomorphic in some wedge with the edge M. We fill the truncated wedge by a family of holomorphic discs .(ht ) provided by Proposition 4.7. By Proposition 4.4, every holomorphic disc .F ◦ ht is of class ∞ up to .bD+ . It is easy to check that all the estimates on the derivatives are uniform with .C respect to parameters t (in the .C m -norm for every m). From this it is also easy to conclude that F is of class .C ∞ up to M. Therefore, the biholomorphism f extends smoothly to .b1 . The local version of Theorem 4.1 also holds. Theorem 4.8 Let .f : 1 → 2 be a biholomorphic map between bounded domains in Cn . Suppose that .1 ⊂ b1 and .2 ⊂ b2 are connected open subsets of the boundaries that are .C ∞ -smooth and strictly pseudoconvex. Assume that for each point .z ∈ 1 the cluster set .C(f, z) is contained in .2 . Then f extends smoothly to .1 .

.

46

4 Boundary Smoothness of Holomorphic Mappings

By Theorem 3.7 the map f extends continuously to the boundary. This gives a localization of the problem. One can verify that all the arguments of this chapter are local and therefore, the same proof applies to Theorem 4.8.

4.4

Further Results

The problem of boundary regularity of biholomorphic (or proper holomorphic, see next chapter) mappings between strictly pseudoconvex domains in .Cn is now well-understood thanks to the contribution of many authors. Fefferman’s theorem [60] asserts that a biholomorphic mapping between strictly pseudoconvex domains with boundaries of class ∞ extends as a .C ∞ diffeomorphism between their closures. The original proof is based .C on the study of asymptotic behaviour of the Bergman kernel near the boundary and is difficult to adapt to the case of finite smoothness. Subsequently, several authors developed different approaches to the problem. One of them, due to Nirenberg et al. [103], uses a smooth version of the reflection principle. Pushing further their approach, Pinchuk and Khasanov [111] proved Theorem 4.1. A similar result was obtained by Lempert [93, 94] using quite a different technique of extremal discs for the Kobayashi metric. Another version of the smooth reflection principle was obtained by Tumanov [145]. Also, Forstneriˇc [66] considerably simplified the original approach of Nirengerg-WebsterYang. Finally, Khurumov [85] (see also Coupet [41] for a detailed exposition) proved that even a better result still remains true with the loss of regularity by .1/2. He also presented the following example confirming that this result is sharp. Let .k > 3/2 be a noninteger, and let f (z1 , z2 ) = (z1 + z2k , z2 ).

.

The map f is biholomorphic from the domain  = {z ∈ C2 : x2 + |z|2 < 0}

.

/ C m () for any onto some bounded domain . ⊂ C2 . It is clearly of class .C k () but .f ∈   k+1/2 . .m > k. One can verify that . is strictly pseudoconvex and .b is of class .C

5

Proper Holomorphic Mappings

In this chapter we extend the results of the previous chapters to proper holomorphic mappings between bounded domains in .Cn . For this we first discuss properties of proper holomorphic correspondences which, in particular, naturally arise as inverses of proper maps. One of the central results of the chapter is Alexander’s theorem (Theorem 5.4) which states, in striking contrast with dimension one, that any proper holomorphic self-map of the unit ball in .Cn , .n > 1, is a biholomorphism. Finally, in Sect. 5.3 we establish local rigidity of CR maps between strictly pseudoconvex hypersurfaces—a problem closely related to boundary regularity of holomorphic maps.

5.1

Holomorphic Correspondences and Algebroid Functions

Let .1 and .2 be two bounded domains in .Cn . Consider an irreducible complex ndimensional analytic subset .A ⊂ 1 × 2 . Assume that its boundary .bA := A \ A is contained in the skeleton .b1 ×b2 . Define the canonical projections .πj : 1 ×2 → j , .j = 1, 2. The triple .(A, π1 , π2 ) is called a proper holomorphic correspondence between .1 and .2 . Given a proper holomorphic correspondence A as above, consider the proper holomorphic map .π1 : A → 1 . This is a branched analytic covering. Therefore, there exists a purely .(n − 1)-dimensional complex analytic subset . ⊂ 1 (the branch locus) such that −1 () →  \  is an m-to-one covering map. For any holomorphic function .π1 : A \ π 1 .f : A → C there exists a monic polynomial P with coefficients holomorphic in .1 such that .P (f ) vanishes identically on A (see, e.g., [74, Ch. 3]). In particular, we may choose f to be the coordinate functions .wj . From this we can see that there exists a system of Weierstrass-type polynomials of the form

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_5

47

48

5 Proper Holomorphic Mappings k −1

k

wj j + a1j (z)wj j

.

+ . . . + amj (z) = 0, kj ≤ m, j = 1, . . . , n,

(5.1)

where the functions .aj k (z) are holomorphic in .1 , such that A is an irreducible component of the complex analytic set defined by (5.1). The latter can be viewed as the graph of an m-valued proper algebroid map. Each component .wj = wj (z) of this map is called an algebroid function in .1 . We will call the multiplicity of the covering m the degree of the algebroid mapping. As an example, if a correspondence .A ⊂ C2z × C2w is defined by .w12 = z1 + z2 , .w2 = w1 , then the corresponding system (5.1) is given by .w12 − z1 − z2 = 0 and 2 .w − z1 − z2 = 0. The set defined by this system consists of two irreducible components, 2 one of which is A. Denote by .Dj the discriminant of each polynomial in (5.1). Clearly,  ⊂ ∪j {z : Dj (z) = 0}.

.

The same considerations apply to the proper projection .π2 : A → 2 , which can be considered as the the inverse algebroid mapping. In general, the Carathéodory or the Kobayashi metrics may increase under proper holomorphic correspondences, but their growth can be controlled. Let .D ⊂ Cn be a domain and . ⊂ D be a subdomain such that . ⊂ D. Let .φ : D → C be an algebroid function such that .0 ∈ φ(p) for some .p ∈ . We define .φ|(,p) to be the correspondence obtained by analytic continuation of the germ .φp of .φ satisfying .φp (p) = 0 along any path in .. The graph of .φ|(,p) can be obtained by intersecting the graph of .w = φ(z) with . × C and considering the irreducible component of the intersection that contains the point .(p, 0). We have the following analog of the Schwarz lemma for algebroid functions. Lemma 5.1 Let D and . be as above, and .p ∈ . Then there exists a real .r < 1, which depends on . and a positive integer m such that the following holds: if .φ : D → C is any algebroid function of degree at most m such that .0 ∈ φ(p) and .|φ(z)| < 1 for any .z ∈ D, then .|φ|(,p) (z)| < r for any .z ∈ . Proof Denote the set of all algebroid functions .φ : D → C satisfying the assumptions of the lemma by .A. Set r(φ) = sup{|w| : w ∈ φ|(,p) (z), z ∈ },

.

and .r = sup{r(φ) : φ ∈ A}. We have to prove that .r < 1. Consider a sequence .φ j in .A such that .r(φ j ) → r. Every .φ j is defined by a monic polynomial of degree .1 ≤ mj ≤ m Pmj (w, z) = w mj + a1j (z)w mj −1 + . . . + amj j (z).

.

j

5.1 Holomorphic Correspondences and Algebroid Functions

49

Since .|φ| < 1, the coefficients .akj are uniformly bounded and we assume that they converge uniformly on compact subsets of D to functions .ak ∈ O(D). Since all polynomials are monic we may further assume that the limit polynomial .P0 (w, z) is of positive degree. Denote by .φ0 : D → C the map corresponding to the polynomial .P0 . Consider the function ψ(z) = max |w|2 .

.

w∈φ0 (z)

Then .ψ(z) is plurisubharmonic in D. Indeed, this is clearly the case at points where φ0 splits into holomorphic branches, and it extends to the branch locus of .φ0 as a plurisubharmonic function by the removable singularity result (Theorem 1.2). Since .ψ(z) ≤ 1 for any .z ∈ D, by the Maximum principle there exists .r0 < 1 such that .|φ0 |(,p) (z)| < r0 for any .z ∈ . But .r = r0 and this proves the lemma.   .

For a bounded domain .1 in .Cn and .p ∈ , let .d = dist(p, b1 ). Denote by .Bn (p, r) c (p, r) the ball centred at p and of radius .r for the Euclidean and the Carathéodory and .B distance respectively. Lemma 5.2 Let .f : 1 → 2 be a proper holomorphic correspondence (algebroid mapping) of degree m between two bounded domains. For every .ε with .0 < ε < 1 there exists .r > 0, independent of .p ∈ 1 , such that    c f Bn (p, εd) ⊂ B (q, r). 2

.

q∈f (p)

(the inclusion is understood in the sense of the previous Lemma). Proof Note that .Bn (p, εd) ⊂ Bn (p, d) ⊂  and .ε > 0 is fixed. By Lemma 5.1, there exists c, .0 < c < 1, such that for any p and any algebroid function a of degree m in .1 , satisfying .a(p) = 0, one has .|a(z)| < c for .z ∈ Bn (p, εd). Therefore, .|h ◦ f (z)| < c for any holomorphic function .h : 2 :→ D satisfying .h(f (p)) = 0. This implies that c .f (z) ∈ B   2 (f (p), r) with .r = ln(1 + c)/(1 − c). Now we can present the main result of this section. We say that a proper holomorphic correspondence given by the Eq. (5.1) extends continuously to .1 if all coefficients .aj k are continuous on .1 . Theorem 5.3 Let .f : 1 → 2 be a proper holomorphic correspondence between bounded strictly pseudoconvex domains with .C 2 -smooth boundaries in .Cn . Then f extends continuously to .1 .

50

5 Proper Holomorphic Mappings

Proof For every point .ζ ∈ b1 consider the segment .[ζ, p0 ] on the inward normal to .b1 of length .d0 small enough; clearly, .p0 = p0 (ζ ). There exists a neighbourhood U of .b1 such that each .z ∈  ∩ U belongs to the unique inward normal .[ζ (z), p0 ]. The projection 1 .π : 1 ∩ U → b1 is a .C -submersion. Hence, the image .π() is of measure 0 in .b1 . It follows that the points .ζ for which .[ζ, p0 ] does not intersect the branch locus . form a set .S ⊂ b1 of full measure. We will prove that for every branch of f the limit .

lim

[p0 ,ζ ]p→ζ

f (p)

exist uniformly with respect to .ζ ∈ S. From this the theorem will follow since S is dense in .b1 . Denote by .pk a point of .[ζ, p0 ] for which .dk := dist(pk , ζ ) = d0 2−k . Let also .qk = f (pk ), and .δk = dist(qk , b2 ). The same argument as for proper holomorphic mappings, implies that .δk ≤ cdk . Note that .pk belongs to .Bn (pk−1 , dk−1 /2). By Lemma 5.2, there c (q exists .r > 0 such that .qk ∈ B k−1 , r) for every .qk . The strict pseudoconvexity of .2 2 and the estimates of the Carathéodory metric imply that  c B (qk−1 , r) ⊂ Bn (qq−1 , R δk ), 2

.

where .R > 0 depends only on r and .2 . Therefore, .dist(qk−1 , qk ) ≤ Rc1 (d0 2−k )1/2 ; this means that the sequence .f (pk ) converges. Now the convergence of .f (p) when .p → ζ , .p ∈ (ζ, p0 ] follows. Indeed,  c f ([pk−1 , pk ]) ⊂ B (qk−1 , r) ⊂ Bn (qk−1 , R δk ). 2

.

This convergence is uniform in .ζ ∈ S because all the estimates are uniform.

5.2

 

Proper Holomorphic Mappings of Strictly Pseudoconvex Domains

We begin with the following result due to Alexander [3]. Theorem 5.4 A proper holomorphic self-mapping of the unit ball .Bn in .Cn , .n > 1, is necessarily an automorphism. For the proof assume that .f : Bn → Bn is a proper holomorphic mapping. Then .f −1 : → Bn is a proper holomorphic correspondence. Therefore, it extends continuously to .Bn in view of Theorem 5.3. It follows that its branch locus . is defined as a zero set of a Bn

5.2 Proper Holomorphic Mappings of Strictly Pseudoconvex Domains

51

n

function which is holomorphic in .Bn and continuous on .B . By the uniqueness theorem, n n −1 splits over a point .p ∈ . ∩ bB is a closed nowhere dense subset of .bB . Hence, .f n bB \  to a union of local biholomorphic maps. These maps satisfy the assumptions of Theorem 4.8, which gives smooth extension to .bBn . Further, by Theorem 2.1 they extend holomorphically to a neighbourhood of p. Finally, by the Poincaré-Alexander theorem (Theorem 2.2) each map extends to an automorphism of .Bn . One of these automorphisms coincides with .f −1 , and this proves the result. Alexander’s theorem is clearly false for polydiscs, for example the map .(z, w) → 2 (z , w 2 ) is a proper holomorphic self-map of .D2 . The determinant of the Jacobian matrix of this map vanishes in the bidisc along the variety .zw = 0. Our second main result of this section shows that this is not possible for strictly pseudoconvex domains. Theorem 5.5 Let .j ⊂ Cn , .n > 1, .j = 1, 2, be bounded strictly pseudoconvex domains with .C 2 -smooth boundaries. Then a proper holomorphic mapping .f : 1 → 2 has a nonvanishing Jacobian determinant. Proof The proof is based on the scaling method. Arguing by contradiction, assume that the Jacobian .Jf vanishes somewhere. Then its zero set is a complex hypersurface S in .1 and its closure necessarily touches the boundary .b1 at some point p. We know that f is .1/2-Hölder continuous in .1 . Let .q = f (p). Consider a sequence of points .{pk } in S converging to p and their images .q k = f (pk ). Now apply the scaling construction along these sequences. We obtain the sequences of domains .kj , converging to .H, and a sequence of proper holomorphic mappings .f k : k1 → k2 converging uniformly on compact subsets to a holomorphic map .F : H → H. Note that by the scaling construction, .F (0, .., −1) = (0, ..0, −1) and JF (0, . . . , 0, −1) = 0

.

(5.2)

because of our choice of the sequence .{pk }. The inverse proper holomorphic correspondences .(f k )−1 : k2 → k1 also converge to a holomorphic correspondence .G : H → H. From this it easily follows that .F : H → H is proper and, therefore, is a biholomorphism   by Theorem 5.4. This contradicts (5.2), which proves the result. It follows that in the assumptions of the theorem above the proper map f is locally biholomorphic, but in general, f is not globally biholomorphic as can be seen in the following example. Let 1 = {z ∈ C2 : |z1 |4 + |z1 |−4 + |z2 |2 < 3},

.

2 = {w ∈ C2 : |w1 |2 + |w1 |−2 + |w2 |2 < 3}.

.

52

5 Proper Holomorphic Mappings

The map .f (z1 , z2 ) = (z12 , z2 ) is proper holomorphic from .1 to .2 , but it is not biholomorphic. However, if .1 is simply-connected, or if .1 = 2 , then f is a biholomorphism, see [105]. Further, Theorem 5.5 cannot be generalized to the case when the target domain is of higher dimension (in which case one could conjecture that a proper holomorphic map .f : 1 → 2 is an immersion). Indeed, consider the domains 1 = {z ∈ C2 : |z1 |2 + |z1 |4 + |z1 z2 |2 < 1},

.

2 = {w ∈ C3 : |w1 |2 + |w1 |2 + |w3 |2 < 1}.

.

The map .f (z1 , z2 ) = (z1 , z22 , z1 z2 ) is proper holomorphic from .1 onto .2 but its rank is 1 at .z = 0. To summarize some of the results of this and the previous chapter we formulate the following. Theorem 5.6 Let .f : 1 → 2 be a proper holomorphic mapping between two smoothly bounded strictly pseudoconvex domains. Then f extends as a mapping of class .C ∞ (1 ). If in addition .bj , .j = 1, 2, are real analytic, then the map f extends holomorphically to a neighbourhood of .1 . Proof By Theorem 3.7 the map f extends continuously to .1 , and By Theorem 5.5 the Jacobian of f does not vanish on .. Because of that, we may apply Theorem 4.8 in a neighbourhood of every boundary point, this gives the smooth extension of f . The second   statement follows by Theorem 2.1.

5.3

Local Invertibility of CR Maps

In the previous section we studied globally proper holomorphic mappings. It turns out that some rigidity phenomena still hold also locally. The following result is due to Pinchuk and Tsyganov [114]. Theorem 5.7 Let . j , .j = 1, 2, be germs of .C 2 -strictly pseudoconvex hypersurfaces in n .C , .n > 1. Then any continuous nonconstant CR mapping .f : 1 → 2 is a local homeomorphism of class .C 1/2 . The proof consists of several steps. First of all, by the Lewy extension theorem (see Sec. 1.2.2), f extends holomorphically to a one-sided pseudoconvex neighbourhood .1 of . 1 . The main difficulty is to choose .1 and a one-sided pseudoconvex neighbourhood .2 of . 2 such that .f : 1 → 2 is proper.

5.3 Local Invertibility of CR Maps

53

Lemma 5.8 In the assumptions of Theorem 5.7, the following holds: (i) There exists .C > 0 such that C −1 dist(z, 1 ) ≤ dist(f (z), 2 ) ≤ Cdist(z, 1 )

.

(5.3)

for all .z ∈ 1 . (ii) The map f is .1/2-Hölder continuous on .1 ∪ 1 . This is already proved in Theorem 3.7. The next step given in the lemma below is straight forward. Lemma 5.9 Suppose that .f −1 (f (p)) is a compact subset of . 1 . Then one can choose .1 such that .f : 1 → f (1 ) is proper. The remaining part of the proof is to show that the assumption of Lemma 5.9 always holds. Without loss of generality assume that .p = 0 and .f (0) = 0. By a suitable choice of coordinates we may assume that the defining functions .ρ1 of . 1 and .ρ2 of . 2 are given by ρ1 (z) = 2 Re zn + |z|2 + o(|z|2 ), ρ2 (w) = 2 Re wn + |w|2 + o(|w|2 ).

.

Let .pk ∈ 1 be a sequence of points converging to 0 and .q k = f (pk ). We apply the scaling construction to the mapping f with respect to these sequences. As usual, we obtain the following data: (a) the sequences .ρjk , .j = 1, 2, of functions converge uniformly on compact sets to the function .ρ(z) = 2 Re zn +|z1 |2 +. . .+|zn−1 |2 , and the domains .kj , .j = 1, 2, converge to .H = {ρ(z) < 0}; (b) the sequence of holomorphic mappings .F k : k1 → k2 converges uniformly on compact subsets to a holomorphic mapping .F : H → H; (c) if the initial mapping .f : 1 → 2 is proper, then F is an automorphism of .H; (d) if the initial mapping f satisfies (5.3), then for all k one has C −1 |ρ1k (z)| ≤ |ρ2k (F k (z))| ≤ |Cρ1k (z)|,

.

and C −1 |ρ(z)| ≤ |ρ(F (z))| ≤ C|ρ(z)|.

.

(5.4)

It follows from (5.4) that for every point .ζ ∈ bH the cluster set .C(F, z) does not intersect .H, i.e., .C(F, ζ ) is contained in .bH∪{∞}. Therefore, we may apply Theorem 3.7,

54

5 Proper Holomorphic Mappings

from which it follows that either F extends .1/2-Hölder continuously to a neighbourhood of ζ in .bH, or .limz→ζ |F (z)| = ∞. Let .Ca : H :→ Bn be Cayley’s transform given by (1.2) that sends biholomorphically .H onto the unit ball. Then we obtain the following

.

(e) the composition .Ca ◦ F : H → Bn extends continuously to .bH. We also need the following Lemma 5.10 Assume that the points .pj belong to the real normal of . 1 at 0. Then .F : H → H is biholomorphic if and only if .f : 1 → 2 is a homeomorphism near 0. Proof If f is a local homeomorphism, then F is a biholomorphism by part (c) above. Assume that F is biholomorphic. Arguing by contradiction, suppose that f is not a homeomorphism. Then by Lemma 5.9 and the convexity, for any .ε > 0 there exists a point .ζ in . 1 ∩ {2 Re zn = −ε} with .f (ζ ) = f (0) = 0. By (d) this property also holds for F , and this is a contradiction.   The above lemma reduces the proof of the theorem to the case when .1 = 2 = H. Since .H is biholomorphic to the unit ball .Bn via the Cayley transformation, the boundary 2n−1 with a deleted point. Thus, it suffices to establish .bH corresponds to the unit sphere .S the following Proposition 5.11 Let .F : Bn → Bn be a holomorphic mapping. Assume that for some n n n n .a ∈ bB the mapping F is continuous on .B \ {a} and .F (bB \ {a}) ⊂ bB . Then F is a n biholomorphic automorphism of .B . To prove the proposition we first establish the following Lemma 5.12 Let .pk be a sequence in .Bn converging to .p ∈ bBn \ {a}. Then the mapping ˜ : H → H, obtained from F and .pk by the scaling construction, is biholomorphic if and .F only if F is biholomorphic. Proof Clearly, if F is biholomorphic, then .F˜ is biholomorphic. For the converse, suppose that .F˜ is biholomorphic. Arguing by contradiction assume that F is not biholomorphic. Then by the Poincaré-Alexander theorem, the restriction .F |bBn \{a} is nowhere diffeomorphic. Let .ζ k ∈ bBn \ {a} be the point closest to .pk . Denote by . k the real hyperplane through .pk parallel to the tangent space .Tζ k bBn . By Lemma 5.9 and the PoincaréAlexander theorem, there exists a point .bk ∈ k ∩ bBn such that .F (bk ) = F (ζ k ). Passing to the limit as .k → ∞, we conclude by part (d) above that there exists a point ˜ (b) = F˜ (0). This is a contradiction. .b ∈ bH ∩ {2 Re zn = −1} such that .F  

5.3 Local Invertibility of CR Maps

55

Now we are able to conclude the proof of Proposition 5.11. After a change of coordinates, we replace .Bn with .{ρ(z) = 2Rezn + |z|2 < 0}. Assume that .a = (0, . . . , 0, −2). Let L = lim sup |ρ(F (z)|/|ρ(z)|.

.

z→0

Then .L < ∞. Consider a sequence .pk along which the limit is attained. Let .F˜ : H → H be a holomorphic mapping obtained from F and .pk by scaling. The maps F and .F˜ are both biholomorphic. It follows from the scaling construction that .F˜ (0, . . . , 0, −1) = (0, . . . , 0, −L), and 2ReF˜n (z) + |F˜1 (z)|2 + . . . + |F˜n−1(z) |2 ≥ L(2 Re zn + |z1 |2 + . . . + |zn−1 |2 ) (5.5)

.

for .z ∈ H. Replacing .F˜ with .(L−1/2 F˜1 , . . . , L−1/2 F˜n−1 , L−1 F˜n ), one may assume from now on that .L = 1. Therefore, .F˜ is a holomorphic mapping of .H into itself with a fixed point .(0, . . . , 0, −1). Moreover, it follows from (5.5) (with .L = 1) that F maps the domain 2 2 .{z : −1 < 2 Re zn + |z1 | + . . . + |zn−1 | < 0} into itself. This implies that ∂ F˜n (0, .., 0, −1)/∂zk = 0, k = 1, . . . , n − 1.

.

It also follows from (5.5) that 2 Re F˜n (0, . . . , 0, zn ) + |F˜1 (0, .., 0, zn )|2 + . . . + |F˜n−1 (0, .., 0, zn )|2 ≥ 2 Re zn ,

.

which implies that .|∂ F˜n (0, . . . , 0, −1)/∂zn (0, . . . , −1)| ≥ 1. Finally, setting .zn = −1 in (5.5), we have n−1  .

j =1

|F˜j (z1 , . . . zn−1 , −1)|2 −

n−1 

|zj |2 ≥ −2 − 2 Re F˜n (z1 , . . . , zn−1 , −1).

j =1

The function on the right side of this inequality is pluriharmonic and vanishes at the origin. This implies that the differential of the mapping .G := (F˜1 , . . . ., F˜n−1 )(z1 , .., zn−1 , −1) at the origin does not decrease the length of any vector. Therefore, .|JG (0, . . . , 0, −1)| ≥ 1 and .|JF˜ (0, . . . , 0, −1)| ≥ 1. Now it follows by Cartan’s theorem (Theorem 1.1) that .F˜ is a biholomorphism of .H. This concludes the proof.

56

5.4

5 Proper Holomorphic Mappings

Comments and Further Results

There exists a different, more analytic approach to the boundary regularity problem for proper holomorphic mappings leading to very general results. We briefly describe it below. Let . be a bounded domain in .Cn . The orthogonal projection operator .P : L2 () → O() ∩ L2 () is called the Bergman projection. A smoothly bounded pseudoconvex domain . is said to satisfy Condition R if .P (C ∞ ()) ⊂ C ∞ (). The following result is due to Bell and Catlin [21] and Diederich and Fornæss [48]. Theorem 5.13 Let .f : 1 → 2 , be a proper holomorphic mapping between smoothly bounded pseudoconvex domains. Suppose that .1 satisfies Condition R. Then f extends as a .C ∞ -smooth mapping to .1 . A general approach to verifying Condition R for a prescribed class of domains relies on the .∂-Neumann problem. Let . be a smoothly bounded pseudoconvex domain in 2 n .C . Let g be a .∂-closed (0,1)-differential form with coefficients of class .L (), i.e., 2 .g ∈ L (0,1) (). The .∂-Neumann problem consists of determining the regularity of the solution u of the equation .∂u = g which is orthogonal to the kernel of the operator .∂, that is, to the class .O() ∩ L2 (). This solution is called the canonical solution. The operator .N : L2(0,1) () → L2 (), N : g → u, is called the .∂-Neumann operator on ∞ .. If the .∂-Neumann operator is globally regular, i.e., maps the space .C () to itself, then Condition R holds. Regularity of the .∂-Neumann operator has been an active area of research that led to the development of many important technical tools. A smoothly bounded domain . ⊂ Cn admits a defining function which is plurisubharmonic along the boundary if there exists a smooth defining function of . whose Levi form is positive semi-definite for all vectors at each boundary point (this condition is stronger than the pseudoconvexity which requires that the Levi form is positive semi-definite only on the holomorphic tangent space). Boas and Straube [25] proved that if . admits a defining function which is plurisubharmonic along the boundary, then it satisfies Condition R. In particular, every smoothly bounded convex domain satisfies Condition R. Christ [40] proved that for the worm-domains of Diederich and Fornæss [43], which are smooth and pseudoconvex, Condition R does not hold. This shows limitations of this approach and raises an important question of finding sufficient and necessary conditions for regularity of the Bergman projection. Condition R holds for bounded domains with real analytic pseudoconvex boundaries as proved by Diederich and Fornæss [45], and this implies smooth extendability to the boundary of proper holomorphic maps between such domains. For domains with real analytic boundaries that are not necessarily pseudoconvex we will present in the subsequent chapters a geometric approach to boundary regularity that is based on the technique of Segre varieties. We already saw the effectiveness of this method in the proof

5.4 Comments and Further Results

57

of Poincaré-Alexander theorem (Theorem 2.2). This approach is natural for this class of domains and is independent of the .L2 -methods or Condition R. We also briefly discuss some progress in the direction originated from Theorem 5.4. One of the most general results in this direction is obtained by Bedford-Bell [12]. Theorem 5.14 Let . be a bounded pseudoconvex domain with real analytic boundary in Cn , .n ≥ 2. Then every proper holomorphic self-mapping .f :  →  is a biholomorphism.

.

The proof is based on a careful analysis of the branch locus of a proper holomorphic mapping from a pseudoconvex domain with real analytic boundary. There are some partial results concerning special classes of domains. For example, we mention the result obtained by Berteloot and Pinchuk [23]. Theorem 5.15 Among bounded, complete, Reinhardt domains in .C2 , the bidiscs are the only ones that admit proper holomorphic self-mappings that are not automorphisms. The work of Berteloot and Pinchuk [23] also contains a detailed description of proper holomorphic maps between complete Reinhardt domains. The general case of Reinhardt domains in .C2 (not necessarily complete) was considered by Isaev and Kruzhilin [83]. They obtained a complete description of proper holomorphic mappings and classified all Reinhardt domains in .C2 admitting proper holomorphic self-maps which are not biholomorphisms. Proper holomorphic mappings between the classical Cartan domains and a wide class of Siegel domains were studied by Tumanov and Henkin [147, 148] and Henkin and Novikov [77]. Finally, Bedford and Bell [14], using the technique of Bergman projection, proved the following result. Theorem 5.16 Let . be a bounded, simply connected, strictly pseudoconvex domain contained in .Cn with a .C ∞ -smooth boundary. Then every irreducible proper holomorphic self-correspondence of . is a biholomorphic mapping. If .f : 1 → 2 is a biholomorphic map from a bounded strictly pseudoconvex domain n .1 into a smoothly bounded pseudoconvex domain .2 in .C , then by Theorem 5.13, the map f extends smoothly to the boundary. Such a map preserves the Levi-form, and therefore, .2 is also strictly pseudoconvex, as observed by Bell [20]. Further generalization of this and Theorem 5.5 is the result of Diederich and Fornæss [49]: if .j are as above, .j = 1, 2, and .f : 1 → 2 is a proper holomorphic map, then f is necessarily unbranched; it extends to an unbranched .C ∞ -smooth covering .f : 1 → 2 and consequently, .2 is also strictly pseudoconvex. If .2 is simply connected, then f is a biholomorphism. Finally, further results on continuous extension of holomorphic correspondences are obtained by Berteloot and Sukhov [24].

6

Uniformization of Domains with Large Automorphism Groups

The Riemann mapping theorem does not hold in complex dimension higher than 1 without additional assumptions. One such assumption is given in the Wong-Rosay theorem: strictly pseudoconvex domains with large automorphism groups are biholomorphic to the unit ball. In this chapter we explore possible extensions of the Wong-Rosay theorem to a wider class of domains. The following is the principal result of this chapter, it is due to Bedford and Pinchuk [16]. Theorem 6.1 Let . ⊂ C2 be a bounded pseudoconvex domain with smooth real analytic boundary. If the group .Aut () is not compact, then . is biholomorphically equivalent to a domain of the form Em = {(z1 , z2 ) ∈ C2 : |z1 |2m + |z2 |2 < 1}, m ∈ N.

.

(6.1)

In particular, we have the following Corollary 6.2 If . satisfies the assumptions of Theorem 6.1, then it admits a proper holomorphic map onto the unit ball .B2 ⊂ C2 . Note that the corollary does not hold in dimension .n = 3. Indeed, consider a domain  = {z ∈ C3 : (|z1 |2 + |z2 |2 )m + |z3 |2 < 1}.

.

Then mappings of the form 1

wj = (1 − ββ) 2m zj (1 − βz3 )−1/m , j = 1, 2; w3 = (z3 − β)/(1 − βz3 ),

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_6

59

60

6 Uniformization of Domains with Large Automorphism Groups

where .β ∈ C, .|β| < 1, are automorphisms of .. The group .Aut() is noncompact, but the set of weakly pseudoconvex points on .b is given by .{z ∈ b : z1 = z2 = 0} is onedimensional. By the result of Diederich and Fornæss [49], any proper holomorphic map of . onto a smoothly bounded domain is a biholomorphism. Therefore, . cannot cover the unit ball .B3 . The proof of Theorem 6.1 is presented in several steps. Throughout this chapter we assume that . satisfies the assumptions of the theorem.

6.1

Existence of Parabolic Subgroups of Automorphisms

We begin with the following technical lemma. Lemma 6.3 Let .0 ∈ b. There exists a holomorphic coordinate system in a neighbourhood of 0 such that the domain . can be locally defined by a real analytic function .ρ of the form ρ(z, z) = 2x2 + ψ(z1 , z1 ) +

∞ 

ak (z1 , z1 )y2k ,

(6.2)

ak (z1 , z1 )y2k = o(|z1 |2m ) + y2 b(z1 , z1 , y2 ),

(6.3)

.

k=0

where ∞  .

k=0

and .ψ is a homogeneous subharmonic polynomial of degree 2m (.m ≥ 1) in .z1 and .z1 that does not contain any harmonic terms (i.e., terms of the form .cz12m + cz2m 1 ). Furthermore, each .ak is a real analytic function satisfying .a0 (z1 , z1 ) = o(|z1 |2m ), .a0 (z1 , 0) ≡ 0, and m .ak (0, 0) = 0, .k = 1, 2, 3, . . .. If .m > 1, then also .a1 (z1 , z1 ) = o(|z1 | ). Proof We first perform a linear change of coordinates such that the outward normal to .b at 0 is in the direction of the .x2 -axis. Then (by the Implicit function theorem) a defining real analytic function of . can be chosen in the from ρ(z, z) = 2x2 + a(z1 , z1 , y2 ).

.

Moreover, without loss of generality, we may assume that .{ρ(z, 0) = 0} = {z2 = 0}; this can be achieved via the change of coordinates .(z1 , z2 ) → (z1 , ρ(z, 0)). Next, consider a map .(z1 , z2 ) → (z1 , φ(z2 )), with .φ(z2 ) = z2 + . . . ., which takes .b ∩ {z1 = 0} to the set .{z1 = 0, x2 = 0}. In these coordinates the function .ρ takes the form

6.1 Existence of Parabolic Subgroups of Automorphisms

ρ(z, z) = 2x2 +

∞ 

.

61

ak (z1 , z1 )y2k ,

k=0

where .a0 (z1 , 0) ≡ 0 and .ak (0, 0) = 0 for .k = 1, 2, . . . . Since .b is a compact real analytic hypersurface in .C2 , it does not contain nontrivial germs of complex analytic curves, see Theorem 9.2. Therefore, .ρ(z, z) does not vanish identically on the set .{z2 = 0}, which implies that .a0 (z1 , z1 ) does not vanish identically. Taking as .ψ the homogeneous part of the Taylor expansion of the lowest degree, we obtain (6.2), and (6.3). To complete the proof, consider the Levi operator   0  .L = −   ∂ρ

∂zμ

∂ρ ∂zμ ∂2ρ ∂zμ ∂zμ

    , μ, ν = 1, 2.  2

ψ ≥ Since . is pseudoconvex, .L ≥ 0 on .b. Putting here .y2 = 0, we conclude that . ∂z∂1 ∂z 1 0; hence the degree of .ψ is even, say, 2m. If .m > 1, the condition .L ≥ 0 implies that .a1 = o(|z1 |). A direct computation shows that, up to the terms of higher order, the 2a 1 . From this we conclude that coefficient of .y2 in the expression for .L is equal to . ∂z∂ 1 ∂z 1 m .a1 = o(|z1 | ). This proves the lemma. 

Lemma 6.4 Let D be a domain in .C2 , and let .fj : D →  be a sequence of holomorphic mappings. Suppose that for some point .z0 ∈ D the sequence .(f j (z0 )) converges to a point 0 j .p ∈ b. Then the sequence .(f ) converges uniformly on compact subsets of D to the 0 constant mapping .f (z) ≡ p . Proof It follows from Bedford and Fornæss [15] that there exists a peak function for the point .p0 , i.e., a function h that is holomorphic on ., continuous on . with .h(p0 ) = 1 and .|h(z)| < 1 for all .z ∈  \ {p0 }. By Montel’s theorem and the Maximum principle the sequence .(h ◦ f j ) converges uniformly on compacts to the constant function 1, which implies the lemma. 

Since the group .Aut() is noncompact, there exist a point .z0 ∈ , .p ∈ b, and a sequence .fj ∈ Aut() such that .fj (z0 ) → p as .j → ∞. By Lemma 6.4, the sequence .(fj ) converges uniformly on compacts in . to a constant function .f ≡ p. Without loss of generality we may assume that .p = 0 and that the defining function .ρ is chosen as in Lemma 6.3 in a neighbourhood U of the origin. Since the sequence .(f j ) converges to 0 uniformly on every compact subset K of ., given any .K   there exists j large enough, so that the image .f j (K) is contained in U , and so the composition .ρ ◦ f j is well-defined. j j j j j Set .zj = f j (z0 ); let .zˆ 2 ∈ C be such that .Imˆz2 = Imz2 and .(z1 , zˆ 2 ) ∈ b. We perform the change of coordinates

62

6 Uniformization of Domains with Large Automorphism Groups j

j

j

ω1 = z1 − z1 , ω2 = z2 − zˆ 2 − bj (z1 − z1 ),

.

where .bj is chosen so that at the point .ω = 0 the tangent plane to .b is defined by the equation .Re ω2 = 0. Let .Sj denote the mappings .Sj (z) = ω, and let .Gj = Sj () ∩ U . In what follows we add, if necessary, suitable higher order terms in the definition of .ω in order to assure that there are no harmonic terms in the defining function for the domain .Gj . This defining function is given by ∞ 

φj = 2 Re ω2 +

.

(k)

ψj (ω1 , ω1 ) + Im ω2 φj (ω1 , ω1 , Im ω2 ) = 0,

k=2

where .ψj(k) is a homogeneous polynomial of degree k in .ω1 , .ω1 . Consider the transforma-

tions .Tj (ω) = w, where .w1 = δj−1 ω1 , .w2 = εj−1 ω2 and .δj , εj → 0. Denote by .Dj the image .Tj (Gj ), and set .w = u + iv. Then the defining function of .Dj near the origin is given by 2εj u2 +

∞ 

.

δjk ψj(k) (w1 , w 1 ) + εj o(1) = 0.

k=2 j

j

The term .o(1) is uniformly convergent to 0 as .j → ∞. Put .εj = |z2 − zˆ 2 |. Clearly, j .Tj ◦ Sj (z ) = (0, −1). Choose .δj such that the absolute value of the largest coefficient of the polynomial −1 .ε j

 2m 

 δjk ψj(k) (w1 , w 1 )

k=2 (k)

is equal to 1. Now we note that .limj →∞ ψj (2m) .limj →∞ ψ j

= ψ. Hence,

−1 2m .supj ε j δj

= 0 for .2 ≤ k ≤ 2m − 1, and

< ∞. Passing to a subsequence, we conclude

that −1 . lim ε j →∞ j

 ∞  (δj w1 , δj w 1 ) + εj v2 φj (δj w1 , δj w 1 , εj v2 ) = P (w1 , w 1 ), k=2

where P is a nonzero polynomial of degree at most 2m. Therefore, as .j → ∞, the domains Dj converge to the domain

.

D = {φ = 2u2 + P (w1 , w 1 ) < 0}.

.

By the result of Bedford and Fornæss [15], there exists a function .G ∈ C(D) holomorphic on D with the following properties:

6.1 Existence of Parabolic Subgroups of Automorphisms

63

(i) there are constants .c1 , c2 > 0, and natural numbers k and N such that c1 (|w1 |2k + |w2 |2 )1/N ≤ |G(w)| ≤ c2 (|w1 |2k + |w2 |2 )1/N ,

.

for .w ∈ D with .|w| large enough. (ii) .arg G(w) ∈ [−π/4, π/4] for .|w| large enough. We will use this peak function in the proof of the following proposition. Proposition 6.5 The domains . and D are biholomorphically equivalent. Proof Set .j = (f j )−1 ( ∩ U ). Since .limj →∞ f j (z) = 0 ∈ U uniformly on compact subsets, it follows that the domains .j ⊂  converge to .: for every compact subset K of ., we have .K ⊂ j for j large enough. Consider the biholomorphic mappings j j :  → D . Since the domains .D converge to D, and . is bounded, the .f˜ = Tj ◦Sj ◦f j j j mappings .g j := (f˜j )−1 form a normal family. Passing to a subsequence, we may assume that these mappings converge to a holomorphic mapping .g : D → . But .g(0, −1) = z0 ∈ , and therefore .g(D) ⊂ . To complete the proof, consider a point .w ∈ Dj and a tangent vector X at w. As usual, denote by .FD (p, X) the Kobayashi-Royden metric. In order to control the convergence of j .g it is convenient to use the Sibony metric as defined in Eq. (3.10). This assigns the length S .F (p, X) to a tangent vector X a point .p ∈ D. The metric decreases under holomorphic D mappings, and FDS (p, X) ≤ FD (p, X).

.

(6.4)

A useful property of this metric is that it is defined in terms of certain plurisubharmonic functions on D. In particular, if D admits a smooth, bounded above, plurisubharmonic function u, and if there exists .δ > 0 such that .dd c u(X, X) ≥ δ|X|2 for all p satisfying S .|p − p0 | < δ, then there exists .ε = ε(δ) > 0 such that .F (p0 , X) ≥ ε|X|. We need the D following Lemma 6.6 There exist a constant .c1 > 0, independent of j , and a point .q0 ∈ D such that .FDj (p, X) ≥ c1 |X| holds at .w 0 = (0, −1). This also follows from the result of Catlin [31], but we present another proof using Sibony’s metric. Proof By Theorem 1.10, there exists .η ∈ (0, 1] such that the function .ρ˜ := −(−ρ)η is a bounded strictly plurisubharmonic exhaustion function for .. Then it is easy to see that .φ˜ j := −(−φj )η is a similar function for .Dj . The functions .φ˜ j converge to

64

6 Uniformization of Domains with Large Automorphism Groups

φ˜ := −(−φ)η . Let .q0 ∈ D be a point where .φ˜ is strictly plurisubharmonic. Then the functions .φ˜ j are uniformly strictly plurisubharmonic in a fixed neighbourhood of .q0 and we have the estimate

.

FDS j (p, X) ≥ ε|X|.

.

This proves the lemma in view of inequality (6.4).



Since .F is strictly positive, and the Kobayashi–Royden metric is nondecreasing, it follows that . dg j ≥ c > 0. Hence, . dg ≥ c. This means that the Jacobian determinant of g does not vanish at .q0 ∈ D. By Cartan’s theorem (Theorem 1.1), .g : D → g(D) is a biholomorphic mapping. It remains to prove that .g(D) = . First we show that the map g sends the boundary of D to the boundary of .. Choose small neighbourhoods .B0 of .q0 and .B1 of .g(q0 ) such that ˜ < −c on .B 0 , and thus .B1 ⊂ gj (B0 ) for j sufficiently large. We choose .c > 0 such that .φ j ˜ < −c on .B 0 for j large. Now let h denote the harmonic function on . \ B1 such that .φ .h = −c on .bB1 and .h = 0 on .b. By the Hopf lemma, there is a constant .ε > 0 such that .

− ε dist(p, b) > h(p)

for .p ∈ . For any .ε1 > 0, we may choose a compact K such that .h > −ε1 outside K, and thus for j sufficiently large and .p ∈ K we have .

− ε dist(p, b) > h ≥ φ˜ j ((g j )−1 (p)) − ε1 .

If .p ∈ g(D), we may pass to the limit as .j → ∞ and obtain the same inequality for .φ˜ with .ε1 = 0. Finally, we note that for .R > 0 there is a constant .η > 0 such that the inequality .

− η dist(q, bD) < φ(q)

holds for .q ∈ D with .|q| < R. We conclude that for .q ∈ D with .|q| < R we have ε1/δ η dist(g(q), b)1/δ ≤ dist(q, bD).

.

(6.5)

Now we suppose that g is not surjective. By (6.5) we must have .limp→p0 g −1 (p) = ∞ for any .p0 ∈  ∩ b(g(D)). Let G be the peak function on D given above. It follows that .

lim

p→∩b(g(D))

G(g −1 (p)) = 1.

Thus, by Rado’s theorem (see Sect. 1.1.2), .G ◦ g −1 extends analytically to . as we set it equal to 1 on . \ g(D). However, .|G ◦ g −1 | < 1 on .g(D), and so the extended function

6.1 Existence of Parabolic Subgroups of Automorphisms

65

is bounded in absolute value by 1 on .. Hence, . \ g(D) is empty by the Maximum principle. This proves the proposition. 

In what follows we change the notation and denote by .f : D →  the biholomorphic mapping constructed in Proposition 6.5. The one-parameter group of transformations .Lt (z1 , z2 ) = (z1 , z2 + it) acts on the domain D. The transformation .f : D →  allows us to define a one-parameter subgroup of .Aut() acting by .ht = f ◦ Lt ◦ f −1 . Lemma 6.7 The subgroup .H = {ht } is parabolic, i.e., there exists a point .p ∈ b such that for all .z ∈ , .

lim ht (z) = lim ht (z) = p.

t→−∞

t→+∞

We call the point p the parabolic point. Proof In view of Lemma 6.4, it suffices to prove that for some point .w 0 ∈ D one has .

lim f (w10 , w2 + it) = lim f (w10 , w2 + it).

t→−∞

t→+∞

One can assume that .w 0 = (0, −1). Let G be the peak function on D given above. By the property (ii) above, the function φ(w) = |(G(w) − 1)/(G(w) + 1)|2 − 1

.

is plurisubharmonic and negative on D. Furthermore, the property (i) above gives the estimate φ(0, w2 ) ∼ |w2 |−2/N

.

(6.6)

on the line .w1 = 0 as .w2 → ∞. It follows from the Hopf lemma that |φ ◦ f −1 (z)| ≤ c3 dist(z, b).

.

Together with (6.6) this gives dist(f (0, w2 ), b) ≤ c4 |w2 |−2/N .

.

(6.7)

The lemma now follows from the estimates on the Kobayashi-Royden metric. On the line w1 = 0 in D we have the obvious estimate from above:

.

FD (w, X) ≤ c6 |X|/|u2 |

.

(6.8)

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6 Uniformization of Domains with Large Automorphism Groups

for any vector .X = (0, X2 ). On the other hand, from Catlin [31] we have the following estimate from below: F (z, Y ) ≤ c7 |Y | dist(z, b)−α

.

(6.9)

for some .α ∈ (0, 1/2]. From (6.7)–(6.9) and the invariance of the Kobayashi-Royden metric, it follows that    ∂fk  c   .  ∂w (0, w2 ) ≤ |u | |w |2α/N , k, j = 1, 2. j 2 2

(6.10)

Now from (6.10) we have  |f (0, −1 ± it) − f (0, −1 − t)| ≤ c

.

0

t

dτ = ct −2α/N ln(1 + t) → 0, (1 + τ )t 2α/N

as .t → ∞. This completes the proof.

6.2



Geometry Near a Parabolic Point

In this section we study the behaviour of the mappings .(ht ) near a parabolic point p. This will reveal some geometric properties of .b in a neighbourhood of p. As we know, each automorphism extends holomorphically to a neighbourhood of .; furthermore, the action of .(ht ) on . extends to a .C ∞ (even real analytic) mapping . : R ×  → . Let .(H1 , H2 ) be the vector field that generates the group .(ht ), i.e., Hj (z) =

.

dhtj dt

(z)|t=0 , htj (z) = exp(tHj (z)), j = 1, 2.

Since .Hj are holomorphic in . and of class .C ∞ (), we conclude from Lemma 6.7 that t .H (p) = 0. Furthermore, since .h (b) ⊂ b, it follows that .H (z) ∈ Hz (b) for all .z ∈ b, i.e., .

Re(ρ1 (z, z)H1 (z) + ρ2 (z, z)H2 (z)) = 0.

(6.11)

A vector field H that is holomorphic on ., of class .C ∞ (), and satisfying (6.11), is called a holomorphic tangent vector field for .. Assume now that .p = 0 and that the defining function .ρ is given as in Lemma 6.3. Furthermore, assume that the function .ψ, that appears in the expansion of .ρ, is a real homogeneous polynomial of degree 2m without harmonic terms. Denote (temporarily, in the present section) the variables .z1 and .z2 by z and w respectively, and set .ψ1 = ∂ψ/∂z.

6.2 Geometry Near a Parabolic Point

67

The following auxiliary result can be verified by a direct computation, which we leave to the reader. Lemma 6.8 For all z we have .

Re(zψ1 (z, z)) ≡ mψ(z, z).

Furthermore, if .Im(zψ1 (z, z)) ≡ αψ(z, z), then .α = 0, and .ψ(z, z) = c|z|2m . We will also need the following lemma. Lemma 6.9 Suppose that for .Re w = (−1/2)ψ(z, z) we have .

Re(azw k−1 ψ1 (z, z) + bw k ) = 0,

(6.12)

where .a, b ∈ C are not both zero. Then k is either 1 or 2. If .k = 1, then .b = 2m Re a, and if .Im a = 0, then .ψ(z, z) = c|z|2m . If .k = 2, then .ψ(z, z) = c|z|2m , .b = ma, and .Re a = Re b = 0. Proof We substitute the expression .w = (−1/2)ψ(z, z) + iv into Eq. (6.12) and we set the coefficients of .v l equal to zero for .l = 0, 1, 2, . . . , k. The lemma now follows by a direct computation as above. 

We assign weight 1 to the variable z and weight 2m to the variable w. Lemma 6.10 Let .Q1 (z, w) and .Q2 (z, w) be weighted homogeneous polynomials with weights q and .q + 2m − 1 respectively. Suppose that .

Re [ψ1 (z, z)Q1 (z, w) + Q2 (z, w)] = 0

(6.13)

holds for .2 Re w = −ψ(z, z). Then q must be either 1 or .2m + 1, and (a) If .q = 1, then .Q1 (z, w) = (α + iβ)z and .Q2 (z, w) = 2mαw holds for some .α, β ∈ R, and if .β = 0, then .ψ(z, z) = c|z|2m . (b) If .q = 2m + 1, then .Q1 (z, w) = iαzw and .Q2 (z, w) = imαw 2 for some .α ∈ R, and 2m . .ψ(z, z) = c|z| Proof Taking into account the homogeneity of .Q1 and .Q2 , the Eq. (6.13) may be rewritten as .

  Re ψ1 (z, z)(a0 zq +a1 zq−2m w + a2 zq−4m w 2 + . . .) + b0 zq+2m−1 + b1 zq−1 w + . . . = 0, (6.14)

68

6 Uniformization of Domains with Large Automorphism Groups

where .aj and .bj are the coefficients of the polynomials .Q1 and .Q2 respectively. Since .ψ has no harmonic terms, it follows that the expression ψ1 (z, z) =

2m−1 

.

aμ zμ z2m−1−μ

μ=1

has no holomorphic terms. Assuming in (6.14) that .v = 0 and .2u = −ψ(z, z), we see that the only holomorphic term is .b0 zq+2m−1 , and it follows that .b0 = 0. Differentiating (6.14) with respect to v and setting .v = 0, we get .b1 = 0 by the same argument. Repeating this process, we get that k .Q2 ≡ 0 if .q + 2m − 1 is not divisible by 2m, and .Q2 (z, w) = bw if .q + 2m − 1 = 2mk. In the case where .Q2 ≡ 0, it follows from (6.14) that .Q1 ≡ 0 as well. In the case .q + 2m − 1 = 2mk, it follows from (6.14) that .

  Re ψ1 (z, z)(a0 z2m(k−1)+1 + a1 z2m(k−2)+1 w + . . .) + bw k = 0.

Setting .v = 0, .2u = −ψ(z, z), and setting the term of the lowest degree in .z equal to zero, we obtain .a0 = 0. Differentiating this equation with respect to v and repeating this procedure, we also obtain that .a2 = . . . = ak−2 = 0. Thus, (6.14) takes the form .

Re(azw k−1 ψ1 (z, z) + bw k ) = 0.

The lemma now follows from Lemma 6.9.



Lemma 6.11 The holomorphic vector field .H = (H1 , H2 ) is given by one of the following expressions (in a neighbourhood of the origin): (a) .H1 (z, w) = (α + iβ)z + . . . and .H2 (z, w) = 2mαw + . . . ., where the dots denote terms of higher weight, .α 2 + β 2 = 0, and if .β = 0, then .ψ(z, z) = c|z|2m ; (b) .H1 (z, w) = iαzw + . . . and .H2 = imαw 2 + . . ., .α ∈ R, .α = 0, and .ψ(z, z) = c|z|2m . Proof First assume that H vanishes to finite order at the origin. Then H may be represented in the form H1 (z, w) = Q1 (z, w) + . . . ., H2 (z, w) = Q2 (z, w) + . . . .,

.

(6.15)

where .Q1 and .Q2 are weighted polynomials with weights q and .q + 2m − 1 respectively, the dots indicate the terms of higher weight, and .(Q1 , Q2 ) does not vanish identically. Substituting (6.2), (6.3), and (6.15) to (6.11), using the fact that .2 Re w = −ψ(z, z), and

6.2 Geometry Near a Parabolic Point

69

collecting the terms of the smallest weight, we see that (6.13) holds. Thus the lemma follows from Lemma 6.10. It remains to consider the case when H vanishes to infinite order at .z = w = 0. But then for any .t ∈ R the transformation .ht (z, w) agrees with the identity transformation to infinite order at the origin. Now .ht is holomorphic in a neighbourhood of the origin for any t, so it follows that .ht (z, w) = (z, w). This means 

that H vanishes identically, which contradicts the parabolicity of .{ht }. Corollary 6.12 One has .ψ(z, z) = c|z|2m . Proof If .ψ does not have the indicated form, then by Lemma 6.11 we have .H1 (z, w) = αz + γ w + . . . and .H2 (z, w) = 2mαw + . . ., where .α ∈ R and .α = 0. Consider the change ˜ By (6.11), the vector field on . can be written in the of coordinates .z = δ z˜ , .w = δ 2m w. ˜ where new coordinates as .H˜ (˜z, w), .

˜ = δ −1 H1 (δ z˜ , δ 2m w) ˜ = α z˜ + γ δ 2m−1 w˜ + . . . , H˜ 1 (˜z, w) H˜ 2 (˜z, w) ˜ = δ −2m H2 (δ z˜ , δ 2m w) ˜ = 2mα w˜ + . . . .

Consider the Euclidean scalar product of the vectors .H˜ (˜z, w) ˜ This is given by ˜ and .(˜z, w). ˜ + ..., α|˜z|2 + 2mα|w| ˜ 2 + 2 Re(γ δ 2m−1 z˜ w)

.

and so for small .δ > 0 the sign of this expression is the same as the sign of .α. This means that the origin is an attracting or a repelling point for .H˜ (depending on the sign of .α), and therefore, the origin is not a parabolic point. 

Corollary 6.13 There is a holomorphic coordinate system in a neighbourhood of .p = 0 in which . is convex. Proof Consider first the coordinates and defining function .ρ as in Lemma 6.3. By Corollary 6.12 one has .ψ(z, z) = c|z|2m . Now we perform the change of coordinates .z = z ˜ , .w = w˜ − (1/2)w˜ 2 . Since in (6.2) we have .a1 (z, z) = O(|z|m+1 ), the boundary .b is given in the new coordinates by the equation .ρ = 0 with ρ(˜z, w) ˜ = 2u˜ + |˜z|2m + v˜ 2 + . . . ,

.

(6.16)

where the dots denote the terms of weight .> 2m. It is immediate that . is convex in these coordinates. 

Corollary 6.14 The domain D is convex.

70

6.3

6 Uniformization of Domains with Large Automorphism Groups

Parabolic Orbits

To complete the proof of Theorem 6.1, we need to investigate the behaviour of an orbit of the group .{ht } near the parabolic point .0 ∈ b. The preimage of 0 under the mapping .f : D →  is a real line parallel to the .Im w2 -axis. We will obtain the necessary information about the behaviour of an orbit in . by studying the map f . After the change of coordinates .(w1 , w2 ) → (w1 , 1/w2 ) the domain D takes the form D = {w ∈ C2 : 2u2 + |w2 |2 P (w1 , w 1 ) < 0},

.

which we still denote by D. The parabolic point .0 ∈ b now corresponds to the complex line .{w2 = 0} ⊂ bD, and the preimages of orbits of .{ht } are the circles .w(t) given by w1 = a1 , w2 = 1/(a2 + it), t ∈ (−∞, ∞).

.

It follows by Corollary 6.14 that .P (w1 , w 1 ) ≥ 0. Hence, .D ⊂ {u2 < 0}. The function φ(w) = u2 is negative and pluriharmonic on D, and for points of the form .w = (0, w2 ) we have .|φ(w)| = dist(w, bD). Applying the Hopf lemma to the function .φ ◦ f −1 , we obtain the estimate

.

dist(f (w), b) ≤ c1 dist(w, bD)

.

(6.17)

for the points of the form .w = (0, w2 ). Let L1 (z) = (∂ρ/∂z1 , −∂ρ/∂z1 ), L2 (z) = (∂ρ/∂z1 , ∂ρ/∂z2 )

.

denote respectively the tangential and the normal vector fields to the boundary .b. Although these vector fields are originally defined on .b, we will consider them in a neighbourhood W of .. Let .X = α1 L1 + α2 L2 be a tangent vector to .C2 at a point .z ∈ W ∩ . By the result of Catlin [31] we have the following estimate for the KobayashiRoyden metric in a neighbourhood of the origin in .b: F (z, X) ≥ c2 (|α1 |/|ρ(z)|1/2m + |α2 |/|ρ(z)|).

.

(6.18)

From this the technique discussed in Chap. 3 gives the following. Lemma 6.15 There exists a neighbourhood V of the origin such that the mapping .f : D →  extends to .V ∩ D as a mapping of class .C 1/2m . If .ρ˜ is any function defined in a neighbourhood of a parabolic point .0 ∈ b, then by shrinking the neighbourhood V if necessary, we can assume that .ρ˜ ◦ f is defined on V .

6.3 Parabolic Orbits

71

For .ρ˜ we take the defining function for the domain . which is defined in a neighbourhood of 0 (such a function exists by Corollary 6.13). Applying the Hopf lemma to the function .ρ ˜ ◦ f , we obtain the estimate dist(f (w), b) ≥ c3 dist(w, bD),

.

(6.19)

for .w ∈ (V ∩ D). Lemma 6.16 There exist .c4 > 0 and .δ0 > 0 such that for all .δ satisfying .0 < δ < δ0 we have |f2 (0, −δ)| ≤ c4 δ,

.

where .f = (f1 , f2 ). Proof It follows from (6.18) and the form of .L1 (z) that |(∂f2 /∂w2 )(w)| ≤ c5 (1 + |ρ1 ◦ f (w)|/|ρ ◦ f (w)|(2m−1)/2m ),

.

(6.20)

where .ρ is the defining function constructed in the proof of Corollary 6.13. Since ∂ρ/∂z1 = O(|z1 |2m−1 + y22 ), and taking into account that .f (0, −δ) = O(δ 1/2m ), as well as (6.17) and (6.19), we obtain

.

|(∂f2 /∂w2 )(0, −δ)| = O(δ (1−m)/m ).

.

This implies that .f2 (0, −δ) = O(δ 1/2m ). Substituting this into (6.20), we obtain in a similar fashion that |(∂f2 /∂w2 )(0, −δ)| = O(δ (3−2m)/2m ),

.

i.e., .f2 (0, −δ) = O(δ 3/2m ). Repeating this process finitely many times yields f2 (0, −δ) = O(δ),

.

which proves the lemma.



Consider the defining function from Corollary 6.13 in a neighbourhood of the origin. Since .x2 < 0 on ., Lemma 6.16 allows us to apply the Julia-Carathéodory theorem (see, e.g., [125]) to the function .z2 = f2 (0, w2 ). We conclude that for any .K > 0 there is an .ε > 0 such that the image of the circle .w(t) = (0, 1/(ε + it)) under .f2 lies in the disc .x2 + K(x22 + y22 ) < 0. The image .z(t) = f ◦ w(t) is an orbit of .(ht ) in ., i.e., each automorphism .ht :  →  maps the point .z(0) to the point .z(t). We introduce the

72

6 Uniformization of Domains with Large Automorphism Groups

notation .| Re z2 (t)| = δ, .Im z2 (t) = s2 and .z1 (t) = s1 . By the choice of .ε > 0 we have s22 /δ ≤ 1/K, and by (6.16) it follows that .|s1 |2m ≤ 4δ. We apply the Scaling method to the mappings .{ht }. Consider the transformation .w = t S (z) of the form

.

w1 = δ −1/2m z1 , w2 = δ −1 (z2 − is2 ).

.

The composition .S t ◦ ht maps the domain . biholomorphically to the domain .t defined by the function ρ t (w) = δ −1 ρ ◦ (S t )−1 (w).

.

Since .ρ has the form (6.16), it follows that ρ t (w) = 2u2 + |w1 |2m + s22 δ −1 + o(1),

.

where .o(1) denotes an expression that together with its derivatives tends uniformly to zero on compact subsets of .C2 as .t → ∞. Thus, since the point .w t = S t ◦ ht ◦ z(0) satisfies the condition .Re w2t = −1, it follows that .{S t ◦ ht } is a normal family. Hence, we may take a sequence .tn → ∞ such that the mappings .S t ◦ ht converge uniformly on compact subsets of . to a mapping .g :  → E˜ m , where E˜ m = {2u2 + |w1 |2m + c < 0}, 0 ≤ c ≤ 1/K.

.

If the point .w ∞ = g ◦ z(0) lies in .E˜ m , then g obviously maps . biholomorphically to ˜ m , which is biholomorphic to .Em . To conclude the proof of the theorem we need the .E following. Proposition 6.17 Let .g j : G → j be a sequence of biholomorphic mappings from a bounded domain .G ⊂ Cn to domains .j ⊂ Cn . Suppose that in some neighbourhood U of a point .p ∈ Cn there are (local) defining functions .ρ j ∈ C 2 (U ) for the domains j 2 .j such that the sequence .(ρ ) converges as .j → ∞ in the .C -metric to a function 2 .ρ ∈ C (U ) satisfying .ρ(p) = 0, .dρ(p) = 0, and . = {z ∈ U : ρ(z) < 0}. If . is strictly pseudoconvex at p and if there exists .z0 ∈ G such that .

lim gj (z0 ) = p,

j →∞

then the domain G (and therefore each .j ) is biholomorphically equivalent to the unit ball .Bn .

6.4 Further Results

73

This is a version of the Wong-Rosay theorem that can be proved using the same method; we leave the details to the reader. Now let .w ∞ ∈ bE˜ m . Since K may be taken sufficiently large, the equation .u∞ 2 = ∞ ∞ ˜ −1 implies that .w1 = 0. This means that .bEm is strictly pseudoconvex at .w . By Proposition 6.17 the domain . is biholomorphic to .Bn . This completes the proof of the theorem.

6.4

Further Results

There are several approaches to the problem of uniformization of domains with large (in a suitable sense: noncompact, co-compact, homogeneous,. . . ) automorphism groups, see, e.g., Greene et al. [73]. For unbounded domains a version of the Wong-Rosay theorem was proved by Efimov [59]. Further, Bedford and Pinchuk [19] proved that the pseudoconvexity assumption in Theorem 6.1 can be dropped. A natural related question is whether it is possible to impose only local restrictions on the boundary near an accumulation point of an automorphism group . A complete answer in .C2 is obtained by Verma [151] who proved the following classification result. Theorem 6.18 Let . be a bounded domain in .C2 . Suppose that there exists a point .p ∈  and a sequence .{φj } ⊂ Aut() such that .{φj (p)} converges to .p∞ ∈ ∂. Assume that the boundary of . is real analytic and does not contain nontrivial germs of complex curves near the point .p∞ . Then exactly one of the following cases holds: (i) If .dim Aut() = 2 then either (a) . is biholomorphic to .1 = {z ∈ C2 : 2 Re z2 +P1 (Re z1 ) < 0} where .P1 (Re z1 ) is a polynomial that depends on .Re z1 , or (b) . is biholomorphic to .2 = {z ∈ C2 : 2 Re z2 + P2 (|z1 |2 ) < 0} where .P2 (|z1 |2 ) is a homogeneous polynomial that depends on .|z1 |2 , or (c) . is biholomorphic to .3 = {z ∈ C2 : 2 Re z2 + P2m (z1 , z1 ) < 0} where .P2m (z1 , z1 ) is a homogeneous polynomial of degree 2m without harmonic terms. (ii) If .dim Aut() = 3 then . is biholomorphic to .4 = {z ∈ C2 : 2 Re z2 + (Re z1 )2m < 0} for some integer .m ≥ 2. (iii) If .dim Aut() = 4 then . is biholomorphic to .5 = {z ∈ C2 : |z1 |2 + |z1 |2m < 0} for some integer .m ≥ 2. (iv) If .dim Aut() = 8 then . is biholomorphic to .B2 . The dimensions .0, 1, 5, 6, 7 cannot occur with . as above.

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6 Uniformization of Domains with Large Automorphism Groups

In higher dimensions the situation is more complicated. We assign to the variables z1 , . . . , zn the weights .δ1 , . . . , δn , where .δj = (2mj )−1 for .mj a positive integer. If .J = (j1 , . . . , jn ) and .K = (k1 , . . . , kn ) are the multi-indices, we set .wt (J ) = j1 δ1 +. . .+jn δn and .wt (zJ zK ) = wt (J ) + wt (K). We consider real polynomials of the form

.

p(z, z) =



.

aJ K zJ zK .

(6.21)

wtJ =wtK=1/2

The reality of p is equivalent to .aJ K = aKJ . The balance of the weights .wt (J ) = wt (K) implies that the domain G = {(w, z1 , . . . , zn ) ∈ C × Cn : |w|2 + p(z, z) < 1}

.

(6.22)

is invariant under the action of the real torus (φ, θ ) → (eiφ w, eiδ1 θ z1 , . . . , eiδn θ zn ).

.

(6.23)

The weighted homogeneity of p implies that the Cayley-type transform .(w, z) → (w ∗ , z∗ ), defined by w = (1 − iw ∗ /4)(1 + iw ∗ /4)−1 , zj = zj∗ (1 + iw ∗ /4)−2δj ,

.

(6.24)

maps G biholomorphically onto the domain D = {(w, z1 , . . . zn ) : C × Cn : Im w + p(z, z) < 0}.

.

(6.25)

The latter is an unbounded realization of G. Note that D is invariant under the translation along the .Re w-direction. Since p is homogeneous, the domain D is invariant with respect to the family of anisotropic dilations. Hence the dimension of .Aut(D) is at least 4. Theorem 6.19 Let . ∈ Cn+1 be a bounded convex domain whose boundary is real analytic and does not contain nontrivial germs of complex curves. If .Aut() is noncompact then . is equivalent to the domain of the form (6.22). This result is obtained in Bedford and Pinchuk [17]. It is also valid for smoothly bounded convex domains of finite type. The general problem of classification of bounded real analytic (pseudoconvex) domains in .Cn , .n > 2, with noncompact automorphism groups remains open.

7

Local Equivalence of Real Analytic Hypersurfaces

There is a rich interplay between the geometry of real hypersurfaces in .Cn , .n > 1, and the complex structure of the ambient space. This gives rise to local biholomorphic invariants of real hypersurfaces, the signature of the Levi form, being, perhaps, the simplest of them. Determining a complete set of local invariants then becomes an important problem, its successful solution should also resolve the local equivalence problem: given the (germs of) real analytic hypersurfaces .(, 0) and .(  , 0) in .Cn , determine if there exists (the germ of) a local biholomorphic map .f : (Cn , 0) → (Cn , 0) sending . to .  . This problem is completely solved in the case when . and .  are Levi nondegenerate. In this chapter we will describe several general approaches to determining local biholomorphic invariants of real analytic hypersurfaces, and develop some tools necessary for applications. Some of these invariants, such as Segre varieties, Moser’s normal form, and Fefferman’s metric, will be used later in the book. We will not go into technical details here as our goal is only to introduce the ideas and formulate some results.

7.1

Complexification, Segre Varieties, and Differential Equations

We already saw in Sect. 2.2 that the family of complex hyperplanes is invariant under biholomorphic mappings of the unit sphere. In fact, any real analytic hypersurface . in .Cn , .n > 1, admits a family of local biholomorphic invariants–a family of complex hypersurfaces, called Segre varieties associated with .. One can view them as (the graphs of) solutions of a holomorphic second order PDE system with a completely integrable prolongation to the space of 1-jets. When .n = 2 such a system becomes a second order holomorphic ODE and the Segre family consists of complex curves. The biholomorphic maps of . are precisely Lie symmetries of its Segre family. Thus, the geometry of real

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_7

75

76

7 Local Equivalence of Real Analytic Hypersurfaces

analytic hypersurfaces is closely related to the geometry of holomorphic ODEs and PDEs. This fundamental correspondence, discovered by Segre [127], inspired Cartan [30] to study the geometry of real hypersurfaces in .C2 in analogy with the geometry of a second order ODE developed by the school of Lie [149]. The approach of E. Cartan is different in that it is based on the equivalence method for Pfaffian systems. All considerations of this section are local so the results should be understood in terms of the germs of the analytic objects involved. To simplify the notation, we will not use the language of germs, so the reader should keep in mind the locality of the assumptions. Let . be a real analytic hypersurface in a neighbourhood U of .0 ∈  in .Cn . Then . admits a canonical complexification, i.e., a complex analytic manifold of dimension .2n−1 that contains . as a totally real submanifold. It can be constructed by taking .U ∗ to be the open set U together with the complex structure .−J , it is opposite to the standard complex structure on .U ⊂ Cn . The map  → ˜ := {(z, z) ∈ U × U ∗ : z ∈ }

.

is a totally real embedding of . into .U × U ∗ . The smallest complex analytic set . c containing .˜ is then the complexification of .. This construction works in general for the germ of any real analytic set in .Cn . When . is a smooth real analytic manifold, then c . is a complex manifold. Let now the hypersurface . be given by a defining function .ρ(z, z) = 0. Then we simply have    c = (z, w) ∈ U × U ∗ : ρ(z, w) = 0 .

.

Note that .ρ(z, w) is a holomorphic function on .U × U ∗ . In applications, we may ignore the change of the complex structure on .U ∗ and treat .ρ(z, w) as antiholomorphic in w on .U × U . For w close enough to the origin we consider the complex hypersurface Qw = {z ∈ U : ρ(z, w) = 0}.

.

(7.1)

This hypersurface is called the Segre variety of the point w (associated with .). The collection of all Segre varieties is called the Segre family of .. More generally, if . is any real analytic set defined as the zero set of the (vector-valued) function .ρ, its Segre varieties are complex analytic sets in .Cn also defined by (7.1). The following are fundamental properties of the Segre family. Lemma 7.1 Let .0 ∈  be a real analytic hypersurface in .Cn and .0 ∈   be a real analytic hypersurface in .CN . Let .F : Cn → CN be a holomorphic mapping such that .F (0) = 0 and .F () ⊂   . Then .F (Qw ) ⊂ QF (w) , where .Qw denote the Segre varieties associated with the hypersurface .  .

7.1 Complexification, Segre Varieties, and Differential Equations

77

This lemma implies, in particular, that the Segre family is invariant under biholomorphic mappings. In the one-dimensional case Segre varieties are points and so the lemma becomes the classical Schwarz reflection principle. Proof Let .  = {z ∈ CN : φ(z , z ) = 0}. Then .φ(F (z), F (z)) = 0 whenever .ρ(z, z) = 0. Therefore, .φ(F (z), F (z)) = λ(z, z)ρ(z, z), where .λ is a real analytic function in a neighbourhood of the origin. It follows that   φ F (z), F (w) = λ(z, w)ρ(z, w),

.

for .(z, w) close to the origin in .Cn × CN . This proves the lemma.



While geometric properties of Segre varieties will be discussed in detail in Chap. 9 and used extensively further on, in this section we draw a connection between the complex geometry of real analytic hypersurfaces and the geometry of analytic differential equations and projective connections. The main idea is that the Segre family is a general set of solutions of some second order PDE system (or a single second order ODE when .n = 2). Let us discuss in some detail the case of dimension 2. We begin with the basic example of . = {z2 + z2 + z1 z1 = 0}, an unbounded realization of the unit sphere in .C2 . The Segre family has the form Qw = {z ∈ C2 : z2 + w2 + z1 w1 = 0}.

.

This is simply the family of all complex lines in .C2 with the exception of lines of the form .z1 = const. We view every .Qw as the graph of a complex affine function .z2 = h(z1 ) that depends on two complex parameters .w1 and .w2 . We treat .z1 as an independent variable and .z2 = z2 (z1 ) as the dependent one. Then the Segre family is the set of graphs of all solutions of the ordinary differential equation .z¨2 = 0 (the dot denotes differentiation with respect to .z1 ). Now for the general case we can assume that  = {z : Re z2 = φ(z1 , z1 , Im z2 )},

.

where .∇φ(0) = 0. By the Implicit function theorem, Qw = {z : z2 = h(z1 , w 1 , w 2 )},

.

for some holomorphic function h. Again, we view .z2 as the dependent variable and .z1 as an j j independent one. Applying the Chain rule we obtain .d j z2 /dz1 = (∂ j h/∂z1 )(z1 , w 1 , w 2 ), .j = 1, 2. By the Implicit function theorem we represent the parameters .(w1 , w2 ) as the functions of .(z1 , z2 , z˙ 2 ) and obtain the holomorphic ODE

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7 Local Equivalence of Real Analytic Hypersurfaces

z¨2 = F (z1 , z2 , z˙ 2 ) .

.

(7.2)

The Segre family of the real hypersurface . is precisely the set of the graphs of solutions of (7.2). The invariance property of the Segre family means that if f is a biholomorphic map with .f () = , then f sends the graph of a solution of (7.2) to the graph of another solution. But this means precisely that f is a (point) Lie symmetry of the Eq. (7.2). Therefore, the classical theory of Lie symmetries can be applied to the study of biholomorphisms between real analytic hypersurfaces. As an example, consider a holomorphic differential equation (S) : u¨ = F (x, u, u), ˙

.

(7.3)

where x denotes an independent complex variable and F is a holomorphic function. A symmetry group .Sym(S) of the equation .(S) is a (maximal) local (Lie) group G acting on a domain in .C2 such that the following holds: for every solution .u(x) of .(S) and every .g ∈ G the image (if defined) of the graph of u by g is the graph of some solution of .(S) which we denote by .g∗ u. A holomorphic vector field X=θ

.

∂ ∂ +η ∂x ∂u

(7.4)

is called an infinitesimal Lie symmetry of .(S) if it belongs to the Lie algebra of .Sym(S), i.e., generates a one-parameter group of point Lie symmetries of .(S). Denote by .jxm (u) the m-jet of u at x. In the most important case .m = 2 we set .u1 = ux and .u11 = uxx . Then 2 .jx (u) = (x, u, u1 , u11 ), and so .(x, u, u1 , u11 ) are natural coordinates on this jet space. Every point Lie symmetry g canonically extends to .J m (1, 1) as a biholomorphic m (g u). In particular, mapping .g (m) defined as follows: .g (m) associates to .Jxm (u) the jet .ju(x) ∗ a one-parameter group of symmetries generated by a vector field X lifts to .J m (1, 1). A vector field .X(m) on .J m (1, 1), which generates this lift, is called the prolongation of order m of X. The classical Lie theory gives powerful tools for the study of Lie symmetries which are particularly convenient in the infinitesimal case. For .m = 2 we have X(2) = X + η1

.

∂ ∂ . + η11 ∂u11 ∂u1

The general Lie theory gives the following expressions for the coefficients: η1 = ηx + (ηu − θx )u1 − θu (u1 )2 ,

.

η11 = ηxx + (2ηxu − θxx )u1 + (ηuu − 2θxu )(u1 )2 − θuu (u1 )3

.

+(ηu − 2θx )u11 − 3θu u1 u11 .

(7.5)

7.2 Equivalence Problem I: Moser’s Approach

79

Equation (S) of the form (7.3) defines a complex hypersurface .(S2 ) in .J 2 (1, 1) by the equation .u11 = F (x, u, u1 ). The fundamental principle of the Lie theory states that X is an infinitesimal symmetry of .(S) if and only if the vector field .X(2) is tangent to .(S2 ), that is, X(2) (u11 − F (x, u, u1 )) = 0 for (x, u, u1 , u11 ) ∈ (S2 ).

.

(7.6)

 Consider the expansion .F (x, u, u1 ) = ν≥0 fν (x, u)(u1 )ν . Plugging it into (7.6) and comparing the coefficients of the powers of .u1 we obtain a system of PDEs of the form LD 2 (θ, η) = G(x, u, D 1 (θ, η)),

.

where .D j denotes the set of the partial derivatives of the map .(θ, η) of order j , G is an analytic function, and L is a matrix with constant coefficients. Applying to this system the partial derivatives in x and u, we obtain after a direct computation that D 3 (θ, η) = H (x, u, D 1 (θ, η), D 2 (θ, η)),

.

(7.7)

for some analytic function H . This implies that every infinitesimal Lie symmetry of .(S) is determined by its second order jet at a given point. In particular, .dim Sym(S) ≤ 8. Consider again the Eq. (7.2) describing the Segre family of .. The group of local biholomorphisms of . is embedded into the symmetry group of (7.2) as a totally real subgroup of maximal dimension (see more details in [140]). As a consequence we obtain that the dimension of the real Lie group of biholomorphisms of . is bounded above by 8. In higher dimensions (.n > 2) the Segre family of a Levi nondegenerate real analytic hypersurface . is described by a PDE system uxi xj = Fij (x, u, ux ),

.

(7.8)

where .x ∈ Cn−1 and .ux denotes the set of the first order partial derivatives of the function .u = u(x). This system is completely integrable: its lift to the first order space of jets is a first order PDE system which satisfies the Frobenius integrability conditions. For more details we refer the reader to papers by Sukhov [140] and Merker [98], which are devoted to the study of Lie symmetries of PDE systems and Segre families.

7.2

Equivalence Problem I: Moser’s Approach

Moser’s approach [33] to the local equivalence problem consists of finding a normal form for the defining function of a real analytic hypersurface. This is achieved by applying local biholomorphic maps (change of coordinates) to bring the power series that represents the defining function of the hypersurface to the simplest possible form. Then the remaining

80

7 Local Equivalence of Real Analytic Hypersurfaces

coefficients of the power series are the biholomorphic invariants. In this section we outline this construction. Let . be a real analytic Levi nondegenerate hypersurface in .Cn containing the origin. All considerations will be local. We use the notation .z = ( z, zn ), . z ∈ Cn−1 . By the Implicit function theorem (after a permutation of coordinates), . can be written as the graph yn = F ( z, z, xn )

.

(7.9)

of a real analytic function F . Given a point .p ∈  consider a real analytic curve .γ in . with a parametrization .z = z(τ ), .τ ∈ (−τ0 , τ0 ). Assume that .γ passes through p along a noncomplex tangential direction, i.e., .z(0) = p and the vector .z˙ (0) is not in .Hp . In a neighbourhood of p there exists a biholomorphic map .z∗ = h(z) taking the curve  .γ to the real interval . z = 0, .zn = τ (we drop the asterisk for simplicity), that is, .h(z(τ )) = ( 0, τ ). Furthermore, in the new coordinates . is given by the equation yn = | z|2 +



Fkl ( z,  z, xn ),

.

(7.10)

k,l≥2

where .Fkl are real homogeneous polynomials of degree k in . z and l in . z with coefficients analytic in .xn . Of course, such a change of coordinates is not unique. First of all this is due to the freedom in the choice of the curve .γ and its parametrization. Moreover, consider a transformation ( z, zn ) → (U (zn ) z, zn ),

.

(7.11)

where .zn → U (zn ) is an .(n − 1) × (n − 1) nondegenerate holomorphic matrix function in zn which is unitary for .zn = τ . This transformation fixes .γ and preserves (7.10). Using the tensor notation we write

.



Fkl =

.

aα1 ...αk β1 ...βl zα1 . . . zαk zβ1 . . . zβl .

1≤αν ,βμ ≤n−1

Here we assume that the coefficients .aαβ do not change under permutations of indices .αν and .βμ . For .k, l ≥ 1, we put trFkl =

.



bα1 ...αk−1 β1 ...βl−1 zα1 . . . zαk−1 zβ1 . . . zβl−1 ,

with bα1 ...αk−1 β1 ...βl−1 =



.

αk =αl

aα1 ...αk β1 ...βl .

7.2 Equivalence Problem I: Moser’s Approach

81

Moser [33] proved that after a biholomorphic change of coordinates one can additionally achieve in (7.10) the conditions trF22 = (tr)2 F32 = (tr)3 F33 = 0.

.

(7.12)

Representation (7.10) and (7.12) is called the (Moser) normal form of .. A real analytic curve .γ , which in the normal form has the equation . z = 0, .yn = 0, is called a chain. We note the following important property of normal forms. Let . be in the normal from (7.10), (7.12) near the origin and let .γ be the chain . z = 0, .yn = 0. Consider the maps of the form 

.

z →

azn |a|1/2 , U ( z), zn → bzn + 1 bzn + 1

(7.13)

where U is a unitary transformation of .Cn−1 , .a, b ∈ R, and .a = 0. The map (7.13) preserves the normal form of . and the chain .γ . The converse to this also holds: if we have any other normal form of ., where .γ is still defined by . z = 0, .yn = 0, then this form is obtained from the initial normal form by a transformation (7.13). Equation (7.13) admits the following geometric interpretation: the unitary transformation U defines the change of basis of .H0 (), while the transformation (7.13) with .U = Id defines a new parametrization of .γ . Conditions (7.12) may be also viewed geometrically. First we note that for a given point .p ∈  there exists a unique chain passing trough p in a prescribed noncomplex tangential direction. Furthermore, if . is given by (7.10), then the line . z = 0, .yn = 0 is a chain iff .(tr)2 F32 = 0. Let now A be a unitary .(n − 1) × (n − 1) matrix. There exists a unique mapping (7.11) such that .U (0) = A and in the new coordinates .trF22 = 0. This matrix A can be viewed as a new choice of an orthonormal basis in .H0 (). Finally, consider an admissible reparametrization  ˙ n )  z, q(zn )) ( z, zn ) → ( q(z

.

of .γ . Here .q(0) = 0 and .q(zn ) = q(zn ). Such a change of coordinates preserves (7.10) and the conditions .trF22 = 0, .(tr)2 F32 = 0. If q additionally satisfies a certain third order ODE then the condition .(tr)3 F33 = 0 also holds. Since .q(0) = 0, the solution to this equation is uniquely determined by its first and second order derivatives at the origin. Thus, the normalization of . depends on the following data: (i) the point .p ∈  corresponding to the origin in the normal form; (ii) a noncomplex tangential direction at p defining the chain .γ which has the equation  . z = 0, .yn = 0 in the normal form; (iii) the choice of an orthonormal basis in .Hp ; (iv) two real parameters fixing the parametrization of .γ . The main result of Moser’s theory can be stated as follows.

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7 Local Equivalence of Real Analytic Hypersurfaces

Theorem 7.2 For each choice of the initial data (i)–(iv) there exists a unique biholomorphic mapping h taking . to the normal form. The normal form of the sphere is .yn = | z|2 . Every choice of the initial condition (i)-(iv) determines a unique linear-fractional automorphism of the sphere. Therefore, by Moser’s theorem every local automorphism is global. We obtain the Poincaré-Alexander theorem, see Sect. 2.2. Suppose that . is given by the normal form (7.10). Every admissible curve in . can be expressed as .(z1 , . . . , zn−1 ) = p(xn ) = (p1 (xn ), . . . , pn−1 (xn )). Moser proved that the vector function p satisfies the second order ordinary differential equation

.

d 2p dp dp , p, = Q x , p, , n dxn dxn dxn2

(7.14)

where Q is a rational function in .dp/dxn , .dp/dxn with the coefficients analytic in .xn , p, p. The denominator of Q vanishes exactly on the complex tangent directions of . (i.e., direction in .H ()). By the existence and uniqueness of solutions of ODE, given a noncomplex tangent direction, there is a unique chain in this direction through a prescribed point. Let .ζ ∈  be a point and .v ∈ Tζ (). Denote by .ω(v) the angle between v and .Hζ (). If .γ is a real curve on ., we denote by .|γ | its Euclidean length. As another consequence of the existence and uniqueness theorem we have the following

.

Lemma 7.3 Let K be a compact subset of . and .ω0 ∈ (0, π/2). There exists .δ = δ(K, ω0 ) > 0 such that for every .ζ ∈ K and every .v ∈ Tζ () with .ω(v) ≥ ω0 , there is a chain .γ : [0, 1] → , satisfying .γ (0) = ζ , .(dγ /dt)(0) = v, .|γ | = δ and .ω((dγ /dt)(t)) ≥ ω0 /2 for all .t ∈ [0, 1].

7.3

Local Equivalence II: The Cartan-Chern Approach

Cartan’s equivalence problem (in local form) can be stated as follows. Let U and V be 1 , . . . , ωn ) and . = (1 , . . . , n ) are coopen subsets in .Rn . Suppose that .ωU = (ωU V U V V frames (bases of 1-forms) on U and V respectively. Consider a prescribed linear group G. The problem is to determine all diffeomorphisms f : U → V,

.

satisfying f ∗ V = ωU γU V with γU V ∈ G.

.

7.4 Fefferman’s Metric

83

In the original work of E. Cartan, the group G is allowed to vary from point to point. Note that many natural equivalence problems in differential geometry (for Riemannian structures, differential equations, CR structures, etc.) can be represented in this form for an appropriate choice of co-frames and the group G. The approach of E. Cartan to the solution of the above equivalence problem is based on the observation that the problem has a (relatively) simple solution in the case of the trivial group .G = {e}. Even then a complete solution of the equivalence problem is not quite explicit. In most cases this method provides a number of geometric invariants of the problem at hand (a common application of these invariants is to show that the equivalence problem does not admit any solution). The equivalence problem is considered to be solved if it is reduced to the problem with .G = {e}. The main idea of this reduction consists of the introduction of a (finite) sequence of larger spaces reducing the group G at every step. The procedure proposed by E. Cartan comprises several iterated steps. First, the coframes .ω and . must be extended in an equivariant way to .U × G and .V × G. There the first structure equations for .dω and .d can be written. These equations contain the so-called torsion terms and the special procedure of absorption of these terms allows one to reduce the group. Note that this requires the expansion of the initial system to a higher bundle. In the case of Levi nondegenerate hypersurfaces in .Cn (not necessarily real analytic but sufficiently smooth) this approach leads to a complete solution of the local equivalence problem. This was achieved by Cartan [30] for .n = 2 and by Chern [33] and Tanaka [142] in all dimensions. The adjacent equivalence problem for systems (7.8) was solved by Hachtroudi [75] using Cartan’s method. This problem can be also viewed in terms of Cartan’s theory of the projective connections. This approach to the Segre geometry is developed by Chern [32] and Burns and Shnider [28].

7.4

Fefferman’s Metric

Finally, we briefly discuss another approach to the equivalence problem that was introduced by Fefferman [61], see also Beals et al. [10]. The main object of study is a conformal Lorentz metric .ds 2 defined on the product .D × S 1 , where .D ⊂ Cn is a strictly pseudoconvex domain. This metric is invariant under biholomorphic maps and features a distinguished family of invariant curves, namely, the null-geodesics, defined by .ds 2 , i.e., those whose tangents are of length 0. In fact, it was shown by Burns et al. [29] and Webster [157] that the Chern-Moser chains on bD discussed above arise precisely as projections from .bD × S 1 onto bD of these geodesics. Informally, Fefferman’s metric is obtained as follows: from the Bergman kernel on a strictly pseudoconvex domain D it is possible to construct a biholomorphically invariant nondegenerate metric on .D × (C \ {0}) = D × S 1 × R. Although this metric cannot be computed inside the domain, one can evaluate its Taylor series to a certain order at bD. Its restriction to the boundary gives a 2-tensor, which can then be further collapsed to a

84

7 Local Equivalence of Real Analytic Hypersurfaces

conformal 2-tensor on .bD × S 1 . This turns out to be a nondegenerate conformal Lorentz metric. The final result can be formulated in the following way. Set S 1 = bD = {eiθ : θ ∈ [0, 2π )},

.

and let r be a negative function on D such that .r = 0 on bD, .dr = 0 near bD, and  J (r) := (−1) det n

.

r ∂r ∂zj

∂r ∂zk ∂2r ∂zj ∂zk

= −1 + o(r 2 ).

(7.15)

j,k=1,..,n

To achieve the latter condition one can start with a defining function r, replace it with r · [J (r)]−1/(n+1) so that .J (r) = −1, and then replace this r with .r + ηr s , where .η is a suitable smooth function. One can verify that its Levi form is positive-definite on each holomorphic tangent space .Hp (bD). Define a Lorentzian metric on .bD × S 1 by

.

ds 2 = −

.

n  ∂ 2r i (∂r − ∂r)dθ + dzj dzk . n+1 ∂zj ∂zk j,k=1

This metric is invariant under biholomorphic maps in the following sense: let .F : D → D  be a biholomorphic map between strictly pseudoconvex domains with the metrics .ds 2 and  2 respectively. Define its lift .F˜ : bD × S 1 → bD  × S 1 by setting .ds F˜ : (z, θ ) → (F (z), θ − arg det DF (z)).

.

Then 2 ds 2 = | det DF |2/(n+1) F˜ ∗ (ds  ).

.

(7.16)

Since the construction is local, it remains invariant under locally biholomorphic mappings defined in a neighbourhood of a point in bD. Fefferman’s metric admits an intrinsic construction, see Burns et al. [29]. In [157] Webster studied further pseudo-conformal invariants of hypersurfaces that can be derived from approximate solutions of the Monge-Ampère equation. Finally, Lee [92] generalized the intrinsic characterization of the metric to abstract CR manifolds of hypersurface type. Each of the methods described in this section gives useful information concerning local properties of biholomorphic mappings between strictly pseudoconvex or Levi nondegenerate hypersurfaces: precise bounds on the dimension of the automorphism group, efficient parametrization of biholomorphisms by their second order jets, etc. We note that the methods of Moser and Cartan-Chern admit some generalizations to a wider class of real hypersurfaces (with special Levi degeneracies) or to real manifolds of higher codimension. However, in these cases the approach based on Segre varieties often turns out to be the most convenient and flexible.

8

Geometry of Real Hypersurfaces: Analytic Continuation

One of the striking phenomena in several complex variables is analytic continuation of germs of holomorphic mappings between real analytic hypersurfaces. Conceptually this can also be viewed as prolongation of local equivalence of two real analytic hypersurfaces. The general problem can be formulated as follows: Let .1 and .2 be connected real analytic hypersurfaces in .Cn , and let f be the germ of a biholomorphic map at a point .ζ ∈ 1 such that .f (1 ) ⊂ 2 . Under what conditions on .1 and .2 , does f extend analytically along any path on .1 ? This questions is only interesting for .n > 1, since for .n = 1 a real analytic curve in .C has no local invariants, and analytic continuation is not possible in general. For example, the function f (z) =

.

e1/z − i e1/z + i

is holomorphic on the negative real axis and maps it to the unit circle, but f does not extend holomorphically to the origin. In the general problem above it is natural to assume that .2 is compact and strictly pseudoconvex. The latter ensures that the extension is single-valued. Indeed, consider the following two examples. The map √ f (z1 , z2 ) = ( z1 , z2 )

.

admits a biholomorphic germ .fζ at any point .ζ on .1 = {|z1 |2 + |z2 |2 = 1} with .ζ1 = 0, and .fζ (1 ) ⊂ 2 , where .2 = {|w1 |4 + |w2 |2 = 1}. The germ .fζ clearly extends as a locally biholomorphic map along any path on .1 \ {z1 = 0}, but does not admit a singlevalued continuation to the points on .1 , with .z1 = 0. This is the spherical case. Note that the set .1 ∩ {z1 = 0} is mapped by f into weakly pseudoconvex points of .2 . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_8

85

86

8 Geometry of Real Hypersurfaces: Analytic Continuation

Another example, this time nonspherical, is given by taking 1 = {z ∈ C2 : y2 = |z1 |2 + |z2 |2 + |z1 |8 },

.

2 = {w ∈ C2 : v2 = |w1 |4 + |w1 |16 + |w2 |2 }.

.

These hypersurfaces are compact and real analytic, .1 is strictly pseudoconvex, and .2 is strictly pseudoconvex except where .w1 = 0. The same map f sends .1 \ {z1 = 0} to .2 but does not extend holomorphically to .1 ∩ {z1 = 0}. The hypersurfaces .1 and .2 are nonspherical, i.e., not locally biholomorphic to the unit sphere .S 3 . To see this, consider the map  (z1 , z2 ) →

.

−iz2 z1 , z2 − i z2 − i

 ,

which sends .1 to a hypersurface given by the equation y2 = |z1 |2 +

.

|z1 |8 . |z2 + i|2

By solving this for .y2 we obtain y2 = |z1 |2 + |z1 |8 φ(z1 , z1 , x2 ),

.

where .φ ≡ 0. This means Moser’s normal form (7.10) for .1 contains nonvanishing terms Fkl , and this implies that it is nonspherical. Consequently, .2 is also nonspherical. The next example is due to Burns and Shnider [27]. It shows that such analytic continuation is not always possible even if both hypersurfaces are compact and strictly pseudoconvex. Consider a hypersurface

.

2 = {z ∈ C2 : sin(ln |z2 |) + |z1 |2 = 0, e−π ≤ |z1 | ≤ 1}.

.

(8.1)

Note that .2 is compact strictly pseudoconvex but not simply-connected. Further, this hypersurface is spherical—at every point it is locally biholomorphic to the unit sphere in .C2 . For example, the map √ g(z1 , z2 ) = (z1 / z2 , ei ln z2 )

.

sends the open subset of the unbounded realization of the unit sphere 1 = {z ∈ C2 : y2 = |z1 |2 , y2 ≥ 0, z2 = 0}

.

8.1 Extension of Germs of Holomorphic Mappings, I: The Spherical Case

87

into .2 (we choose the branch of the logarithm so that .0 ≤ Im ln z2 ≤ π ). Nevertheless, no local equivalence from the unit sphere to .2 can be extended holomorphically along all paths. In particular, the map g above cannot be extended to .z = 0. As it turns out the reason for nonextension is that the hypersurface .2 is spherical and not simply-connected. In the next section we show that any germ of a biholomorphic map from a real analytic strictly pseudoconvex hypersurface .1 into the unit sphere admits analytic continuation along any curve in .1 . This is the spherical case. In particular, this implies that a strictly pseudoconvex hypersurface is spherical if and only if it is spherical near one of its points. In Sect. 8.2 we show that in the nonspherical case, that is, when .j are not locally equivalent to the unit sphere, the germ .f : 1 → 2 admits analytic continuation along .1 without the assumption that .2 is simply-connected. Both results are due to Pinchuk [107, 108]. Finally, in Sect. 8.3 we show how the extension of the map f along curves on .1 can be extended to a map between domains bounded by .j .

8.1

Extension of Germs of Holomorphic Mappings, I: The Spherical Case

In this section we prove the analytic continuation result in the spherical case, following the argument of Pinchuk [107]. Theorem 8.1 Let . ⊂ Cn , .n > 1 be a smooth real analytic strictly pseudoconvex hypersurface . Let .U ⊂ Cn be a connected neighbourhood of a point .p ∈ , and let .f : U → f (U ) be a biholomorphic mapping such that .f (U ∩ ) is contained in the unit sphere .bBn . Then f extends analytically along any path in . as a locally biholomorphic mapping from . to .bBn . The proof of this theorem, which is the content of Sect. 8.1, consists of several steps, one of which is a meromorphic extension of the map f . We first note that meromorphic maps on real hypersurfaces with values in the unit sphere have special properties. Lemma 8.2 Let .S ⊂ G be a connected real analytic strictly pseudoconvex hypersurface in a domain .G ⊂ Cn with a defining function .ρ and .S0 ⊂ S be an nonempty open subset. Suppose that a mapping .f = (f1 , . . . , fn ) is meromorphic on G and holomorphic on .S0 . Furthermore, assume that .f (S0 ) is contained in the unit sphere .bBn . Denote by .S ∗ the set of all points in S where f is holomorphic. Then the following holds: (a) .f (ζ ) ∈ bBn for every .ζ ∈ S ∗ ; (b) For any component .fk the intersection of its set of poles with S is empty; (c) for every point .ζ ∈ S there exists a neighbourhood U such that f is holomorphic on .U ∩ {z ∈ G : ρ(z) < 0}.

88

8 Geometry of Real Hypersurfaces: Analytic Continuation

Proof (a) The identity n  .

fk f k = 1

(8.2)

1

holds on .S0 . Since S is connected and the poles of f do not divide S, by uniqueness this also holds on .S ∗ . (b) For any component .fk , .|fk (z)| → +∞ as z tends to a pole; but by part (a), .|fk | ≤ 1 on a dense subset .S ∗ of S; (c) The claim is obvious if .ζ ∈ S ∗ . Assume that .ζ ∈ S \ S ∗ . It follows by part (b) that .ζ is an indeterminacy point for some component .fk . We may assume that .fk = φ/ψ, where .φ(ζ ) = 0, .ψ(ζ ) = 0, the functions .φ, .ψ are coprime and holomorphic in a neighbourhood V of .ζ . Consider the decomposition .ψ = ψ1 . . . ψl of .ψ into irreducible factors. Set P = {φ = 0}, Q = {ψ = 0}, Qj = {ψj = 0}, j = 1, . . . , l.

.

If (c) does not hold, there exists j such that for any neighbourhood U of .ζ the intersection .U ∩ Qj ∩ {z ∈ G : ρ(z) < 0} is not empty. Note that the function .1/ψj cannot be holomorphic on .U ∩ {ρ > 0}. Indeed, in this case it would extend holomorphically to .ζ by the strict pseudoconvexity of S. Hence, .U ∩ Qj ∩ {z ∈ G : ρ(z) > 0} is also nonempty. Consider some points z1 ∈ U ∩ Qj ∩ {z ∈ G : ρ(z) < 0}, z2 ∈ U ∩ Qj ∩ {z ∈ G : ρ(z) > 0}.

.

Since the functions .φ and .ψ are coprime, the dimension of the complex analytic set l l / P , .l = 1, 2. Since .P ∩ Q is at most .n − 2. Therefore, one can choose .z such that .z ∈ .Qj is irreducible, one can assume that .(Qj \ P ) ∩ U is connected. Thus, there exists a path .γ in .(Qj \ P ) ∩ U joining .z1 and .z2 . At any point of .γ ∩ S we have .φ = 0, .ψj = 0. But this contradicts (b). The lemma is proved. 

8.1.1

The Reflection Principle Revisited

Here we give a different proof of the Reflection principle (Theorem 2.1) for strictly pseudoconvex domains. In the proof we will also setup some technical tools used in the proof of Theorem 8.1.

8.1 Extension of Germs of Holomorphic Mappings, I: The Spherical Case

89

Theorem 8.3 Let . j ⊂ Cn , .j = 1, 2, be domains and .j ⊂ b j be (relatively open) real analytic strictly pseudoconvex hypersurfaces. Let .f : 1 → 2 be a holomorphic mapping of class .C 1 ( 1 ∪ 1 ) such that .f (1 ) ⊂ 2 . Then f extends holomorphically to a neighbourhood of .1 in .Cn . Proof The result is local. Without loss of generality assume that .0 ∈ 1 and .f (0) = 0. Denote by .ρj a defining function of .j near the origin. It suffices to consider the case when f is not constant. Then .f : 1 → 2 is a local diffeomorphism and .Jf (0) = 0. We view the defining function .ρ1 (resp. .ρ2 ) as a power series in z and .z (resp. w and .w). By strict pseudoconvexity we may assume that ρ1 = yn + |z|2 + o(|z|2 ),

.

and ρ2 = vn + |w|2 + o(|w|2 ).

.

Consider the tangential conjugate Cauchy-Riemann operators on .1 : Xj =

.

∂ρ1 ∂ ∂ρ1 ∂ − , j = 1, . . . , n − 1. ∂zn ∂zj ∂zj ∂zn

(8.3)

Then, since .f (1 ) ⊂ 2 , we have ρ2 (f (z), f (z)) = 0, z ∈ 1 .

.

(8.4)

Since the restriction of f to .1 is CR, we have .Xj f = 0. Applying the operators .Xj and using the chain rule, we obtain additional .n − 1 equations

.

n  ∂ρ2 (f (z), f (z))Xj fk (z) = 0, j = 1, . . . , n − 1. ∂wk

(8.5)

k=1

In the vector-valued function form we have Xf = (X1 f1 , . . . , X1 fn , . . . , Xn−1 f1 , . . . , Xn−1 fn ).

.

Since .2 is strictly pseudoconvex and the Jacobian of f is nondegenerate, we can apply the Implicit function theorem to the system (8.4) and (8.5). This implies that there exists a 2 function H holomorphic in a neighbourhood of the point .(0, Xf (0)) ∈ Cn such that f (z) = H (f (z), Xf (z)).

.

(8.6)

90

8 Geometry of Real Hypersurfaces: Analytic Continuation

Now consider a family .lc (zn ) = (c, zn ) of complex lines, depending on a parameter .c ∈ Cn−1 . Every intersection .γc = lc ∩ 1 is a real analytic curve in .1 . The coefficients of the operator .Xj are real analytic, and therefore, their restrictions on each .γc extend holomorphically to a neighbourhood of .γc in the line .lc . Hence, one can extend every restriction .H (f, Xf )|γc holomorphically to .lc ∩ 1 . Using (8.6) and applying the Schwarz reflection principle in one variable, we conclude that f extends holomorphically along every .lc . By the Hartogs lemma, f extends holomorphically to a neighbourhood of the origin. This concludes the proof. 

8.1.2

Extension Across Generic Submanifolds

The next step in the proof of Theorem 8.1 is to study the problem of meromorphic extension of f across generic submanifolds. We assume that the map f is meromorphic on a domain .V ⊂  with values in the unit sphere, and bV is a smooth .2n − 2-dimensional submanifold of ., which is generic near a point .ζ0 ∈ bV , i.e., .Tζ0 bV is a generic subspace of .Cn . We will show that f extends to a neighbourhood of .ζ0 as a meromorphic mapping. Since . is strictly pseudoconvex, there exists a neighbourhood G of .ζ0 and a local change of coordinates such that in the new coordinate system the hypersurface .S =  ∩ G is strictly geometrically convex. Then we can find a real hyperplane . ⊂ Cn given by the  equation .Re A(z) = 0, where .A(z) = nk=1 ak zk + bk , with the property that .ζ0 ∈  and the set S + = S ∩ {z ∈ Cn : Re A(z) > 0}

.

is contained in V . Then .M := bS + = S∩ is also generic near the point .ζ0 . The following holds. Lemma 8.4 Assume that .f = (f1 , . . . , fn ) is a meromorphic mapping on .S + and + where f is holomorphic. Then f extends .|f (z)| = 1 for every point .z ∈ S meromorphically to .ζ0 . Proof Let .ρ be a defining function of S in G. It follows from Lemma 8.2 that f extends holomorphically to a one-sided (where .ρ < 0) neighbourhood of .S + . Then, using the strict convexity of S and the Kontinuitätssatz, one can extend f holomorphically to the set .{z ∈ G : Re A(z) > 0, ρ(z) < 0}. Let .ζ 1 ∈ S + . Consider the hyperplane .1 containing .Tζ 0 (M) and .ζ 1 . We may assume  that it is given in the form .1 = {z ∈ Cn : Re A1 (z) = 0}, where .A1 (z) = nk=1 ak zk +b1 . Clearly, we have the inclusion 1 ∩ S ⊂ S + ∪ {ζ 0 }.

.

(8.7)

8.1 Extension of Germs of Holomorphic Mappings, I: The Spherical Case

91

Replacing .A1 with .−A1 if necessary, we may also assume that {z ∈ G : ρ(z) < 0, Re A1 (z) > 0} ⊂ {z ∈ G : ρ(z) < 0, Re A(z) > 0}.

.

(8.8)

Denote by L the complex hyperplane in .1 containing .ζ 0 . Then L is transverse to S and by (8.7) we have .L ∩ S ⊂ S + ∪ {ζ 0 }. By Lemma 8.2 the map f does not have poles on .S + . Since the set of indeterminacy of f has the complex codimension at least 2, one can find a point .ζ ∗ ∈ S + ∩ L such that f is holomorphic in a neighbourhood U of .ζ ∗ . Applying to the identity (8.2) the tangential operators .Xj given by (8.3) we obtain a linear system with respect to the components of .f . As in the proof of Theorem 8.3, using Cramer’s rule we may solve this system so that for .z ∈ S + ∩ U we have fk =

.

k , k = 1, . . . , n,

(8.9)

where . k , . are polynomials in .fl and .Xj fl . Let l be the complex line through .ζ 0 and .ζ ∗ . Fix .ω ∈ Cn \ {0} so that the complex lines l(ζ ) = {z ∈ Cn : z = λω + ζ, λ ∈ C}

.

are parallel to l. In particular, .l(ζ ∗ ) = l. Then there exists .ε > 0 such that for all .ζ ∈ U the restriction of partial derivatives .∂ρ/∂zk to the curves .γ (ζ ) = S ∩ l(ζ ) extend holomorphically to .l(ζ ) ∩ {z ∈ G : |ρ(z)| < ε}. Note that l(ζ ) ⊂ {z ∈ Cn : Re A1 (z) > 0},

.

and in view of (8.7) the restriction .f |l(ζ ) is holomorphic on .l(ζ ) ∩ {z ∈ G : ρ(z) < 0}. Therefore, for all .ζ the restrictions of the functions .Xj fk , . k , and . to .l(ζ ) ∩ S + extend holomorphically to .l(ζ ) ∩ {z ∈ G : −ε < ρ(z) < 0}. Now it follows by the one-dimensional Schwarz reflection principle and (8.9) that the functions .f k (already antimeromorphic on .l(ζ ∩ {z ∈ G : ρ(z) ≤ 0}) extend antimeromorphically to .l(ζ ) ∩ {z ∈ G : ρ(z) < δ} for some .δ > 0. We apply the extension of the classical Hartogs lemma to the class of meromorphic functions due to Rothstein [122] (see also Shiffman [134]). This implies that the functions .fk extend meromorphically to the set   {z ∈ G : ρ(z) < δ} ∩ ∪ζ l(ζ ) ,

.

where the union is taken over all .ζ ∈ U ∩ {z ∈ Cn : Re A1 (z) > 0}. Therefore, there exists a neighbourhood .U1 of .ζ 0 such that all functions .fk extend meromorphically to + .U 1 = {z ∈ U1 : Re A1 (z) > 0}.

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8 Geometry of Real Hypersurfaces: Analytic Continuation

Choose a point .ζ 2 ∈ S + \ 1 and consider the hyperplane 2 = {z ∈ Cn : Re A2 (z) = 0},

.

containing .Tζ 0 (M) and .ζ 2 . Repeating the previous argument, we conclude that there exists a neighbourhood .U2 of .ζ 0 such that all functions .fk extend meromorphically to the set + .U 2 = {z ∈ U2 : Re A2 (z) > 0}. Consider now the domain .U1+ ∪U2+ . The affine space .Tζ 0 (M) = 1 ∩2 is not complex analytic, and therefore, by Kneser’s theorem (see, e.g., [129]), any function holomorphic in .U1+ ∪ U2+ admits holomorphic extension to a full neighbourhood of .ζ0 . This implies that for a suitable neighbourhood V of .ζ0 the envelope of holomorphy of .V ∩ (U1+ ∪ U2+ ) is V . It is well-known (see, e.g., Fuks [70, Ch II]) that the envelope of holomorphy of a domain in .Cn agrees with its envelope of meromorphy, and so we conclude that each .fk extends 

meromorphically to a neighbourhood of .ζ0 . This proves the lemma.

8.1.3

Holomorphic Extension

In this section we show that the meromorphic extension obtained in Lemma 8.4 is, in fact, holomorphic on .. Let . be a (small) neighbourhood of the point .ζ0 where the extended map, which we denote again by f for simplicity, is meromorphic. By Lemma 8.2 f is holomorphic on the strictly pseudoconvex component of . \ , which we denote by . + . Further, f admits the following properties: (a) (b) (c) (d)

f (z) ∈ bBn if f is holomorphic at .z ∈  ∩ ; f doest not have poles on . ∩ ; f is locally biholomorphic on . + , if . is sufficiently small; + n .f ( ) ⊂ B . .

Indeed, properties (a) and (b) follow from Lemma 8.2. We prove (c) arguing by contradiction. Let E = {z ∈ : det Df (z) = 0}.

.

Without loss of generality we may assume that there exists .z0 ∈ E ∩  ∩ . By Theorem 2.3, E does not contain any point .z ∈ , where f is holomorphic, and so the set .E ∩  is contained in N , the indeterminacy locus of f . By the dimension considerations it follows that .E ∩ + = ∅, which proves the claim. Finally, for (d) we need to prove that .|f (z0 )| < 1 for each point .z0 ∈ + . Consider a complex line . through .z0 whose intersection with . + is compactly contained in . . The restriction of f to . is holomorphic on . + ∩ and meromorphic on . ∩. Furthermore, .∩ is not contained in N , as otherwise we would have . + ∩  ⊂ N . By part (a), .|f (z)| = 1 on .( \ N) ∩ . Therefore, .f | is

8.1 Extension of Germs of Holomorphic Mappings, I: The Spherical Case

93

holomorphic on . ∩ , and by the Maximum principle for subharmonic functions and local biholomorphicity of f we obtain .|f (z0 )| < 1. We now show that the meromorphic extension is actually holomorphic near . ∩ . To see this consider the map .g = f + → Bn . We claim that g extends continuously to . ∩ . To apply Theorem 3.7 to the map g we need to verify two conditions: (i) The cluster set of any point .z ∈  ∩ cannot intersect .Bn . This is obvious if .z ∈  is a point where f is already holomorphic. Suppose now that .z ∈  ∩ N. The cluster set .C(g, z) is contained in the fibre of the point z in the graph of the meromorphic map f , which is a complex analytic set in .Cn × CPn of some positive dimension. We claim that if .  ζj → z, then there exists a unique point .z ∈ bBn such that .f (ζj ) → z . By the invariance of Segre varieties given by Eq. (2.2) it follows that the germ of .Qζj at .ζj is mapped by f into .Hf (ζj ) bBn , the complex tangent to the unit sphere at the point .f (ζj ). Therefore, by continuity of the fibres, the fibre in the graph of f of the complex tangent n n  .Hz bB must be contained in the limit of .Hf (ζj ) bB for any limit point .z of the sequence  .f (ζj ). But this implies that there exists only one such .z . This shows that the projection of the fibre of z in the graph cannot intersect the interior of the unit ball in the target space. In particular, .C(g, z) ∩ Bn = ∅. (ii) For any .z ∈  ∩ , the set .C(g, z) contains a strictly pseudoconvex point. This is clearly the case since .C(g, z) ⊂ bBn . Thus, by Theorem 3.7, the map g extends as a .1/2-Hölder continuous map to . ∩ . In fact, by Theorem 5.7, the CR map .g :  ∩ → bBn is a local homeomorphism, and therefore the map f is locally proper near any point .z ∈  ∩ . Then we may apply Corollary 5.6 to conclude that the extension to . is, in fact, smooth. Finally, the holomorphic extension follows by the Reflection principle (Theorem 8.3).

8.1.4

Analytic Continuation Along Curves

We now show that extension of the map f across generic submanifolds on . implies analytic continuation along any curve on .. In fact, it suffices to consider only CR-curves. A smooth curve .γ : [0, 1] →  is called a CR-curve if its tangent vector is contained in .Hγ (t) , .t ∈ [0, 1]. To prove this assertion we recall Sussmann’s theorem [141]: if .D is a collection of smooth vector fields in a domain . ⊂ Rn , and .p ∈ , then there exists a unique smallest submanifold .S ⊂ passing through p such that every vector field in .D is tangent to S at every point of S. By taking .D to be the collection of all smooth CR vector fields on . (i.e., those with values in .H ), we see that .S = , as otherwise S would be a complex hypersurface contained in ., which is impossible because . is strictly pseudoconvex. Further, it follows from Sussmann’s theorem that if . is connected, then any two points in . can be connected by a piecewise smooth curve which is CR on every smooth component. Therefore, for analytic continuation of the map f we only need to consider CR curves.

94

8 Geometry of Real Hypersurfaces: Analytic Continuation

Suppose now that .γ ⊂  is a CR-curve, and f is holomorphic in a neighbourhood of .γ (0). We show that the map f extends analytically along .γ . Suppose that q is the first point on .γ to which f does not extend holomorphically. Near the point q there exists a smooth CR vector field L such that .γ is contained in an integral curve of L. By integrating L we obtain a smooth coordinate system .(t, s) ∈ R×R2n−2 on . such that for any fixed .s0 the segments .(t, s0 ) are contained in the trajectories of L. We may further choose a point .p ∈ γ sufficiently close to q, so that f is holomorphic near p. After a translation, assume that .p = (0, 0). Following [99], for .ε > 0 we define the family of ellipsoids on . by Eτ = {(t, s) : |t|2 /τ + |s|2 < ε},

.

(8.10)

where .ε > 0 is so small that for some .τ0 > 0 the ellipsoid .Eτ0 is compactly contained in the portion of . where f is holomorphic. Then .bEτ is generic at every point except the set  = {(0, s) : |s|2 = ε}.

.

Let further .τ1 > 0 be such that .q ∈ bEτ1 . To prove that f extends holomorphically to a neighbourhood of q we argue by contradiction. For that we assume that .τ ∗ is the smallest positive number such that f does not extend holomorphically to some point on .bEτ ∗ , and assume that .τ ∗ < τ1 . By construction, .τ ∗ > τ0 . Also by construction, near any point .b ∈ bEτ ∗ to which f does not extend holomorphically, the set .bEτ ∗ is a smooth generic submanifold of ., since the nongeneric points of .bEτ ∗ are contained in ., where f is already known to be holomorphic. Then by Lemma 8.4 the map f extends holomorphically to a neighbourhood of b. This shows that .τ ∗ cannot be smaller than .τ1 , which proves that the map f extends holomorphically to q, and therefore along any CR-curve on .. This completes the proof of Theorem 8.1.

8.2

Extension of Germs of Holomorphic Mappings II: The Nonspherical Case

Let . ⊂ Cn , .n > 1, be a connected strictly pseudoconvex real analytic hypersurface. We call . nonspherical if no open subset of . can be mapped biholomorphically onto an open subset of the unit sphere in .Cn . This is equivalent to the condition that at every point .p ∈  in the Moser normal (7.10) and (7.12) for . at p, some coefficient of .Fkl , .k, l ≥ 2, is nonzero. In view of Theorem 8.1 this condition holds on all of . if it holds at some point of .. In this section we prove analytic continuation of germs of biholomorphic maps between nonspherical hypersurfaces.

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

95

Theorem 8.5 Let .j , .j = 1, 2, be connected real analytic strictly pseudoconvex nonspherical hypersurfaces in .Cn , .n > 1. Assume that .2 is compact. Let U be an open neighbourhood of a point .p ∈ 1 such that .1 ∩ U is connected. Suppose that .f : 1 ∩ U → 2 is a nonconstant holomorphic mapping. Then f extends analytically along any path in .1 as a locally biholomorphic mapping with the extension sending .1 to .2 . Note that in the assumption of the theorem, .f is locally biholomorphic if it is not constant. This follows, for example, from Theorem 2.3. From the argument in Sect. 8.1.4 we already know that for the proof of analytic continuation of f along curves on .1 is suffices to consider holomorphic extension of f across generic submanifolds in .1 . So we may consider the following situation: let U be a domain where the map f is defined, .bU ∩ 1 is smooth and .0 ∈ bU ∩ 1 is a generic point, i.e., .bU ∩ 1 is a generic submanifold near the point 0 of real codimension 2. After a biholomorphic change of coordinates we may assume that the one-sided neighbourhood . of .1 near the origin is a strictly convex domain. Consider a real hyperplane given by L = {z ∈ Cn : Re l(z) = 0}

.

for some complex linear function .l(z). Using strict convexity of . , we may choose .l(z) in such a way that the manifold M := L ∩ 1

.

is tangent to .bU ∩ 1 and that the domain + = {z ∈ Cn : Re l(z) ≥ 0} ∩

.

is contained in U . Our goal is to show that .f | + extends holomorphically to a neighbourhood of the origin.

8.2.1

Continuous Extension to M

In this section we establish the following key result. Proposition 8.6 .f | + extends continuously to M. Our proof uses the normal form of Moser and properties of Fefferman’s metric, see Sect. 7.4. Let .ds2 be Fefferman’s metric associated with the strictly pseudoconvex hypersurface ..

96

8 Geometry of Real Hypersurfaces: Analytic Continuation

Lemma 8.7 Let .γ : [a, b] →  be a piecewise smooth curve everywhere transverse to H  (at nonsmooth points the limit directions on .γ are also assumed to be transverse to 2 1 .H ). Then there exists a piecewise smooth lift .γ˜ : [a, b] →  × S such that .ds  = 0 along .γ˜ . .

Observe that it follows from (7.16) that if .F : 1 → 2 is biholomorphic, and .ds2 1 = 0 along .γ˜ , then .ds2 2 = 0 along .F˜ ◦ γ˜ . This will be used in applications. Proof of Lemma 8.7 The desired lift has the form .γ˜ (t) = (γ (t), θ (t)). We need to find a piecewise smooth function .θ (t) satisfying ⎞ ⎛ n n  ∂ 2 r dzj dzk 2i ⎝ ∂r dzj ⎠ dθ + = 0. .− ∂zj dt dt ∂zj ∂zk dt dt n+1 j =1

j,k=1

By transversality of .γ to .H , we have

.

n  ∂r dzj  0. = ∂zj dt j =1

Therefore, the above equation can be solved for .θ on each subinterval of .[a, b] where .γ is smooth. Choosing suitable initial data at the end points of these subintervals, we obtain 

the required function .θ (t) The proof of Proposition 8.6 consists of two steps. First we need to control dilations of chains under biholomorphic mappings. By .ωj (v) we denote the angle between a vector .v ∈ Tp j and .Hp j . Lemma 8.8 Let .j , .j = 1, 2, be real analytic nonspherical hypersurfaces in .Cn , and let n .2 be compact. Consider a bounded domain .U ∈ C such that .1 ∩ U is a nonempty connected set and its closure is contained in .. Assume that .f : U → Cn is a nonconstant holomorphic mapping and .f (1 ∩ U ) ⊂ 2 . Then, given .ε > 0 and .ω0 > 0, there exists .δ > 0 with the following property: for each chain .γ : [−1, 1] → 1 ∩ U and its image .λ = f ◦ γ , the conditions |γ | ≤ δ,

.

ω1 (dγ /dt (0)) ≥ ω0 ,

.

ω2 (dλ/dt (0)) ≥ ω0

.

imply .|λ| ≤ ε.

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

97

Proof Arguing by contradiction, suppose that there exist .ε > 0 and .ω0 > 0 , and a sequence of chains .γk : [−1, 1] → 1 ∩ U , .k = 1, 2, .., such that .ω1 (dγk /dt (0)) ≥ ω0 , .ω2 (dλk /dt (0)) ≥ ω0 , .limk→∞ |γk | = 0, but .|λk | ≥ ε. It follows by Lemma 7.3 that one can parametrize the chains .γk and .λk in such a way that the points .λk (0) divide the chains .λk into two segments of the same length. Without loss of generality we may assume that .

lim γk (0) = a 1 ∈ 1 , lim λk (0) = a 2 ∈ 2 ,

k→∞

k→∞

and dλk (0) dλk (0) −1 dγk (0) dγk (0) −1 1 . lim = v ∈ Ta 1 (1 ), lim = v 2 ∈ Ta 2 (2 ). k→∞ k→∞ dt dt dt dt Since the vectors .v j are not complex tangent, there exist chains .γ0 : [−1, 1] → 1 and .λ0 : [−1, 1] → 2 satisfying the conditions dγ0 (0) dγ0 (0) −1 .γ0 (0) = a , = v1, dt dt 1

and λ0 (0) = a 2 ,

.

dλ0 (0) dλ0 (0) −1 = v2, dt dt

respectively. For every .k = 0, 1, 2, . . . choose local holomorphic coordinate systems z∗ = φk (z), w ∗ = ψk (w)

.

near the points .γk (0) and .λk (0) such that .φk (γk (0)) = 0, .ψk (λk (0)) = 0, and such that with respect to these new coordinates the hypersurfaces .j are given by their normal forms: 1 : yn∗ = | z|2 +



(k)  ∗  Flm ( z , z, xn ),

(8.11)

 ∗  G(k) lm ( w , w, un ).

(8.12)

.

l,m≥2

2 : vn∗ = | w|2 +



.

l,m≥2

We may assume that the parameters defining these normal forms at the limit points γ0 (0), .λ0 (0) are the limits (as .k → ∞) of the parameters at the points .γk (0) and .λk (0)

.

98

8 Geometry of Real Hypersurfaces: Analytic Continuation

respectively. Slightly shrinking the chains .γk and .λk we may further assume that the following conditions are satisfied: (i) the mappings .ψk take the chains .λk to the segments .{ w ∗ = 0, vn∗ = 0, u∗n ∈ [−ε1 , ε1 ]}, with .ε1 > 0 independent of k; (ii) the mappings .φk transform the chains .φk , .k = 1, 2, . . ., into the segments { z∗ = 0, yn∗ = 0, xn ∈ [−tk1 , tk2 ]},

.

where .tk1 tk2 > 0 and .tk1 , tk2 → 0 as .k → ∞; (k) (iii) the coefficients of the polynomials .Glm converge as .k → ∞ to the coefficients (0) of the polynomials .Glm on the segment .[−ε1 , ε1 ]; similarly, the coefficients of the (k) (0) polynomials .Flm converge to the coefficients .Flm on some segment .[−ε2 , ε2 ] with .ε2 > 0. In the coordinates .z∗ = φk (z) and .w ∗ = ψk (w) the inverse mapping .f −1 is a linearfractional transformation 1/2

 ∗

.

z =

ak ak wn∗  ∗ ∗ U . ( w ), z = k n bk wn∗ + 1 bk wn∗ + 1

(8.13)

Here U is a unitary operator on .Cn−1 , .ak , .bk are real, and .ak > 0. Since the transformation (8.13) takes the segment .[−ε1 , ε1 ] of the axis .u∗n to the segment .[tk1 , tk2 ] of the .xn∗ -axis, we have .

− tk1 =

ak ε1 −ak ε1 . , tk2 = bk ε1 + 1 −bk ε1 + 1

Then t1 − t2 k k .|bk | = ≤ ε1−1 , ε1 (tk1 + tk2 ) which implies that .ak → 0 as .k → ∞. Substituting (8.13) into (8.11) we represent the local defining equation of .2 with respect to the coordinates .w ∗ = ψk (w) in the form |bk wn∗ + 1|2  (k) ∗  2 .vn = | w| + Flm ak l,m≥2

 1/2 1/2 ak Uk ( w ∗ ) ak Uk ( w ∗ ) ak wn∗ , , Re . bk wn∗ + 1 bk wn∗ + 1 bk wn∗ + 1

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

99

(k) We have .ak (bk wn∗ + 1)−1 → 0 uniformly as .|wn∗ | < ε1−1 . Furthermore, .Flm = 0 if ∗ .min(l, m) < 2. Passing to the limit, we obtain that in the coordinates .w = ψ0 (w) the hypersurface .2 has the form 1/2

vn∗ = | w ∗ |2 .

.

This is a contradiction because .2 is nonspherical.



The assumptions of Lemma 8.8 impose some restrictions on the direction of the chain λ = f ◦ γ . This is an obstacle in proving Proposition 8.6. To overcome it we use Fefferman’s metric. Without loss of generality we may assume that f is holomorphic on the set

.

V + = {z ∈ V : ρ(z) ≥ 0, y1 > 0},

.

where .ρ(z) = yn + o(|z|)} in a neighbourhood of the origin; furthermore, .M = 1 ∩ {y1 = 0}. In the coordinates .( z, xn ) we consider the vector fields X=

.

∂ ∂ ∂ ∂ − , Y = + . ∂y1 ∂xn ∂y1 ∂xn

Our considerations are local (in a neighbourhood of the origin), so these vector fields are not complex tangent. Given .p ∈ M, denote by .γ(p,X) and .γ(p,Y ) the chains through p in the direction .Xp and .Yp respectively. Fix .α > 0 and a neighbourhood . ∈ 1 of the origin such that . ⊂  ∩ {y1 < α}, and for each .p ∈ ∩ M the chains .γ(p,X) , .γ(p,Y ) are parametrized by .y1 ∈ [−2α, 2α]. The chains .γ(p,X) and .γ(p,Y ) are differentiable with respect to the parameter p. By the Implicit function theorem, we may assume that every point .ζ ∈ + = ∩ {y1 > 0} is contained in a unique chain .γζ1 (y1 ) of the form .γ(p,X) and a unique chain .γζ2 of the form .γ(p,Y ) . Consider a piecewise smooth curve .γζ formed by the segments of the chains .γζ1 and .γζ2 determined by the condition .y1 ∈ [Im ζ1 , α]. Choose a parameter .t ∈ [−1, 1] on .γζ such that .γζ (−1) = γζ1 (α), .γζ (0) = ζ and .γζ (1) = γζ2 (α). Let now r be a function that defines Fefferman’s metric for .1 , see Sect. 7.4. Then it is easy to check that i∂r, dγζ /dt > 0.

.

Furthermore, there exists .ω0 > 0, .c1 > 0 such that for all .ζ ∈ + and all .t ∈ [−1, 1] we have ω1 (dγζ /dt (t)) ≥ ω0 , |γζ | ≤ c1 .

.

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8 Geometry of Real Hypersurfaces: Analytic Continuation

Lemma 8.9 Let .λ = f ◦ γζ be the image of .γζ . Then i∂ r˜ , dλζ /dt (t) > 0, t ∈ [−1, 1],

.

where .r˜ is a function defining Fefferman’s metric for .2 . Proof We have .r˜ ◦ f = βr, where .β > 0 on .1 . Hence, i∂ r˜ , dλζ /dt = i∂(˜r ◦ f ), dγζ /dt = β∂r, dγζ /dt > 0.



.

Lemma 8.10 The mapping f is equicontinuous on the family of curves .{γζ }, .ζ ∈ + . Proof Since the hypersurface .2 is strictly pseudoconvex and compact, there exists .c2 > 0 such that for each .w ∈ 2 and each vector .v ∈ Hw (2 ) one has n  .

j,k=1

∂ 2 r˜ vj v k ≥ c2 |v|2 . ∂wj ∂wk

(8.14)

Fix .ω0 > 0 small enough so that this estimate holds for every .w ∈ 2 and every .v ∈ Tw (2 ) with .ω2 (v) < ω0 . Given .ζ ∈ + we divide the domain .[−1, 1] of the parameter t into two parts, .Iζ1 and .Iζ2 , as follows: .ω2 (dλζ /dt (t)) > ω0 when .t ∈ Iζ1 and .ω2 (dλζ /dt (t)) ≤ ω0 when j

j

t ∈ Iζ2 . We use the notation .γζ for the pieces of .γζ parametrized by .Iζ , .j = 1, 2. By Lemma 8.7 every curve .γζ admits a lift

.

γ˜ζ = (γζ , θζ ) : [−1, 1] → 1 × S 1 ,

.

such that .ds1 vanishes along .γ˜ζ . Then we have .

−

 ∂ 2 r˜ 2i ∂ r˜ ∂ θ˜ζ + dwj dw k = 0 ∂wj ∂wk n+1

(8.15)

j,k

on .λ˜ ζ = (λζ , θ˜ζ ), where .θ˜ζ = θζ − arg Jf . By the compactness of .2 , there exists .c3 > 0 such that  ∂ 2 r˜ vj v k ≤ c3 |v|2 , w ∈ 2 , v ∈ Cn . . j,k ∂wj ∂wk

(8.16)

On the other hand, the form .∂ r˜ vanishes on .H (2 ). Therefore, there exists .c4 > 0 such that along each curve .λ1ζ (where .ζ ∈ + ) one has

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

101

i∂ r˜ > c4 |dw|.

.

From this and (8.15) and (8.16) we conclude that |d θ˜ζ | < c5 |dw|

.

(8.17)

on .λ1ζ , with .ζ ∈ + . We claim that the lengths of the curves .λ1ζ are uniformly bounded. Indeed, given .ζ ∈ + consider the set of points of .λζ where .ω1 (dλζ /dt) ≥ ω0 /2. Denote by .λ3ζ the union of components of this set intersecting .λ1ζ . Clearly, .λ1ζ ⊂ λ3ζ . It follows from Lemma 7.3 that there exists an .ε > 0 such that the length of each component of .λ3ζ is not less that 3 .ε. Now divide .λ into arcs of length bounded from below and from above by .ε and .2ε ζ respectively. It follows from Lemma 8.8 that the length of the preimage by f of every arc is not less than some .δ = δ(ε) > 0. But the length .|γζ | is uniformly bounded. Therefore, the number of the arcs above in .λ3ζ is also bounded uniformly in .ζ . Therefore, the length of .λ3ζ (and so the length of .λ1ζ ) are uniformly bounded. Together with (8.17) this implies the estimate VarI 1 θ˜ζ < c6 ,

.

ζ

where .Var denotes the total variation. The set .1+ = 1 ∩ {y1 > 0} is simply connected; therefore, the function .arg Jf is well-defined on .1+ . The function θ˜ζ (t) = θζ (t) − arg Jf (γζ (t))

.

satisfies |θ˜ζ (1) − θ˜ζ (−1)| ≤ |θζ (1) − θζ (−1)| + | arg Jf (γζ (1)) − arg Jf (γζ (−1))|.

.

By construction of the curves .γζ , the expressions .|θζ (1) − θζ (−1)| are uniformly bounded. Furthermore, the end points of the curves .γζ belong to the set .1 ∩ {y1 = α} compactly contained in .1+ . This implies the estimate |θ˜ζ (1) − θ˜ζ (−1)| < c7 , ζ ∈ + .

.

(8.18)

Now arguing by contradiction, assume that Lemma 8.10 is false. Then there exist .ε > 0, a sequence of points .(ζm ) in . + , and a sequence of arcs .γm ⊂ γζm such that .|γm | < 1/m, and the length of the image of every such curve is equal to .ε, .m = 1, 2, . . .. One can further assume that .γm : [am , bm ] → 1 , where .γm (t) = γζm (t) for .t ∈ [am , bm ] ⊂ [−1, 1]. By Lemma 8.8 we have .min[am ,bm ] ω1 (dλm /dt (t)) → 0 when .m → ∞. In view of Lemma 7.3 one may assume that

102

8 Geometry of Real Hypersurfaces: Analytic Continuation

ωm = max ω1 (dλm /dt (t)) → 0,

.

[am ,bm ]

as .m → ∞. Along each .λm we have the estimate |∂ r˜ | < c8 ωm |dw|.

.

Without loss of generality we may assume that .λm ⊂ λ2ζm . It follows from (8.14) and (8.15) that

|d θ˜m | > (c9 /ωm )|dw|

.

along .λm . Since .|λm | = ε, and .ωm → 0, we obtain that .Var[am ,bm ] θ˜m → ∞ as .m → ∞. On the other hand, we have along .λ2ζm : i∂ r˜ > 0,



.

j,k

∂ 2 r˜ dwj dw k > 0. ∂wj ∂w k

This implies that .d θ˜ζm /dt > 0. Therefore, the functions .θ˜ζm are increasing on the sets .Iζ2m , and .Varθ˜ζm → ∞ as .m → ∞. Recall that .Var θ˜ζm are uniformly bounded on .Iζ1m . Thus, .

˜ θζm (1) − θ˜ζm (−1) → ∞, m → ∞.

But this contradicts (8.18). The lemma is proved.



Now we are able to conclude the proof of Proposition 8.6. Consider a local system of coordinates .(y1 , u), .u = (u1 , . . . , u2n−1 ) in a neighbourhood of the origin on .1 , such that the chains .γ(p,X)) have the equations .u = const. It follows by Lemma 8.10 that the restriction .u → f (y1 , u)|y1 =c converge uniformly (in u) as .c → 0+ . The limit .f (0, u) is a continuous mapping in a neighbourhood U of the origin. Given .ε > 0 choose a neighbourhood .U˜ ⊂ U of the origin such that for each .u ∈ U˜ one has .|f (0, u)−f (0, 0)| < ε/2. By Lemma 8.10 there exists .δ > 0 such that for .u ∈ U˜ and .0 ≤ y1 < δ we have .|f (y1 , u) − f (0, u)| < ε/2. Therefore, .|f (y1 , u) − f (0, 0)| < ε. This proves the proposition.

8.2.2

Analytic Extension

In this section we prove holomorphic extension of .f | + to the origin. We still need some preliminary results.

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

103

Lemma 8.11 Let .Ut , .t ∈ [0, t0 ], be a family of bounded domains in .C, let .ϕt and .ψt be holomorphic functions on .Ut for every t, and set .ht = ϕt /ψt . Suppose that the following conditions are satisfied: (i) the boundaries .bUt are piecewise smooth curves, continuously depending on t; (ii) the functions .ϕt and .ψt depend continuously on t; (iii) for .t > 0 the functions .ϕt and .ψt are continuous on .U t . Further, .ψt = 0 on .bUt for .t ∈ (0, t0 ], and there exists .K > 0 such that .|ht | < K on .bUt for all .t ∈ (0, t0 ]. (iv) there exists a point .p ∈ bU0 such that .ϕ0 , .ψ0 are holomorphic on .U 0 \ {p} and .ψ0 = 0 on .U 0 \ {p}. Then .|h0 | < K on .U0 . Proof It follows by the assumptions that for every .t = 0 the functions .ψt attain in .Ut the same number of zeros (counting multiplicity). Consider the functions .bt , holomorphic in .Ut and continuous on .U t , such that their zeros precisely coincide with the zeros of .ψt , and .|bt | = 1 on .bUt . (After a conformal parametrization of .Ut by the unit disc, the functions .bt are the Blaschke products.) The products .bt ht are holomorphic on .Ut and continuous on .U t ; therefore, .|bt ht | < K on .Ut . Since .ψ0 = 0 on .U0 , the zeros of .ψt converge to p when .t → 0. Using the explicit form of the Blaschke product we see that .limt→0 |bt (z)| = 1 for every .z ∈ U0 . Therefore, |h0 (z)| = lim |ht (z)| = lim |bt (z)ht (z)| < K,

.

t→0

t→0



which proves the lemma.

Lemma 8.12 Suppose that a family of real analytic strictly pseudoconvex hypersurfaces t , .t ∈ [a, b], is defined in a neighbourhood U of the origin by

.

yn = |  z|2 + F t ( z,  z, xn ),

.

(8.19)

where the functions .F t = o(| z|2 + xn2 ) and depend continuously on t. Let .

d 2p = Qt (xn , p, p, dp/dt, dp/dt) dt 2

be the differential equations that define chains on .t . Then .Qt (0, 0, 0, dp0 /dt, dp 0 /dt) depend continuously on t for any fixed direction .dp0 /dt ∈ Cn−1 . Proof A choice of a direction .dp0 /dt ∈ Cn−1 determines the direction of a chain .γt in .t at the origin. One can assume that .dp0 /dt = 0. Indeed, it suffices to apply a suitable automorphism of the hypersurface .yn = | z|2 taking the above direction to be the direction

104

8 Geometry of Real Hypersurfaces: Analytic Continuation

of the .xn -axis. Clearly, this preserves (8.19) and continuous dependence on t. By the construction of Moser’s normal form, see Sect. 7.2, there exists a local biholomorphic change of coordinates 

.

z =  g(z∗ ), zn = gn (z∗ ),

satisfying the conditions .

∗

Re gn (0, xn∗ ) = xn∗ ,  g(z∗ ) =  z + o(|z∗ |),

(8.20)

that sends .γt to a segment of the real line . z∗ = 0, yn∗ = 0. Furthermore, .t in the new coordinates is defined by ∗

yn∗ = | z |2 +

.



∗

Fklt ( z ,  z∗ , xn∗ ),

k,l≥2 t t where .trF22 = 0, .(tr)2 F32 = 0. This equation is in the normal form up to a parametrization of the chain .γt . Suppose that in the initial coordinates z the chain was defined by the equation . z = p(xn ). Then it follows from (8.20) that .p(xn ) =  g(0, xn ); hence,

Qt (0, 0, 0, dp/dt, dp/dt) =

.

d 2p ∂ 2 ( g) (0) = (0, 0). dt 2 ∂zn2

It follows from the computation of Moser’s normal form that .∂ 2 ( g)/∂zn2 (0, 0) is a combination of the coefficients of the expansion of .F t in . z, . z, and .xn of degree not higher than 5. This proves the lemma. 

Now assume that we are in the hypothesis of Theorem 8.5. Using the same notation we have the following Lemma 8.13 For every generic point .ζ ∈ M the exist a chain .γ through .ζ and a neighbourhood U of .ζ , such that the following holds: (a) .γ ∩ U ⊂ 1+ ∪ {ζ }, where .1+ = + ∩ 1 ; (b) if S is a complex surface containing .γ (that is, .γ = 1 ∩ S), then S ∩ U ∩ {ρ ≤ 0} ⊂ + ∪ {ζ }.

.

Proof (a) Set .ζ = 0 and choose a chain .γ˜ (t) such that .γ˜ (0) = 0, .d γ˜ /dt (0) ∈ T0 (M). Consider a local system of coordinates near the origin such that .1 is in Moser’s normal form, and .γ˜ is defined by . z = 0, .yn = 0. In these coordinates we have

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

105

+ = {z ∈ : ρ(z) ≤ 0, φ(z) > 0}.

.

As before, .ρ is a defining function of .1 and .φ is a real (in general, nonlinear) function with .φ(0) = 0 and .dφ(0) = 0. The above choice of the chain .γ˜ implies that .dφ/dxn (0) = 0. After a suitable transformation . z → U  z, .zn → zn , with some unitary U (recall that such a transformation preserves the normal form), one can assume that φ(z) = y1 + ayn + r(z), r(z) = O(|z|2 ).

.

Consider the transformation zn → tzn ,  z →

.

√  t z, t ∈ R+ .

In the new coordinates .1 is defined by yn = | z|2 + (1/t)



.

√ √ Fkl ( t  z, t  z, txn ),

(8.21)

kl

and .1+ is defined by the condition y1 +

√

.

√ √ tayn + (1/ t)t ( t  z, tzn ) > 0.

(8.22)

We claim that the chain .γ can be defined for .t > 0 small enough by the equation = p(xn ) satisfying the conditions .p(0) = 0, .dp1 /dt (0) = 1, .dpj /dt (0) = 0, .j = 2, . . . , n − 1. It suffices to check that for small .xn = 0 one has  . z

.

√ √ √ Im(p1 (xn ) + a t yn (xn )) + (1/ t) r( tp(xn ), tzn ) > 0,

where .yn (xn ) is defined by the equation of .1 and the condition .yn (xn ) = O(|p(xn )|2 ). From the Taylor expansion of .p(xn ) and the initial conditions, we obtain that .

d 2 yn (0) = 2|dp/dt (0)|2 = 2, dt 2

and .

Im pj (xn ) =

d 2p 1 Im 2 (0)xn2 + o(xn2 ), j = 1, . . . , n − 1. 2 dt

Along .γ we have the estimate √ √ √ |(1/ t) r( tp(xn ), tzn )| < tCxn2 + o(xn2 ).

.

106

8 Geometry of Real Hypersurfaces: Analytic Continuation

Hence, it suffices to verify that (1/2) Im

.

√ d 2 p1 (0) > tC1 . 2 dt

(8.23)

The coefficients of the expansion on the right-hand side of (8.21) continuously depend on t; furthermore, we obtain the hypersurface .yn = | z|2 in the limit as .t → 0. The chains on this hypersurface are obtained by intersections with complex lines. In particular, a chain . z = p(xn ) satisfying the conditions .p(0) = 0, .dp1 /dt (0) = 1, .dpj /dt (0) = 0, .j = 2, . . . , n − 1, is the intersection of the hypersurface with the complex line {z1 = zn , zj = 0, j = 2, . . . , n − 1}.

.

2

A direct computation shows that in this case . ddtp21 (0) = 2i. It follows now from Lemma 8.12 that the inequality (8.23) holds for .t > 0 small enough. (b) We introduce on S the complex parameter .ξ = α + iβ such that .γ = {β = 0} and .(∂xn /∂α)(0) = 1. Then the set .S ∩ {ρ < 0} corresponds to the set .{β > 0} on the plane .ξ . Therefore, in a neighbourhood of the origin S admits a parametric representation z1 = ξ + o(ξ ), zn = ξ + o(ξ ), zj = o(ξ ), j = 2, . . . , n − 1.

.

√ Substituting this into (8.22) we obtain that .S∩ + is defined by .β(1+ tα)+o(ξ ) > 0. By part (a), this inequality holds when .β = 0 and .α = 0. Furthermore, the derivative √ of the left-hand side with respect to .β at .ξ = 0 is equal to .1 + tα > 0. Hence, there exists a neighbourhood V of 0 in the plane .ξ such that this inequality holds when .ξ ∈ V ∩ {β ≥ 0} and .ξ = 0. 

Now assume that a chain .γ satisfies the properties (a) and (b) of the previous lemma. One can assume that .ζ = 0 and . is given in the normal form. Also .γ = { z = 0, yn = 0} and .S = { z = 0}. Then there exists .r > 0 such that {z ∈ S : ρ(z) < 0, |zn | < r} ⊂ + .

.

Therefore, f is defined and holomorphic on the set .{z ∈ S : ρ(z) < 0, |zn | < r}. Lemma 8.14 There exists .r1 > 0 such that f extends holomorphically along the set S to the set .S ∩ {|zn | < r1 }.

8.2 Extension of Germs of Holomorphic Mappings II: The Nonspherical. . .

107

Proof We already know that f extends continuously to .ζ = 0. Assume that .f (0) = 0 and .2 = {ρ˜ = 0}, where .ρ˜ is a real analytic strictly plurisubharmonic function in ˜ 2 = 0. Choose local coordinates such that the a neighbourhood of .2 satisfying .d ρ| expansion of .ρ˜ in w, .w has the form ρ(w, ˜ w) = (wn − w n )/2i +

n 

.

wj w j + o(|w|2 ).

(8.24)

j =1

Since .f (1+ ) ⊂ 2 , we have on .1+ ρ(f ˜ (z), f (z)) = 0.

.

(8.25)

Consider the vector fields on .1 Xj =

.

∂ρ ∂ ∂ρ ∂ − , j = 1, . . . , n − 1. ∂zj ∂zn ∂zn ∂zj

Applying these vector fields to (8.25), we obtain .(n − 1) equations on .1+ :

.

n  ∂ ρ˜ Xj fk = 0, j = 1, . . . , n − 1. ∂wk

(8.26)

k=1

To simplify the notation we write .∂ ρ/∂w ˜ k = ρ˜k . Then one can rewrite (8.26) in the form

.

n−1  (ρ˜k /ρ˜n )Xj fk = −Xj fn .

(8.27)

k=1

We view these equalities as a system of linear equations with respect to .(ρ˜k /ρ˜n ), .k = 1, . . . , n − 1. Recall that that the differential .df (0) is nondegenerate and .∂ρ/∂zn (0) = 0. This easily implies that the vector fields Xj f = (Xj f1 , . . . , Xj fn ), j = 1, . . . , n − 1,

.

are .C-linearly independent at every point .z ∈ 1+ close enough to the origin. Therefore, in a neighbourhood W of the origin in .Cn the determinant . of this system does not vanish on .1+ ∩ W and we can write the solutions in the form ρ˜k /ρ˜n = k / .

.

Here . k , . are polynomials in .Xj fk , .j, k = 1, . . . , n − 1.

(8.28)

108

8 Geometry of Real Hypersurfaces: Analytic Continuation

As above, one can assume that . + = {z ∈ : ρ(z) ≤ 0, φ(z) > 0}, where .φ(z) = y1 + ayn + o(|z|). Set St = {z1 = it, zk = 0, k = 2, . . . , n − 1},

.

where t is real. Shrinking .r > 0 if necessarily, one can assume that the sets .Ut = St ∩{ρ < 0, |zn | < r} are contained in .W ∩ + for all t from some segment .[0, t0 ] with .t0 > 0. One can represent the boundaries .bUt in the form .bUt = L1t ∪ L2t , where L1t = 1 ∩ {|zn | ≤ r, z1 = it, zk = 0, k = 2, . . . , n − 1},

.

L2t = {ρ < 0, |zn | = rn , z1 = it, zk = 0, k = 2, . . . , n − 1}.

.

The restriction of the functions .∂ρ/∂zn to the curves .L1t are real analytic. One can assume that they extend holomorphically along .St to the set .St ∩ {|zn | < r} , .t ∈ [0, t0 ]. Hence the restriction to .L1t of the functions .Xj fk , as well as . k and . , extend holomorphically along .St on .U t for .t = 0, and on .U 0 \ {0} for .t = 0. Denote these holomorphic extensions of . k and . by . tk and . t respectively. After a small perturbation of r one can further assume that . t does not vanish on all of .L2t for each .t ∈ [0, t0 ]. Since . t also does not vanish on .L1t , we conclude that for .t > 0 every . t has a finite number m of zeros on .Ut ; and m is independent of t. Therefore, for .t = 0 the function . 0 admits at most m zeroes in .U0 . Shrinking .r > 0, one can assume that . 0 = U0 . It follows from (8.24), (8.28) that + 0 0 . / → 0 when .z → 0, .z ∈  . k 1 By Lemma 8.11, the functions . 0k / 0 are bounded on .U0 . Therefore, they extend continuously to .U 0 . Given .ε > 0, one can shrink .r > 0 such that .| 0k / 0 | < ε on .U 0 . In view of (8.24), one can apply the Implicit function theorem to the Eqs. (8.25) and (8.28). We obtain f = P (f, 01 / 0 , . . . , 0n−1 / 0 ),

.

(8.29)

in a neighbourhood where .|f | < ε, .| 0k / 0 | < ε. Here P is a holomorphic function. The expressions on the right-hand side of (8.29) are holomorphic on .U0 and continuous on .U 0 , while the expressions on the left-hand side are antiholomorphic on .U0 and continuous on  .U0 . It follows by the Schwarz reflection principle that for . z = 0 the mapping f extends  holomorphically in .zn to a neighbourhood of the point .zn = 0. This proves the lemma. Now we conclude the proof of the theorem. In a neighbourhood of .1+ one has .ρ˜ ◦ f = βρ, where .β > 0 on .1+ . Hence .ρ˜ ◦ f < 0 on .U0 ⊂ + . By the Hopf lemma, |ρ˜ ◦ f (z)| ≥ Cdist(z, bU0 ).

.

Therefore,

8.3 From Local to Global Extension

∂(ρ˜ ◦ f )/∂zn (0) =

109 n 

.

ρ˜k ◦ f (0)(∂fk /∂zn )(0) = 0.

k=1

Note that the derivatives .(∂fk /∂zn )(0) exist by Lemma 8.14. Since .ρ˜k (0) = 0 for .k = 1, . . . , n − 1, we conclude that .(∂fn /∂zn )(0) = 0. Therefore, the image .λ of the chain  .γ = { z = 0, yn = 0} in a neighbourhood of the point .w = 0 is a nondegenerate curve, everywhere transverse to the holomorphic tangent space of .2 . But .λ is a chain everywhere except .w = 0; hence, it is a chain also in a neighbourhood of the origin. Put .2 into the normal form such that .λ is given by . w = 0, .Im wn = 0. Note that .1 is already in the normal form. According to [33], in these coordinates f is a rational mapping of the form 

.

w=

 dg(zn )/dzn U ( z), wn = g(zn ),

where .g(zn ) = (azn + bn )/(czn + d), and U is a unitary operator on .Cn−1 . We already proved that the restriction of f to .S ∩ {|zn | < r1 } is holomorphic. Therefore, g does not have a singularity at the origin. Thus, f is holomorphic in a full neighbourhood of .0 ∈ M. The proof is complete.

8.3

From Local to Global Extension

Theorems 8.1 and 8.5 give analytic continuation of a biholomorphic germ between strictly pseudoconvex real analytic hypersurfaces along any path in the source hypersurface. In this section we are interested in establishing global extension when the hypersurfaces are boundaries of domains. More precisely, suppose that D and .D  are bounded domains with real analytic strictly pseudoconvex boundaries, where .D  is either the unit ball or is a domain with nonspherical boundary. Suppose that .f : (bD, p) → (bD  , p ) is a germ of a biholomorphic map at a point .p ∈ bD sending bD to .bD  and .f (p) = p . Does f extend holomorphically to a global map .f : D → D  ? If the boundary bD is simply connected, then by the Monodromy theorem, the extension of the germ f along paths on bD gives a globally defined locally biholomorphic map in a neighbourhood of bD. By the Hartogs Kugelsatz, f extends holomorphically to D. The Jacobian of f does not vanish anywhere in D (as otherwise, its zero locus necessarily intersects bD which contradicts Theorem 2.3), and since D is also simply connected, it follows that f extends to a globally biholomorphic map .f : D → D  . If bD is not simply connected, then the extension is in general only multiple-valued, √ for example the map .f (z1 , z2 ) = (z22 , z1 ) maps the boundary of the domain D = {z ∈ C2 : |z1 |2 + |z1 |−2 + |z2 |4 + |z2 |−4 < 5}

.

into itself, but it is not single-valued in D.

110

8 Geometry of Real Hypersurfaces: Analytic Continuation

Nevertheless, the following global extension result, due to Nemirovski and Shafikov [101, 102], holds. Theorem 8.15 Suppose that D and .D  are bounded domains in .Cn with strictly pseudoconvex real analytic boundaries, and there exists a local biholomorphic equivalence    .f : (bD, p) → (bD , p ). Then the universal coverings of D and .D are biholomorphic. In particular, if D is simply connected, then the germ of a local equivalence extends to a biholomorphic map .f : D → D  . As a special case, we note the following higher dimensional analogue of the Uniformization theorem. Corollary 8.16 A bounded strictly pseudoconvex domain with spherical boundary is universally covered by the unit ball. For the proof of the theorem we will need the following.  be its envelope of holomorphy. Then Lemma 8.17 Let .D ⊂ Cn be a domain and .D n every locally biholomorphic map .f : D → C extends to a locally biholomorphic map  → Cn . .F : D Proof Every component .fj , .j = 1, . . . , n, of the map f extends holomorphically to a  → D  , so we only need to show that the map .F = (F1 , . . . , Fn ) : D → function .Fj : D n  : det JF (z) = 0}. C is locally biholomorphic. Arguing by contradiction, let .E = {z ∈ D   and therefore, the set .D\E is a Stein manifold. Then E is a complex hypersurface in .D\D,  But then there exists a function in .O(D) that cannot be extended holomorphically to .D,  

which contradict the definition of the envelope of holomorphy .D. Proof of Theorem 8.15 First we prove that local equivalence of the boundaries implies biholomorphic equivalence of the universal coverings. If bD is not simply connected, then the analytic continuation of a local equivalence germ .f : (bD, p) → (bD  , p ) along homotopically nonequivalent paths starting at .p ∈ bD with the same end point .q ∈ bD may produce different extensions, which may have different domain of convergence at q. However, since bD is strictly pseudoconvex, for every point .q ∈ bD there exists a neighbourhood .Vq such that any function holomorphic on .Vq ∩ bD extends holomorphically to .Vq ∩ D (this can be seen, for example, from the fact that a strictly pseudoconvex domain admits a local change of coordinates in which it is strictly convex). It follows that there exists a neighbourhood V of bD such that the germ f extends as a locally biholomorphic map along any path in .V ∩ D with values in .D  . This extension defines a locally biholomorphic map .f : V˜ → D  from the universal covering .V˜ of V . The envelope of holomorphy of V is precisely D by the Hartogs theorem. By Kerner’s theorem [84], the envelope of holomorphy of the universal covering .V˜ → V is the

8.3 From Local to Global Extension

111

universal covering .D˜ → D. And so by Lemma 8.17 the map .f : V˜ → D  extends to a locally biholomorphic map .F : D˜ → D  . In the nonspherical case we may complete the proof by applying the above reasoning to the germ .f −1 . In fact, this argument shows that a local equivalence between bD and .bD  extends to a biholomorphism from the universal covering of .D to the universal covering  of .D . Consider now the spherical case. In view of the example by Burns and Shnider (see the beginning of this chapter), the germ of the inverse map may not extend along any path in .bD  , and so a different argument is needed. Any local equivalence between spherical hypersurfaces factors through the sphere, and therefore, we only need to consider the case when .D  = Bn . By the argument above, there exists a locally biholomorphic map .F : Y → Bn from the universal covering .Y of .D into the closed unit ball. The following important equivariance property holds. Lemma 8.18 Let .G = π1 (D) = π1 (D) be the group of deck transformations of the universal covering .π : Y → D. There exists a representation .σ : G → Aut(Bn ) such that σ (g) ◦ F (x) = F ◦ g (x)

.

for all x ∈ Y and g ∈ G.

Proof The existence of the representation .σ such that the required relation holds for all x ∈ bY is a direct consequence of the Poincaré–Alexander theorem (Theorem 2.2). The 

extension to the entire .Y follows by the uniqueness theorem.

.

Although the inverse map may not extend along every path in .bBn , we show that it does extend along every path in the open ball .Bn . To see this, it suffices to prove Lemma 8.19 There exists .ε > 0 such that every point .x ∈ Y has an open neighbourhood V ⊂ Y with the following properties:

.

(1) the restriction .F |V : V → F (V ) is biholomorphic, (2) .F (V ) contains the ball of radius .ε centred at .F (x) with respect to the Poincaré metric on .Bn . Proof Let h be the Euclidean metric in .Cn . The Poincaré metric (defined by (3.1) with n n n .ζ ∈ B and .v ∈ Tζ B ) dominates the Euclidean metric in the ball .B . In particular, the n Euclidean ball of radius .R > 0 centred at a point .b ∈ B contains the Poincaré ball of the same radius centred at b. ∗ Denote by .F h the pull-back of the Euclidean metric to the manifold .Y . Let further . ⊂ Y be a fundamental domain for the action of G on Y . Notice that . is a relatively compact subset of .Y . It follows that there exists an .ε > 0 such that for every point .x ∈  the map .F is a biholomorphism of the ball of radius .ε centred at x with respect to the

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8 Geometry of Real Hypersurfaces: Analytic Continuation ∗

metric .F h onto the intersection of the Euclidean ball of the same radius centred at .F (x) n with the closed ball .B . Since the Euclidean ball about an interior point of .Bn contains the Poincaré ball with the same centre and radius, we have just shown that every point in . does indeed have a neighbourhood with properties (1) and (2). Now let .x ∈ Y be an arbitrary point. By the definition of a fundamental domain, there exists a deck transformation .g ∈ G such that .g(x) ∈ . Let W be the neighbourhood of −1 (W ). By Lemma 8.18, we have .g(x) constructed above and set .V = g F = σ (g)−1 ◦ F ◦ g.

.

It follows that F is biholomorphic in V if and only if it is biholomorphic in W . Furthermore, the image .F (V ) = σ (g)−1 (F (W )) contains the Poincaré ball of radius .ε about .F (x) because .F (W ) contains the ball of this radius about .F (g(x)) and the 

automorphism .σ (g)−1 is an isometry of the Poincaré metric. Now the proof of the theorem immediately follows: since .Bn is simply connected, the 

map .F : Y → Bn of open manifolds is biholomorphic. The converse to Theorem 8.15 also holds. Theorem 8.20 Let D and .D  be bounded strictly pseudoconvex domains with real analytic boundaries. Then the following holds. (i) If D is covered by the unit ball, then its boundary is spherical, that is, locally biholomorphically equivalent to the unit sphere. (ii) If the universal coverings of the open domains D and .D  are not biholomorphic to the unit ball, then any biholomorphism between them extends to a biholomorphism of the universal coverings of the closed domains .D and .D  . In particular, bD and .bD  are locally equivalent. Note that in this theorem the conclusion for domains with spherical boundaries is weaker than in the nonspherical case. In fact, the statement in (ii) does not hold in general for spherical domains. For example, let .D = D  be the domain whose boundary is given by (8.1). The closure of D is universally covered by a region of the form .Bn \ A, where A is a nonempty closed subset of the unit sphere. Take any biholomorphism of the unit ball whose extension to .bBn does not preserve A, and we obtain an example of a biholomorphism of the universal covering of D that does not extend to a biholomorphism of the universal covering of .D. Before we give the proof of Theorem 8.20 we need to sharpen some results that we discussed in Chaps. 2 and 3. Recall that a bounded strictly pseudoconvex domain in n .C is Kobayashi complete and hyperbolic. Further, if .π : U → D is an unbranched covering, then the complex manifold U is also complete hyperbolic, see Eastwood [58].

8.3 From Local to Global Extension

113

Montel’s theorem also holds for holomorphic maps into complete hyperbolic spaces, see Kiernan [86], from which we conclude that a sequence of holomorphic maps .fν : N → U from a connected complex manifold N is relatively compact if and only if the sequence of points .fν (z) ∈ U is relatively compact for some point .z ∈ N. The following proposition is a version of the Wong-Rosay theorem for coverings of strictly pseudoconvex domains. Proposition 8.21 Let .π : U → D be a covering of a strictly pseudoconvex domain in .Cn with .C 2 -smooth boundary. Assume that there exist a point .z0 ∈ U and a sequence of biholomorphic maps .gν ∈ Aut(U ) such that .π(gν (z0 )) → ζ0 ∈ bD as .ν → ∞. Then U is biholomorphic to the unit ball. Proof The scaling method discussed in Sect. 4.1 can be applied to the sequence of maps .fν := π ◦ gν : U → D in almost exactly the same way as in the proof of Theorem 2.4. The only new point concerns the definition and convergence of the sequence of inverse maps used at the end of the proof. Namely, if .V  ζ0 is a neighbourhood such that .V ∩ D is convex (and therefore simply connected), then there exists a uniquely defined sequence of inverse maps (fν )−1 : V ∩ D → U,

.

such that .(fν )−1 (fν (z0 )) = z0 . Composing these maps with the re-scaling maps gives a sequence of maps from (increasing subsets of) the ball to U taking a fixed point in the ball to the point .z0 . By Montel’s theorem above we may conclude that this sequence of maps is relatively compact. 

We will also need the following corollary for the proof of Theorem 8.20. Corollary 8.22 Let .π : U → D be a covering of a strictly pseudoconvex domain with C 2 -smooth boundary. Assume that U is not biholomorphic to the unit ball. Then for every sequence of biholomorphic maps .gν ∈ Aut(U ) and for every compact set .K  U , the projections .π(gν (K)) ⊂ D lie in a compact set in D that does not depend on .ν.

.

Proof Pick a point .z0 ∈ K and let .d < ∞ be the diameter of K with respect to the Kobayashi metric on U . By Proposition 8.21 the points .π(gν (z0 )) lie in a compact set inside D. The image of any point .z ∈ K under any of the maps .π ◦ gν : U → D lies within Kobayashi distance d from this compact set. Since D is complete with respect to the Kobayashi distance, this proves the corollary. 

Proof of Theorem 8.20 (i) Let D be a strictly pseudoconvex domain with real analytic boundary, and let .π : Bn → D be a covering of D by the unit ball .Bn . Let .q ∈ bD be an arbitrary point and V a neighbourhood of q such that .V ∩ D is simply connected. Then the germ of the map .π −1 extends to a biholomorphic map g from .V ∩ D to an open set in .Bn .

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8 Geometry of Real Hypersurfaces: Analytic Continuation

Let .φ : D → (−∞, 0] be a smooth plurisubharmonic defining function for D. Applying the Hopf lemma to the negative plurisubharmonic function .φ ◦ π in the ball, we see that c1 dist(g(x), bBn ) ≤ |φ ◦ π(g(x))| = |φ(x)| ≤ c2 dist(x, bD)

.

for any point .x ∈ V ∩ D and for some positive constants .c1 and .c2 that do not depend on x and .bD. Using the estimate above, we may repeat the proof of Theorem 3.4 to conclude that the map g extends to .bD ∩ V as a Hölder continuous map sending .bD ∩ V to the unit sphere. Note that by the boundary uniqueness theorem the extension to the boundary is not constant. Hence, the extension of g to bD is in fact smooth by Theorem 4.1. Finally, by Theorem 2.1, the map g extends biholomorphically to a neighbourhood of q. Hence, .bD is locally biholomorphically equivalent to the unit sphere at every boundary point .q ∈ bD. (ii) Let D and .D  be strictly pseudoconvex domains with real analytic nonspherical boundaries. Denote by .π : Y → D the universal covering of the closure of D. Then .Y is a complex manifold with (possibly noncompact!) real analytic boundary .bY = π −1 (bD). Since D is homotopy equivalent to its closure, the interior .Y = Y \ bY , equipped with the projection .π = π |Y , is the universal covering of the open domain D. Similar notation will be used for the domain .D  . Suppose that .F : Y → Y  is a biholomorphic map. To prove that any such F extends to a biholomorphic map .F : Y → Y  it is enough to show that the map .π  ◦ F : Y → D  extends to a locally biholomorphic map .π  ◦ F : Y → D  . Indeed, by the Monodromy theorem, this extension can be lifted to a locally biholomorphic map .F : Y → Y  that coincides with F on Y . Applying this result to the inverse map .F −1 : Y  → Y , we conclude that .F is one-to-one. Equip .D and .D  with the Euclidean metric, and lift it to metrics on .Y and .Y  , respectively. The distance with respect to any of these metrics will be denoted by .dist(·, ·). Let .φ : D → (−∞, 0] be a smooth plurisubharmonic defining function for D. We define a real-valued function .ψ0 in .D  by setting ψ0 (z) =

.

sup

{y∈Y  |π  (y)=z}

φ ◦ π ◦ F −1 (y).

Notice that .ψ0 is locally the supremum of a family of negative plurisubharmonic functions. Hence, its upper semicontinuous regularization ψ(z) = ψ0∗ (z) = lim sup ψ0 (ζ )

.

ζ →z

is a nonpositive plurisubharmonic function in .D  (see Klimek [87]).

8.3 From Local to Global Extension

115

The crucial point is that if Y is not biholomorphic to the unit ball, then .ψ is, in fact, negative everywhere in .D  . Indeed, take a point .z ∈ D  and a small closed ball Q about  ⊂ Y  under the action of the it. The preimage of this ball in .Y  is the orbit of a ball .Q   countable group .G = π1 (D ) of the deck transformations of the universal covering .π  :  Y  → D  . Hence, the set .F −1 (π −1 (Q)) ⊂ Y is the orbit of the compact set .K = F −1 (Q) −1  under the action of the countable subgroup .F G F ⊂ Aut(Y ). Corollary 8.22 shows that the projection of this orbit to the domain D, i. e., the set {x ∈ D | x = π ◦ F −1 (y), π  (y) ∈ Q}

.

must be contained in a compact subset .E  D. It follows from the definition of .ψ and .ψ0 that ψ(z) ≤ sup ψ0 (ζ ) ≤ max φ(x) < 0.

.

ζ ∈Q

x∈E

Now we can apply the Hopf lemma to the negative plurisubharmonic function .ψ in .D  . Namely, for any point .x ∈ Y we have .

− c1 dist(π  ◦ F (x), bD  ) ≥ ψ(π  ◦ F (x))

for a positive constant .c1 . On the other hand, ψ(π  ◦ F (x)) ≥ ψ0 (π  ◦ F (x)) ≥ φ ◦ π ◦ F −1 (F (x)) = φ(π(x)),

.

by the definition of .ψ. Since .φ is a smooth defining function for D, there exists a positive constant .c2 such that φ(π(x)) ≥ −c2 dist(π(x), bD).

.

Finally, if .x ∈ Y is sufficiently close to .bY , then dist(π(x), bD) = dist(x, bY ).

.

Therefore, we have proven the inequality dist(π  ◦ F (x), bD  ) ≤ c3 dist(x, bY )

.

(8.30)

with some positive constant .c3 that does not depend on .x ∈ Y provided that the point x is close to the boundary. With the inequality (8.30) at hand, the existence of a locally biholomorphic extension of the map .π  ◦ F : Y → D  across any boundary point .q ∈ bY is established by the

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8 Geometry of Real Hypersurfaces: Analytic Continuation

argument given in the proof of part (i) of the theorem. And this completes the proof of part (ii). 

We further note that if D and .D  are covered by the unit ball, then the function .ψ constructed in the proof of Theorem 8.20 can vanish identically.

8.4

Further Results

Theorem 8.1 can be generalized to the case when the target hypersurface is the unit sphere in .CN , .N ≥ n. It was proved by Pinchuk [109] by pushing further the method discussed in Sect. 8.1. Further generalizations can be obtained using a different technique that involves Segre varieties, these will be discussed in the next chapter. Theorem 8.5 can be generalized to the case when the strictly pseudoconvex nonpsherical real analytic hypersurfaces .1 and .2 are contained in some complex manifolds and .2 is closed. This was proved by Vitushkin et al. [154], see also [152] and [153] for an overview of related results. Similarly, Theorems 8.15 and 8.20 remain true if D and .D  are assumed to be connected Stein manifolds with real analytic strictly pseudoconvex boundaries. The proof is essentially the same, see Nemirovski and Shafikov [101, 102].

9

Segre Varieties

Segre varieties, introduced in Chap. 7 as a locally biholomorphically invariant family of complex hypersurfaces associated with a real analytic hypersurface, is a powerful tool in the problem of analytic continuation and boundary regularity of holomorphic mappings. In this chapter we develop the necessary machinery for working with the Segre family and give some immediate applications. In Sect. 9.2 we show that compact real analytic sets in .Cn do not contain nontrivial germs of complex analytic sets–an important tool in itself in identifying the structure of the Segre family. In Sect. 9.3 we give an alternative proof of analytic continuation of a local equivalence in the spherical case that essentially uses the properties of Segre varieties. Finally, in the last section we show that a germ of a biholomorphic map between Levi nondegenerate algebraic hypersurfaces is necessarily algebraic.

9.1

Geometry of Segre Varieties

Let . be a smooth real analytic hypersurface, .z0 ∈ , given in a neighbourhood U of .z0 by a real analytic defining function .ρ(z, z). Recall that a Segre variety of a point .w ∈ U with respect to . is defined as Qw = {z ∈ U : ρ(z, w) = 0}.

.

It is important to note that Segre varieties are defined only locally in a domain of convergence of the power series representing .ρ(z, w). In the special case when the defining function of . can be chosen to be a polynomial, the Segre varieties are algebraic hypersurfaces in .Cn . This situation will be considered in Sect. 9.4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_9

117

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9 Segre Varieties

For points .z ∈ Cn we write .z = ( z, zn ) ∈ Cn−1 × C. We call a standard coordinate system for .z0 with respect to . any local holomorphic coordinate system such that .z0 = 0 and the defining function r can be written in these coordinates in the form ρ(z) = 2xn + o(|z|).

.

(9.1)

A pair of open neighbourhoods .U1 and .U2 , .U 1 ⊂ U2 , is called a standard pair of neighbourhoods for .z0 if it admits the following properties: (a) The complexification .r(z, w) is well-defined on .U2 × U1 , such that for any .w ∈ U1 the Segre variety of w, Qw = {z ∈ U2 : ρ(z, w) = 0},

.

is a smooth closed real analytic hypersurface; (b) With respect to a suitable coordinates system for .z0 = 0 one has .U2 =  U2 × U2n where . U2 is an open neighbourhood of 0 in .Cn−1 and .U2n is an open neighbourhood of 0 in the .zn -axis; (c) .Qw can be written as a graph. More precisely, the exists a function .h( z, w) = hw ( z), depending holomorphically on . z ∈  U2 and antiholomorphically on .w ∈ U1 , such that Qw = {( z, zn ) ∈  U2 × U2n : zn = hw ( z)}.

.

(9.2)

Clearly, each point .z0 ∈  admits a standard pair of neighbourhoods and .{U2 } form a neighbourhood basis for .z0 . By Lemma 7.1, Segre varieties are invariant under biholomorphic maps. Further, Segre varieties are independent of the choice of a defining function .ρ. Indeed, if .ρ1 is another defining function of . near .z0 , then .ρ1 (z) = α(z) · ρ(z) for a suitable real analytic function .α(z) with .α(z0 ) = 0. After complexification of this equation we get .ρ1 (z, w) = α(z, w)ρ(z, w). It follows that ρ1 (z, w) = 0 ⇐⇒ ρ(z, w) = 0,

.

which proves the claim. The Segre map is formally defined as λ : U1  w → Qw .

.

(9.3)

9.1 Geometry of Segre Varieties

119

It is important to understand the structure and set-theoretical properties of this map. In particular, we will be interested in the following sets Aw = {z ∈ U1 : Qz = Qw }, w ∈ U1 .

.

(9.4)

First consider several examples. (1) Let . ⊂ Cn be a strictly pseudoconvex hypersurface. Then in a suitable coordinate system . can be written in the form (1.18), so after its complexification we obtain Qw =

.

⎧ ⎨ ⎩

z = (z1 , . . . , zn ) ∈ U2 : zn + wn +

n−1  j =1

⎫ ⎬ zj wj + o(|z| + |w|) . ⎭

This clearly implies that .Aw are singletons and the Segre map .λ is injective. The same holds if the hypersurface . has nondegenerate but not necessarily positive-definite Levi form. (2) For . = {z ∈ C2 : 2x2 + |z1 |4 = 0}, .Qw = {z ∈ C2 : z2 + w 2 + z12 w1 2 }, which implies that .Aw = {(w1 , w2 ), (−w1 , w2 )} and so the map .λ is 2-to-1. (3) This example appeared in [90]. Consider a hypersurface in .C2 given by r(z, z) = x2 + |z1 |8 +

.

15 |z1 |2 x 6 = 0. 7

The hypersurface is strictly pseudoconvex except at the origin. It was shown in [90] that if a holomorphic function f vanishes at the origin, then for an arbitrarily small neighbourhood U of the origin the set .{z ∈ U : f (z) = 0} intersects both .{r > 0} and .{r < 0}. This means, in particular, that the Segre variety .Q0 , while being tangent to ., necessarily intersects both components of .U \. Note that at any strictly pseudoconvex point .p ∈ , .Qp ∩  = {p}, at least near p. (4) Let . be a strictly pseudoconvex hypersurface in .Cm and let .M =  × Ck , with m+k associated .m, k ≥ 1. Using part (1) it is easy to see that for any w, the set .Aw ⊂ C with M is a k-dimensional complex submanifold, in particular, the map .λ has fibres of dimension .k > 0. Lemma 9.1 Let . be a smooth real analytic hypersurface, and .U1 ⊂ U2 be a standard pair of neighbourhoods of a point .z0 ∈ . Then (a) (b) (c) (d)

z ∈ Qz if and only if .z ∈ ; For .z, w ∈ U1 one has .z ∈ Qw if and only if .w ∈ Qz ; For any .w ∈ U1 , .Aw = z∈Qw Qz . In particular, .Aw is a complex analytic set; For .w ∈  ∩ U1 , one has .Aw ⊂ ;

.

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9 Segre Varieties

Proof (a) This follows from the definition of .ρ: .z ∈  iff .ρ(z, z) = 0. (b) This follows from the identity .ρ(z, w) = ρ(w, z). (c) If .ζ ∈ Aw , then .Qζ = Qw . For an arbitrary .z ∈ Qw we have .z ∈ Qw = Qζ , and so by part (b), .ζ ∈ Qz . We showed that .ζ belongs to .Qz for all .z ∈ Qw . The reverse implications also hold, and this proves the assertion. (d) Suppose that .w ∈  and .z ∈ Aw . Then .Qz = Qw . By part (a), .w ∈ Qw , and so .w ∈ Qz . By part (b), the latter implies that .z ∈ Qw , which means .z ∈ Qz , and so .z ∈ . 

Now, by the Implicit function theorem, the local defining function of . at .z0 = 0 can be written in the form ρ(z, z) = 2xn +



.

ρj ( z,  z)(2yn )j .

(9.5)

j =0

The complexification of .ρ then becomes ρ(z, w) = zn + wn +

∞ 

.

ρj ( z,  w)(−i)j (zn − wn )j .

(9.6)

j =0

From this it follows that ρ(z, w) = 0 ⇐⇒ zn + wn +



.

λk (w)  z = 0, k

(9.7)

|k|>0

where .λk are holomorphic functions on .U1 , .k = (k1 , . . . , kn−1 ), .kj ≥ 0. It follows from (9.7) that ⎛ ρ(z, w) = (1 + α(z, w)) ⎝zn + wn +



.

⎞ k λk (w)  z ⎠ ,

(9.8)

|k|>0

where .α(z, w) is a function holomorphic in z and antiholomorphic in w that vanishes at the origin. We set .λ0 := wn . Then   U1  w → λ(w) = λk (w) : k ∈ Nn−1 0

.

9.2 Compact Real Analytic Sets

121

is a coordinate representation of the Segre map. By the Noetherian property, there exists a positive integer .L > 0 associated with . such that the terms of the total order at most L completely determine the map .λ. With this choice of L, we may assume that the Segre map is given by U1  w → λ(w) := (λk (w) : |k| ≤ L) ∈ CN ,

.

(9.9)

for an appropriate N . If the map .λ is proper, then by Remmert’s Proper Mapping theorem we conclude that the space of Segre varieties admits the structure of a complex analytic variety of finite dimension, which we denote by S = S(U1 , U2 ).

.

In the local setting this happens precisely when the Segre map .λ : U1 → S is finite-toone, i.e., when the sets .Aw are discrete. In this case the hypersurface . is called essentially finite at the point .z0 .

9.2

Compact Real Analytic Sets

The following is a well-known result that was first proved by Diederich and Fornæss [45]. Theorem 9.2 Suppose that X is a closed (compact without boundary) real analytic subset of .Cn of arbitrary dimension. Then X does not contain any nontrivial germs of complex analytic subsets. Before the proof we draw one conclusion from the theorem that is relevant to us in the context of Segre varieties. Corollary 9.3 Let . ⊂ Cn be a compact real analytic hypersurface, and .z0 ∈ . Then for a sufficiently small standard pair of neighbourhoods .U1 ⊂ U2 of .z0 , the sets .Aw are discrete for all .w ∈ U1 , the corresponding Segre map .λ is finite-to-one, and the family of Segre varieties has the structure of a complex analytic set. Proof of Corollary 9.3 The proof is immediate from the discussion in the previous section. Indeed, by Lemma 9.1(d), for any .w ∈ , the set .Aw , which is a complex analytic set by part (c) of that lemma, is contained in .. It follows then by Theorem 9.2 that .Aw are discrete for all .w ∈ , and therefore, there exists a neighbourhood of .z0 in which all fibres of the Segre map .λ are discrete. Hence, .λ is locally proper, and therefore .S = λ(U1 ) is a complex analytic set by Remmert’s theorem. 

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9 Segre Varieties

Proof of Theorem 9.2 First assume that X is a compact smooth real analytic hypersurface, the case most relevant to us. Arguing by contradiction assume that there exists a complex analytic set Y , a representative of the nontrivial germ, such that .0 ∈ Y ⊂ X, .dim Y = k > 0. Let .U1 ⊂ U2 be a standard pair of neighbourhoods of .0 ∈ X. The neighbourhood of the origin, where Y is a priori defined, can be arbitrarily small, but after shrinking .U1 we may assume that Y is a complex analytic subset of .U1 . Our goal is to show that there exists a complex analytic subset .Y2 of .U2 such that .Y ⊂ Y2 ⊂ X. This gives an “analytic extension” of Y to a domain that depends only on the geometry of X. The proof of this consists of three steps below. Note that all Segre varieties are considered with respect to the selected pair .U1 ⊂ U2 . Step 1. We claim that .Y ⊂ Q0 , the Segre variety of the point 0. For the proof it suffices to assume that .dim Y = 1, as the general proof follows from a slicing argument. Let .D  t → z(t) ∈ Y be the Puiseux parametrization of Y near 0 (see, e.g., [37]) . Consider the Taylor series expansion of the function .ρ(z(t), z(t)) around .(z(t), 0). We have ρ(z(t), z(t)) − ρ(z(t), 0) =

.

 1 α ∂wα ρ(z(t), 0) z(t) . α!

|α|≥1

The left-hand side of the above identity is holomorphic in t, while all the terms on the right-hand side contain .t. This means that .ρ(z(t), 0) ≡ 0, i.e., .Y ⊂ Q0 . Step 2. Define the set 

Y1 =

.

Qz .

z∈Y

Clearly, .Y1 is a complex analytic subset of .U2 . By Step 1, if .z ∈ Y , then .Y ⊂ Qz , and therefore, Y ⊂



.

Qz = Y1 .

z∈Y

In particular, .Y1 is nonempty. Step 3. Define 

Y2 =

Qw .

.

w∈Y1 ∩U1

Again, .Y2 is a complex analytic subset of .U2 . Since .Y ⊂ Y1 ∩ U1 , we have  .

z∈Y1 ∩U1

Qz ⊂

 z∈Y

Qz ,

9.3 Analytic Continuation Using Segre Varieties

123

which means that .Y2 ⊂ Y1 . We claim that .Y ⊂ Y2 , and .Y2 ⊂ X. For the first assertion, let .z ∈ Y be arbitrary. Then by the definition of .Y1 , for any .w ∈ Y1 we have .w ∈ Qz . By Lemma 9.1(b), the latter implies .z ∈ Qw for any .w ∈ Y1 , which means .z ∈ Y2 . As for the second assertion, let .z ∈ Y2 be arbitrary. Then .z ∈ Qw for every .Y1 ∩ U1 , and so .w ∈ Qz for all .w ∈ Y1 ∩ U1 . In particular, this is true for .z ∈ Y2 ⊂ Y1 , which means .z ∈ Qz , and by Lemma 9.1(a) this means .z ∈ X. Let now W be the union of all nontrivial germs of complex analytic sets contained in X. Assuming that .W = ∅, let .p ∈ W be the point furthest away from the origin in .Cn . By the argument above, there exists a pair of neighbourhoods .U1 ⊂ U2 , centred at p, and a complex analytic subset .Y2 of .U2 such that .p ∈ Y2 ⊂ X. Then the plurisubharmonic function .φ(z) = |z|2 restricted to .Y2 attains its maximum at p, which contradicts the Maximum principle. This proves that W is empty. Suppose now that X is a compact real analytic set in .Cn of arbitrary dimension. Then X can be locally defined just by one equation, indeed, if .rj (z, z), .j = 1, . . . , k, are realvalued real analytic functions that define X near one of its points, then we may take k 2 .r(z, z) = j =1 rj . We may define Segre varieties of X by complexifying this function: .Qw = {w : r(z, w) = 0}. It is easy to see that Segre varieties defined this way are either complex analytic hypersurfaces or agree with .Cn , and that properties (a) and (b) in Lemma 9.1 still hold for this family. Then the argument with the construction of the set .Y2 and the Maximum principle, that we used in the hypersurface case, can be repeated verbatim in this situation. This gives a complete proof of the theorem. 

9.3

Analytic Continuation Using Segre Varieties

The invariance of Segre varieties under biholomorphic maps gives a powerful technique that can be used in many questions related to the geometry of real analytic hypersurfaces. In this section we will illustrate this by giving a different proof for the analytic continuation of the germ of a biholomorphic map from a real analytic hypersurface into the unit sphere (Theorem 8.1). Theorem 9.4 Let . be a connected real analytic strictly pseudoconvex hypersurface in Cn . Let .p ∈ , and U be a neighbourhood of p. Suppose that .f : U → Cn is a biholomorphic map such that .f (U ∩) ⊂ bBn . Then f extends as a locally biholomorphic map along any path on ..

.

As a first step in the proof we establish the following phenomenon of analytic continuation along Segre varieties. This technique will be also used in Chap. 11. Lemma 9.5 In the setting of Theorem 9.4, let .U1  U2 be a standard pair of neighbourhoods of p. Then there is a neighbourhood V of the set .Qp ∩ U1 , such that the map f extends as a locally biholomorphic map from V into the projective space .CPn .

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9 Segre Varieties

The neighbourhood V in general depends on the set U where f is defined. However, the set .Qp ∩ U1 is independent of U , and so the lemma gives the extension of f to the set .Qp ∩ U1 , which is independent of the size of the original domain of the map. This will be important later for the extension of f further along .. Proof The lemma is nontrivial only if .U  U1 . By Lemma 9.1(b), for any .w ∈ Qp ∩ U1 , the Segre variety .Qw passes through the point p. Therefore, there exists a sufficiently small neighbourhood V of .Qp ∩ U1 such that for all .w ∈ V , .Qw ∩ U = ∅. Further, the neighbourhoods U and V can be adjusted so that for any .w ∈ V , the set .Qw ∩ U is connected. Denote by .Qw Segre varieties with respect to the unit sphere in the target space. Now consider the set A = {(w, w ) ∈ V × Cn : f (Qw ∩ U ) ∈ Qw }.

.

(9.10)

Then A is not empty because by Lemma 7.1, for w sufficiently close to p we have .f (Qw ∩ U ) ⊂ Qf (w) , and so A contains the graph of f . We show that A is a complex analytic set of dimension n, which ultimately gives the required extension of the map f . A point .z = (z1 , . . . , zn ) belongs to the Segre variety .Qw of a point .w  with respect  to the unit sphere in the target space if . nj=1 zj w j = 1. Let .f (z) = (f1 (z), . . . , fn (z)). Then .f (Qw ∩ U ) ∈ Qw is equivalent to n  .

fj (z)fj (w) = 1, for all z ∈ Qw ∩ U.

j =1

By (9.2), .z ∈ Qw can be described as .zn = h( z, w), where the function h depends antiholomorphically on w. Then the set A can be described by the following (infinite) system of equations A=

.

⎧ ⎨ ⎩

(w, w  ) ∈ V × Cn :

n  j =1

fj (z)wj =

n  j =1

fj ( z, h( z, w)) w j = 1,  z ∈  U

⎫ ⎬ ⎭

.

(9.11) The system can always be reduced to a finite subsystem, and this shows that A is a complex analytic set in .V × Cn . The equation defining Segre varieties of the unit sphere can be projectivized, which extends the Segre map .λ defined by (9.3) to the projective space: every point in .CPn is mapped to the corresponding projective hyperplane. Thus A can be defined as a (closed) complex analytic subset of .V × CPn . Indeed, if a sequence of points j j 0 0 n .(w , w ) ∈ A converges to a point .(w , w ) ∈ V × CP , then by analytic dependence of the Segre varieties on w and .w  , and the description of the set A it follows that 0 0 .(w , w ) ∈ A.

9.3 Analytic Continuation Using Segre Varieties

125

It is easy to see that .dim A = n. Indeed, the set .f (Qw ∩ U ) has complex dimension n − 1, and therefore, can be contained in at most one Segre variety .Qw . Since the Segre map .λ is globally injective, A is the graph of a holomorphic map, which is defined on all of V because the projection from A to V is proper by compactness of .CPn . Finally, the extended map is locally biholomorphic because the Segre map .λ is also bijective in .U1 since . is strictly pseudoconvex. This proves the lemma. 

.

The next step is to connect points on the hypersurface . with curves that are contained in the intersection of certain Segre varieties with .. More precisely, the following holds. Lemma 9.6 Let . ⊂ Cn be a strictly pseudoconvex real-analytic hypersurface. Let .M ⊂  be a generic submanifold of dimension .2n−2, and let .p ∈ M. Let U be a neighbourhood of p such that .U ∩ ( \ M) consists of two connected components, which we denote by − and . + . Then .Q ∩ U contains an open subset .ω such that for any point .b ∈ ω there . p exists a closed path .γ satisfying: (i) .γ ⊂ (Qb ∩  + ) ∪ {p} and (ii) .γ ∩ M = {p}. Note that the same set .ω also satisfies the lemma with . + replaced with . − . Proof It suffices to prove the lemma for .n = 2. In higher dimensions the result follows by a slicing argument. If .n = 2, then M is totally real, and after an appropriate change of coordinates we may assume that .p = 0 and in a small neighbourhood U of the origin . is given by the defining function ρ(z, z) = z2 + z2 +



.

ρkl (y2 )z1k zl1 ,

(9.12)

k,l

and M is given by  .

x1 = 0, ρ(z, z) = 0.

(9.13)

Assume that . + = {z ∈  ∩ U : x1 > 0}. By Moser’s construction of the normal form, see Sect. 7.2, the holomorphic change of variables  z1∗ = z1 , . z2∗ = z2 + g(z1 , z2 ), where .g(z1 , z2 ) is some holomorphic function satisfying .g(0, z2 ) ≡ 0, transforms the defining function of . into

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9 Segre Varieties



ρ(z, z) = z2 + z2 + |z1 |2 +

.

ρkl (y2 )z1k zl1 .

(9.14)

k+l>1

It is clear that M in these coordinates is also given by (9.12). Then Q0 = {ρ(z, 0) = 0} = {z2 = 0},

.

and for .b ∈ Q0 , say, .b = (b1 , 0), we have 



Qb = z ∈ U : z2 + z1 b1 +

.

ρkl

z 

k+l>1

2

2i

 l z1k b1

=0 .

By solving this equation for .z2 near the origin, we obtain z2 = ηz1 + α(z1 ),

.

(9.15)

where .η depends holomorphically on .b1 , .η = const, and .α(z1 ) = o(|z1 |). The set .Qb ∩  is given by the system  .

 z2 + z2 + z1 z1 + k+l>1 ρkl (y2 )z1k zl1 = 0, z2 = ηz1 + α(z1 ).

(9.16)

By plugging the second equation into the first, we obtain 2Re(ηz1 + α(z1 )) + |ηz1 + α(z1 )|2 +



.

ρkl (Im(ηz1 + α(z1 ))) z1k zl1 = 0.

(9.17)

k+l>1

Choose .ω ⊂ Q0 such that for .b ∈ ω, .Re η = 0, .Im η = 0. By the Implicit function theorem ˜ 1 ), where .α(y ˜ 1 ) = o(|y1 |) and equation (9.17) can be rewritten in the form .x1 = cy1 + α(y .c = 0. For .b ∈ ω, .γ =  ∩ Qb is then given by  .

˜ 1 ), x1 = cy1 + α(y z2 = ηz1 + α(z1 ).

Therefore, .γ intersects both . + and . − , and .γ ∩ M = {0} for small .y1 . This proves the lemma. 

Next we show that f extends across generic submanifolds. Lemma 9.7 Let . be a strictly pseudoconvex real-analytic hypersurface. Let M be a generic submanifold of dimension .2n − 2 and .p ∈ M. Let U be a neighbourhood of

9.3 Analytic Continuation Using Segre Varieties

127

p. Denote by . − and . + the connected components of .U ∩ ( \ M). Suppose that f is a locally biholomorphic mapping defined on . + , .f ( + ) ⊂ bBn . Then f extends to a neighbourhood of p as a biholomorphic mapping. Proof Let .U1 , .U2 be a standard pair of neighbourhoods of p. Since . is strictly pseudoconvex, we may assume that the Segre map .λ is one-to-one in .U1 . By Lemma 9.6, there exists an open set .ω ⊂ (Qp ∩ U1 ) such that for any point .b ∈ ω, .Qb ∩  contains a path .γ in . + with the end point at p. The choice of .b ∈ ω, .γ and a point .a ∈ γ ∩ U1 will form a triple, which we will denote by .(b, γ , a). We can choose a so close to p that, possibly after a small perturbation, .U1 , .U2 will also be a standard pair of neighbourhoods for a. Let .Ua be a neighbourhood of a so that f is biholomorphic in .Ua . By Lemma 9.5, f extends analytically along any path in V , where V is a neighbourhood of .Qa ∩ U1 . Let .τ be a simple path .τ ⊂ V , connecting a and b. Then .f |Ua extends holomorphically to .Uτ , a neighbourhood of .τ . Denote by F the extension of .f |Ua to .Uτ obtained by Lemma 9.5. Choose a small neighbourhood .Ub of the point b, .Ub ⊂ Uτ such that for any z in some small neighbourhood .Uγ of .γ , .Qz ∩ Ub is nonempty and connected. (Since .γ ⊂ Qb , we have .Qz  b, for all .z ∈ γ .) Thus F is holomorphic in .Ub . Consider the set   A∗ = (w, w  ) ∈ Uγ × Cn : F (Qw ∩ Ub ) ⊂ Qw .

.

(9.18)

The same proof as in Lemma 9.5 shows that .A∗ is a closed complex-analytic subset of n .Uγ × C . We claim that there exists a small neighbourhood . of a such that A∗ ∩ ( ×  ) = f | ,

.

(9.19)

where .  = f ( ). Indeed, choose some small neighbourhood . containing a and a point z in . . Let .w ∈ Qz ∩ Ub be an arbitrary point and .w  = F (w). It follows from the definition of F that .f (Qw ∩ Ua ) ⊂ Qw . Also .z ∈ Qw . This implies that .f (z) ∈ Qw = QF (w) . But then .F (w) ∈ Qf (z) . Since .w ∈ Qz was arbitrary, we deduce that .F (Qz ∩ Ub ) ⊂ Qf (z) . This means that .(z, z ) ∈ A∗ iff .z = f (z), and so .A∗ ∩ ( ×  ) = f | . This proves the claim. Consider the irreducible component of .A∗ which coincides with .f in . ×  . For simplicity denote this component again by .A∗ . Then .dimC A∗ = n. Let .zj → p as .j → ∞, . zj ∈ γ . By passing to a subsequence if necessary, we may assume that there exists  n  j ∗ .p ∈ bB such that .p = limj →∞ f (z ). Since the graph of .f |U ∩ + is contained in .A , γ we have .(zj , f (zj )) ∈ A∗ and thus .(p, p ) ∈ A∗ . Let .π : A∗ → Uγ and .π  : A∗ → Cn be the natural projections. We now show that there exist neighbourhoods .Up  p and .Up   p such that .fˆ := π  ◦ π −1 (z) is a holomorphic mapping in .Up which extends f . Here .π −1 : Up → A∗ ∩ (Up × Up  ). For

128

9 Segre Varieties

this, choose .Up   p so small that the Segre map .λ is one-to-one in .Up  and let .Up be a small neighbourhood of p such that .Up ⊂ π(π  −1 (Up  )). Let us show that .π : A∗ ∩ (Up ×

Up  ) → Up is one-to-one. If not, then we can find .z ∈ Up and .z 1 , z 2 ∈ Up  , z 1 = z 2 , such that (z, z ), (z, z ) ∈ A∗ ∩ (Up × Up  ). 1

.

2

(9.20)

Then .F (Qz ∩ Ub ) ⊂ Q  j , j = 1, 2. It follows from the definition of F that for any z .w ∈ Ub we have f (Qw ∩ Ua ) ⊂ QF (w) .

.

Since .λ : z → Qz injective in .Ub , there exist a unique point in .Ub whose Segre variety is Qw . Thus, .dimC F (Qz ∩ Ub ) = n − 1. But then, since .λ is injective, there exists at most one point .z ∈ Up  such that .F (Qz ∩ Ub ) ⊂ Qz . This contradicts (5.3) and therefore .π is one-to-one. By [37, Prop 3, sec. 3.3], .π : A∗ ∩ (Up × Up  ) → Up is a biholomorphic mapping and hence, .fˆ := π  ◦ π −1 (z) is holomorphic in .Up and extends f . By analyticity we also have n .fˆ( ∩ Up ) ⊂ bB . 

.

Finally, the same argument as in Sect. 8.1.4 shows that extension across generic submanifolds in . implies analytic continuation along any CR-curve on .. And that completes the proof of Theorem 9.4.

9.4

Holomorphic Mappings of Real Algebraic Hypersurfaces

A real hypersurfaces . ⊂ Cn is called algebraic if it is defined by the equation  = {z ∈ Cn : P (z, z) = 0},

.

where .P : Cn → R is a polynomial. In this section we prove the following result due to Webster [158], it can be viewed as a generalization of Theorem 2.2. Theorem 9.8 Let .j , .j = 1, 2, be two real algebraic hypersurfaces in .Cn , .n > 1. Let f be a locally biholomorphic mapping in a neighbourhood .U ⊂ Cn of a Levi nondegenerate point .p ∈ 1 such that .f (1 ∩ U ) ⊂ 2 . Then f extends to .Cn as an algebraic mapping (i.e., the graph of f is contained in a complex algebraic variety of dimension n in .Cn ×Cn ).

9.4 Holomorphic Mappings of Real Algebraic Hypersurfaces

129

Proof Without loss of generality assume that .p = f (p) = 0. Since the Jacobian of f is nondegenerate, 0 is also a Levi nondegenerate point of .2 . As a first step, we repeat the argument used in the proof of the Reflection principle (Theorem 8.3). Assume that .j = {z ∈ Cn : Pj (z, z) = 0}, .j = 1, 2, and .∂P1 /∂zn (0) = 0. Let .X k be the vector fields on .1 given by (1.10) that near the origin represent tangential Cauchy-Riemann operators: Xj =

.

∂P1 ∂ ∂P1 ∂ − , j = 1, . . . , n − 1. ∂zn ∂zj ∂zj ∂zn

Then .X k are the operators of type .(0, 1), with coefficients that are polynomials in .z, z. Consider the equations P2 (f (z), f (z)) = 0,

(9.21)

Xk P2 (f (z), f (z)) = 0, k = 1, . . . , n − 1.

(9.22)

.

and .

Recall that .Xk fj = 0, so the above equations are polynomial in z and .f (z). As in the proof of Theorem 8.3, since .2 is Levi nondegenerate and the Jacobian of f is nonsingular, we may apply the Implicit function theorem to the system (9.21) and (9.22) near the origin. This gives f (z) = H (z, z, f (z), Df (z)),

.

(9.23)

where H is a holomorphic function of its variables, and is algebraic in z. Hence, f (z) = R(z, z), z ∈ 1 ∩ U,

.

(9.24)

where R is a real analytic function that is algebraic in z. Lemma 9.9 The restriction of f to any Segre variety .Qζ = {z ∈ Cn : P1 (z, ζ ) = 0}, for .ζ close enough to the origin, is an algebraic function. This means that the set {(z, w) ∈ Cn × Cn : w = f (z), z ∈ Qζ }

.

is contained in a complex algebraic variety of dimension .n − 1.

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9 Segre Varieties

Proof Consider the manifold ˆ 1 = {(z, z) ∈ C2n : z ∈ 1 },

.

which is a generic submanifold of dimension .(2n − 1) in the complexification 1C = {(z, ζ ) ∈ Cn × Cn : P1 (z, ζ ) = 0}.

.

(We formally view the polynomial .P1 as holomorphic in .(z, ζ ) and consider the corresponding complex structure). Since .ˆ 1 is the uniqueness set for functions holomorphic on C . , we easily obtain from (9.24) that .f (z) = R(z, ζ ) for any .ζ ∈ U and any .z ∈ Q(ζ ), 1 which proves the lemma. 

Since .1 is Levi nondegenerate at the origin, one can find n transverse families of Segre varieties, such that each family depends algebraically on .(n − 1) complex parameters and their union fills a neighbourhood of the origin in .Cn . Indeed, near the origin the family of Segre varieties of .1 is a small deformation of the Segre family for a Levi nondegenerate quadric. The latter is a union of complex hyperplanes, and this shows that such families can be found among Segre varieties of .1 as well. By Lemma 9.9, the restriction of f to every such hyperplane is algebraic. By the Implicit function theorem there exists an algebraic change of coordinates, which is biholomorphic near the origin, such that the families selected above become the families of complex hyperplanes that are parallel to the coordinate subspaces. By the classical separate algebraicity theorem, see Theorem A.8 of the Appendix, the map f is an algebraic 

map on .Cn . As an application of Webster’s theorem we prove the following result, which can be viewed as a generalization of the Poincaré-Alexander theorem (recall that the conclusion of that result is that a local biholomorphism between the unit spheres is the restriction of an automorphism of the ball—a linear-fractional map!). Theorem 9.10 Let D and .D  be smoothly bounded domains in .Cn , .n > 1, with algebraic boundary. Suppose that .f : D → D  is a proper holomorphic map. Then f is algebraic and extends holomorphically across a dense open subset of bD. Proof First observe that a smoothly bounded domain in .Cn necessarily contains strictly pseudoconvex points. This can be seen by taking a large sphere that contains D and then shrinking it until the sphere intersects bD at a point p. Then bD is strictly convex near the point p, and therefore, strictly pseudoconvex. Further, it is easy to see that the set E of Levi degenerate points on bD is a real algebraic set. By the previous argument, .E = bD, and therefore, .dim E < 2n − 1. The same argument applies to .D  . We now claim that the map f is algebraic. Indeed, if bD is not pseudoconvex, then the components of f extend holomorphically across the portion of the boundary of D where

9.5 Comments and Further Results

131

the Levi form of a defining function of bD has at least one negative eigenvalue. Since the map f is proper, the Jacobian of the extension does not vanish near some point of / E. By the extendability q on bD. We further may choose q such that in addition .q ∈ invariance of the Levi form, near the image of q under the extended map the hypersurface  .bD is also Levi nondegenerate. Thus, we are under the assumptions of Theorem 9.8, and the map f is algebraic. Suppose now that D is pseudoconvex. Then .D  is also pseudoconvex, as otherwise, the components of the correspondence .f −1 would extend holomorphically on some nonempty open subset of .bD  , which leads to a contradiction. Let p be a strictly pseudoconvex point of bD, and let .C(f, Up ) be the cluster set of f in a neighbourhood .Up of p. By the uniqueness theorem, .C(f, Up ) cannot be contained in .E  –the set of Levi-degenerate points of .dD  , since .dim E  < 2n − 1. Therefore, there exists a sequence .{zj } ⊂ D converging to a strictly pseudoconvex point .z0 ∈ bD such that the sequence .{f (zj )} converges to a strictly pseudoconvex point .z0 ∈ bD  . Therefore, by Theorem 3.7, f extends continuously to a neighbourhood of .z0 , and by Theorem 4.8 the extension is holomorphic. Again, the Jacobian of the extension does not vanish identically on bD, and therefore, Theorem 9.8 applies. This proves the claim. Let A be the n-dimensional irreducible affine algebraic variety in .Cn × Cn that contains the graph of f . Let .π : A → Cnz and .π  : A → Cnz be the projections to the domain and the target space respectively, and set .F = π  ◦ π −1 . Then outside a proper complex algebraic set .S ⊂ Cnz , F locally splits into holomorphic maps. In particular, if .z ∈ bD \ S, then one of the branches of F gives a holomorphic extension of f . This completes the 

proof, since .S ∩ bD is closed and nowhere dense in bD. In fact, one can show that the map f in the above theorem extends holomorphically everywhere but this requires more work. One needs to consider two cases: the map F does not split into holomorphic maps but is a (locally proper) holomorphic correspondence; this case is considered in Theorem 10.1 of the next chapter in a substantially more general situation when the boundaries are real analytic. The second case is when the fibres of the projection .π may have positive dimension over some points in bD. This case may be eliminated in the algebraic case by using properties of Segre varieties, see [131]; the general situation for real analytic boundaries was considered in [54].

9.5

Comments and Further Results

Segre varieties were first defined by Segre [127] for strongly pseudoconvex hypersurfaces in .C2 , and later introduced in full generality in the context of the theory of holomorphic mappings by Webster [158] as part of the original proof of Theorem 9.8. As mentioned in the previous chapter, the method of the proof of Theorem 9.4, namely, holomorphic extension along Segre varieties, yields a number of generalizations of Theorem 8.1. The important condition used in the proof is the fact that the Segre varieties associated with the hypersurface in the target space are globally defined through

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9 Segre Varieties

projectivization. Therefore, analytic continuation of a holomorphic germ .f : 1 → 2 can be established in the case when .2 is a compact strictly pseudoconvex real algebraic hypersurface, see Shafikov [130]. The assumption that .2 is strictly pseudoconvex ensures that the extension of f remains to be single-valued. The real analytic hypersurface .1 , on the other hand, only needs to admit a locally finite Segre map .λ and need not be strictly pseudoconvex. In this case, at points where .1 is weakly-pseudoconvex, the extended map will not be locally biholomorphic. If one merely assumes that .1 is compact real analytic and .2 is compact real algebraic in .Cn , then the germ .f : 1 → 2 extends as a proper holomorphic correspondence, see Shafikov [131]. For further generalization, one may assume that .2 is a real algebraic hypersurface that is closed in .CPn , for example, Theorem 8.1 remains true if the unit sphere is replace with a Levi-nondegenerate hyperquadric given in homogeneous coordinates by Hk = {[z0 , . . . , zn ] ∈ CPn : |z0 |2 + |z1 |2 + · · · + |zk |2 − |zk+1 |2 − · · · − |zn |2 = 0},

.

where .0 < k < n (for .k = n−1 we obtain the standard sphere), see Hill and Shafikov [78]. Still further generalization can be obtained if we assume that the hypersurface .1 is nonminimal, i.e., it contains a complex hypersurface X. In this situation X is necessarily nonsingular and so it locally divides .1 into two connected components. From the previous discussion, any holomorphic germ .f : 1 → Hk extends holomorphically to one of the connected components of .1 \ X, but it was proved in Kossovskiy and Shafikov [91] that the map f also extends across X to the other component. However, interestingly enough, in this case it may happen that the other component is mapped by the extension of f into .Hk for a different k, see [91] for an explicit example. Webster’s theorem (Theorem 9.8) admits several generalizations, see for example [8, 42, 64, 81, 97, 133, 163], and other authors.

Holomorphic Correspondences

10

Holomorphic correspondences, the multiple-valued analogue of holomorphic mappings, naturally appear as the inverses of holomorphic mappings and play a fundamental role in the problem of analytic continuation of maps. The central result of this chapter is the following theorem proved by Diederich and Pinchuk [51]. Theorem 10.1 Let .D, D  be bounded domains in .Cn , .n ≥ 2, with smooth real analytic boundaries and let .f : D → D  be a proper holomorphic map that extends as a proper holomorphic correspondence to a neighbourhood U of a point .z0 ∈ bD. Then f extends holomorphically to a (possibly smaller) neighbourhood of .z0 . The proof is largely based on the invariance of Segre varieties under holomorphic maps and correspondences. In the first three sections we establish important properties of holomorphic correspondences that will be used in the proof of Theorem 10.1, which is given in Sect. 10.4. Another ingredient of the proof–a general result on the critical sets of holomorphic mappings–is proved in the last section of this chapter.

10.1

Proper Holomorphic Correspondences

Let U , .U  be open subsets in .Cn . Let .π : U × U  → U and .π  : U × U  → U  be the coordinate projections. A holomorphic correspondence is a closed complex analytic subset  .F ⊂ U × U of pure dimension n, such that .π : F → U is proper. The correspondence is called irreducible if F is irreducible as a complex analytic set. F is called proper if in addition the projection .π  : F → U  is also proper. For simplicity of notation the multiple-valued map .π  ◦ π −1 will also be denoted by F .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_10

133

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10 Holomorphic Correspondences

Let D and .D  be domains in .Cn and .z0 ∈ bD. Let .f : D → D  be a proper holomorphic mapping. We say that f extends as a holomorphic correspondence to a neighbourhood U of .z0 , if there exists an open set .U  in .Cn and an irreducible holomorphic correspondence  .F ⊂ U × U , such that f ∩ [(D ∩ U ) × D  ] ⊂ F.

.

As before, .f denotes the graph of f . We use the same terminology for the extension of multiple-valued maps. The above definitions deserve some attention. A correspondence F assigns to any point .z ∈ U a finite number of image points, namely, the points of the set .F (z) = (π  ◦ π −1 )(z) ⊂ U  . Counting with multiplicity, the number of images is equal to the sheet number m of the branched covering .π : F → U . After an arbitrarily small perturbation of the coordinates in the target space, we can make the following assumption: Generic Situation There is a proper closed complex analytic subset .B ⊂ U such that for all .z ∈ U \ B, the following holds: if .z j ∈ F (z), .j = 1, 2, and .z 1 = z 2 , then 1 2 .z k = z k for .k = 1, . . . , n. The components of the image points from .F (z) , written as  .z k = fk (z), .k = 1, . . . , n, are algebroid functions on U . This means that they satisfy polynomial equations of the form z k + ak1 (z)z k

.

m

m−1

+ . . . + akm (z) = 0, k = 1, . . . , n,

(10.1)

with the coefficients .akj holomorphic on U . By genericity and irreducibility of F , these polynomials are also irreducible in the ring of polynomials with the coefficients that are holomorphic on U . Proposition 10.2 If f extends as a holomorphic correspondence to a neighbourhood U of a point .z0 ∈ bD, then f extends continuously to .D ∩ U . Proof Recall that the cluster set .C(f, z) of any point .z ∈ bD is connected. On the other hand, it follows from (10.1) that for any .z ∈ bD ∩ U there are only finitely many possibilities for the image points. Hence, .C(f, z) is a singleton and f extends continuously 

to .D ∩ U . Note also that (10.1) implies the Hölder continuity of f up to the boundary. Let D be a domain in .Cn . Its envelope of holomorphy .D˜ exists in the category  its of Riemann domains over .Cn . For any compact subset .K ⊂ D denote by .K holomorphically convex hull (with respect to the class of functions holomorphic on D). In what follows we will use the prime to identify the objects in the target space. The following property of the envelope of holomorphy will be used several times.

10.2 Invariance of Segre Varieties for Correspondences

135

Lemma 10.3 Let .f : D → D  be a proper holomorphic mapping, and .a ∈ bD. Let   ˜ if .a  ∈ D˜  . .a ∈ bD be a point in the cluster set .C(f, a). Then .a ∈ D ˜ Then for any compact subset K of D one has .dist(K,  a) > 0. Proof Suppose that .a ∈ / D. j j  j We choose a sequence .(a ) in D converging to a, such that .a = f (a ) → a  , and a function h holomorphic on D such that .h(a j ) → ∞. Since f is a proper mapping, there is an algebroid function .h on .D  defined by the relation h (z ) := h ◦ f −1 (z ).

.

In other words, there are functions .ak holomorphic on .D  , .k = 1, . . . , m, such that for any     . z ∈ D and .w ∈ C, one has .w ∈ h (z ) if and only if w m + a1 (z )w m−1 + . . . + am (z ) = 0.

.

Since .a  ∈ D  and all the .ak extend holomorphically to .D˜  , .h has an algebroid extension to a neighbourhood .U  of .a  . Hence, .h is uniformly bounded near .a  , which is a contradiction. 

10.2

Invariance of Segre Varieties for Correspondences

In this section we prove the invariance of Segre varieties under holomorphic correspondences, an analogue of Lemma 7.1. Let D, .D  be bounded domains with real analytic boundaries in .Cn given by the defining functions .ρ(z, z) and .ρ  (z , z ) respectively. Consider a proper holomorphic mapping .f : D → D  which extends as a holomorphic correspondence F to a neighbourhood of a point .z0 ∈ bD. Choose the standard coordinates such that .z0 = 0, .f (z0 ) = 0, with the standard neighbourhoods .U1 ⊂ U2 (resp. .U1 ⊂ U2 ) of 0. We use the notation .U1∗ = {w : w ∈ U1 }, and similarly for .U2∗ and .Uj ∗ . For ∗  .z ∈ U2 and .w ∈ U we put .r(z, w) = ρ(z, w), and similarly for .r . We may suppose that 1   .F (U1 ) ⊂ U and .F (U2 ) ⊂ U . 1 2 Proposition 10.4 Suppose that F is an irreducible holomorphic correspondence. Then for all .(w, w  ) ∈ F ∩ (U1 × U1 ), the relation .F (Qw ) ⊂ Qw holds. What the proposition says is that every branch of F maps any point in .Qw into .Qw for any .w  ∈ F (w). Before giving a rigorous proof of this we outline a simple informal argument explaining why this result should hold. Let M = {(z, z , w, w  ) ∈ U2 × U2 × U1∗ × U1∗ : (z, z ) ∈ F, (w, w  ) ∈ F ∗ , r(z, w) = 0}.

.

136

10 Holomorphic Correspondences

It suffices to show that the function .r  (z , w  ) vanishes on .M. In a suitable neighbourhood V of a point of biholomorphic extendability of the map f we have .ρ  (f (z)) = α(z)ρ(z), where .α(z) is a nonvanishing real analytic function. After complexification we obtain r  (f (z), f ∗ (w)) = α(z, w)r(z, w),

.

where .f ∗ (w) = f (w). Note that .(z, f (z), w, f ∗ (w)) ∈ M for .z ∈ V and .w ∈ V ∗ . Therefore, the function .r  (z , w  ) vanishes on an open nonempty subset of the complex variety .M. If .M were irreducible, then this would imply the required property .ρ  |M ≡ 0. For the rigorous proof we first establish some technical results. Lemma 10.5 Let D be a bounded domain in .Cn with real analytic boundary. Let also U be a connected neighbourhood of a point .z0 ∈ bD and .F ⊂ U × U  be an irreducible holomorphic correspondence. Set Fc = F ∩ ((U \ D) × U  ).

.

If .z0 ∈ / D˜ and if U is chosen small enough, then .Fc is irreducible. Proof We may assume that F is defined by (10.1). Suppose that .Fc is reducible. Then .Fc would have an irreducible component .G ⊂ (U \ D) × U  , such that the sheet number of .π : G → U is less than m. Hence the polynomials (10.1) would be reducible over 0 / D, ˜ the decomposition of .O(U \ D). By Trépreau’s theorem (Theorem 1.3) and since .z ∈ (10.1) would extend to a decomposition over .O(U ) (after possibly shrinking U ). This is a contradiction. 

Lemma 10.6 Let .0 ∈ bD, and let .U1 =  U1 × U1n , .U2 =  U2 × U2n be a standard pair of neighbourhoods centred at 0. For .w = ( w, wn ) define w = { z ∈  U2 : ( z, hw ( z)) ∈ D}.

.

Then for any two points .w 1 , w 2 ∈ U1 \ D with the same .Im wn -coordinates and the same and with .Re wn1 < Re wn1 the relation .w1 ⊂ w2 holds. Further, there exists .δ > 0 such that for any . w ∈  U1 , we have

 . w-coordinates

U1δ := U1 ∩ {w = ( w, wn ) ∈ / D : Re wn > δ} = ∅,

.

and the set .w is connected for any .w ∈ / U1δ . Proof We may choose the coordinate system in such a way that the .Re zn -axis is the outer normal to bD at the origin. Then for any .w = (0, Re zn ) with .Re zn > 0, the function  .hw ( z) can be written in the form

10.2 Invariance of Segre Varieties for Correspondences

137

hw = −Re wn + β( z, Re wn ), β(0) = 0, and dβ(0) = 0.

.

Therefore, ∂β( z, Re wn ) = −1 + o(1). ∂Re wn

.

From this the required statements hold.



Proof of Proposition 10.4 As in Lemma 10.5 we use the notation Fc := F ∩ [(U1 \ D) × U1 ].

.

(10.2)

The proof consists of three steps. (i) First assume that .(w, w  ) ∈ Fc . We show that .f (Qw ∩D) ⊂ Qw ∩D  . Let .a ∈ bD∩U1 be a point of biholomorphic extendability of the map f ; in a neighbourhood of a we have f (s w Qw ) ⊂ Qf (w) ∩ D  .

.

(10.3)

Here the symmetric point .s w is defined as the unique intersection of the complex line through a point w normal to M and .Qw , and .s w Qw denotes the germ of .Qw at .s w. We use Lemmas 10.5 and 10.6 while traveling with w in .U1 \ D. More specifically: if w moves sufficiently far away from bD, the variety .Qw ∩ D becomes connected. Then, it follows by (10.3) that the inclusion f (Qw ∩ D) ⊂ Qw ∩ D 

.

(10.4)

holds for any .w  with .(w, w  ) ∈ Fc . Moving with w down again preserves this relation, since .Fc is irreducible. (ii) Next we show that, for any two points in F of the form .(w, w  1 ) and .(w, w  2 ), one has .Q  1 = Qw 2 . This is a consequence of (10.4) when the above points belong to w .Fc . The general case follows again by analytic continuation because F is irreducible. (iii) Viewing F as a multiple-valued map, we write .F = {f 1 , . . . , f m } and put .w  j = f j (w). We need to show that f k (Qw ) ⊂ Qw j , j, k = 1, . . . , m.

.

(10.5)

Consider first the case .w ∈ U1 \ D. Then from part (i) we conclude that for all .z ∈ D ∩ Qw one has .f k (z) ∈ Q  j . Hence, by part (ii) one has .w  j ∈ Qf k (w) = Qf k (w) , w

138

10 Holomorphic Correspondences

and therefore, .f k (z) ∈ Q  j . Together, we obtain that .f k (Qw ) ⊂ Q  j . By analytic w w 

continuation it follows that (10.5) holds for all .w ∈ U1 and for all j and k. Proposition 10.7 If a holomorphic mapping .f : D → D  extends as a holomorphic correspondence .F ⊂ U × U  to a neighbourhood U of a point .a ∈ bD, then .π  : F → U  is also proper (for suitably chosen U , .U  ). Proof We may assume .a = a  = 0. It suffices to show that the set .E := F −1 (0) is discrete. For this, note that E is an analytic set. For any point .z ∈ E ∩ D, we have .Qf (z) = Q0 by Proposition 10.4. Since f is finite-to-one, we conclude that .E ∩ D is discrete. But then   .dim(E ∩ (U \ D)) = 0. Indeed, by Proposition 10.4 one has .f (Qw ∩ D) ⊂ Q ∩ D for 0 any .w ∈ E \ D. Finally, no open non-empty subset of E can be contained in bD. This proves the claim. 

Finally, the invariance of the Segre varieties also holds for the inverse multiple-valued map (correspondence) .G := F −1 . Namely, for suitably chosen .U1 and .U1 one has    .G(Q  ) ⊂ Qw for all .(w, w ) ∈ U1 × U . We leave the proof to the reader. It is quite 1 w similar to that of Proposition 10.4 but one needs to take into account that in this setting the correspondence G extends a map which is itself multiple-valued.

10.3

From Multiple-Valued to Single-Valued Extension

In this section we prove additional results that are needed for the proof of Theorem 10.1. In the notation of Sect. 9.1 and under the assumptions of Theorem 10.1 we assume that the Segre maps .λ and .λ associated with bD and .bD  are given in the form (9.9). Proposition 10.8 Let .F : U → U  be the correspondence extending a proper holomorphic map .f : D ∩ U → D  ∩ U  , where .U  0, .U   0 are small enough. Then (i) there exists a single-valued (even injective) map .φ : λ(U ) → λ (U  ) such that the following diagram commutes. φ

λ(U ) −−−→ λ (U  )   ⏐  ⏐ ⏐λ ⏐λ U .

F

−−−→

U . (10.6)

For the (multiple-valued) correspondence F this means that for any .z ∈ U , it commutes with any value of .F (z).

10.3 From Multiple-Valued to Single-Valued Extension

139

(ii) (a) .F (∂D ∩ U ) ⊂ ∂D  ∩ U  , (b) .F (U \ D) ⊂ U  \ D  , (c) .F (D ∩ U ) ⊂ D  ∩ U  . (iii) Suppose that normal coordinates are chosen with respect to .0 ∈ bD and .0 ∈ bD  . Denote by .fn and .Fn the n-th components of the map f and the correspondent F . Then in a sufficiently small neighbourhoods of the origin, .Fn is single-valued and .fn (z) = b(z)zn , where .b(z) is some holomorphic function, .b(0) = 0. Proof (i) If .w  ∈ F (w), then by Proposition 10.4 we have .F (Qw ) ⊂ Qw . This implies the existence of the required map .φ. (ii) (a) Choose .w ∈ bD ∩ U1 and .w  ∈ F (w). It follows from Proposition 10.4 that      .Q  = Q f (w) . Since .f (w) ∈ bD , we conclude that .w ∈ bD . w (b) Let .w ∈ U1 \ D be arbitrary, and choose a point .a ∈ bD ∩ U1 of biholomorphic extendability of f . Connect a and w by a curve .w(t) in .(U1 \ D) ∪ {a}. If F (w)  w  ∈ / U1 \ D  ,

.

then there is a point .w 0 = w(t0 ) ∈ U1 \D such that .F (w 0 )  w 0 ∈ bD  , .w 0 = f (z0 ) with .z0 ∈ bD. By Proposition 10.4 applied to the correspondence .F −1 , we obtain .Qw 0 = Qz0 , which is a contradiction. (c) This is a consequence of parts (a) and (b) and irreducibility of .Fc as defined in (10.2). To see this, we first examine (w 0 , w 0 ) ∈ F ∩ ((U1 ∩ D) × (U1 \ D  )).

.

By part (b) there exists a point .(w 1 , w 1 ) ∈ F ∩ ((U1 ∩ D) × (U1 \ D  )). We connect this point with .(w 0 , w 0 ) by a curve in .Fc . On this curve there is a point (w 2 , w 2 ) ∈ F ∩ ((bD ∩ U1 ) × (U1 \ D  )).

.

But this contradicts part (a). The remaining case, in which there might exist a point (w 0 , w 0 ) ∈ F ∩ ((U1 ∩ D) × (U1 ∩ bD  )), can be treated in a similar way. (iii) Assume that near the origin the defining functions of bD and .bD  are given as in (9.5). Then by the Implicit function theorem, the Segre varieties .Qw and .Qw are given by .

zn = −w n +



bk (w)  z , .

(10.7)

bk (w )  z .

(10.8)

k

.

|k|>0

zn = −w n +



|k|>0

k

140

10 Holomorphic Correspondences

Therefore, .Qw1 = Qw2 implies .wn1 = wn2 , and similarly, .Qw1 = Qw2 implies 1 2 .wn = wn . Then it follows from Proposition 10.4 that for any .w ∈ U , where U is a sufficiently small neighbourhood of the origin, and any .w 1 , w 2 ∈ F (w) we have     1 .F (Qw ) ⊂ Q  and .F (Qw ) ⊂ Q  . Hence, .Q 1 = Q 2 , and therefore, .wn = w w w w 1

2

wn2 . This shows that .Fn is single-valued on U . In particular, it gives the required holomorphic extension of .fn to U by setting .fn = Fn . Applying the same argument to the inverse correspondence .G := F −1 we obtain analogous statements for .gn , the n-th component of the inverse map .g := f −1 . It follows from (10.7) that .Q0 = {zn = 0} and .Q0 = {zn = 0}. From Proposition 10.4 we obtain fn ( z, zn ) = 0 ⇐⇒ zn = 0,

.

gn ( z , zn ) = 0 ⇐⇒ zn = 0. Therefore, p fn ( z, zn ) = zn f˜n ( z, zn ),

.

gn ( z , zn ) = zn g˜ n ( z , zn ), q

where .p, q ∈ N, .f˜n (0) = 0, .g˜ n (0) = 0. From .f ◦ G = Id we obtain p q zn = zn f˜n ( z, zn ) = [zn g˜ n ( z , zn )]p f˜n ( g(z ), gn (z )) = (zn )pq g˜ n ( z, zn )p f˜n (g(z )).

.

Since .g˜ n (0) = 0 and .f˜n (0) = 0, we have .pq = 1, and so .p = q = 1. This completes the proof. 

Finally, we will need the following general result. Proposition 10.9 If in the situation of Theorem 10.1 .a ∈ bD and every irreducible component .E  a of the branch locus of the correspondence that extends f enters the domain D at a, then f extends holomorphically to a. Proof Let U be a small neighbourhood of a, and .F : U → Cn be the correspondence that extends the map f near the point a. Let E be the branch locus of F in U . Then E is a complex analytic set of pure dimension .n − 1. Since every component of E enters the domain D at a, we may choose the neighbourhood U so small that for every irreducible ˜ component .E˜ of E, the set .E˜ ∩ D is nonempty and open in .E. Let .S = E \D. We claim that .U \S is simply connected. For the proof we will show that every nontrivial cycle in .U \ E is null-homotopic in .U \ S, from this simple connectivity of .U \ S follows. By the classical van Kampen-Zariski Theorem, see, e.g., [57], the fundamental group of .U \ E is generated by the cycles that generate the fundamental

10.4 Proof of Theorem 10.1

141

group of .L \ (E ∩ L), where L is a complex line intersecting E transversely and avoiding singular points of E. Let .γ be a generator of .π1 (L \ (E ∩ L)). Then .γ is homotopic to a small circle in L around a point p of the intersection of L with an irreducible component .E˜ ˜ and .γ ∩E = ∅. Since the locus of regular of E. Further, the point p is a regular point of .E, ˜ ˜ points of .E is connected and .E ∩ D contains an open subset of .E˜ by the assumptions of the lemma, we can move the cycle .γ along the locus of smooth points of .E˜ avoiding points in E until .γ is entirely contained in D. This means that .γ is null-homotopic in .U \ S, and hence the latter is simply connected. We next show that the map f defined in .D ∩ U extends as a holomorphic map along any path in .U \ S. Indeed, on .U \ E the correspondence F splits into a finite collection of holomorphic mappings, the branches of F . Fix a point .b ∈ (U ∩ ∂D) \ E. Then one of the branches of the correspondence F at b gives the extension of the map f to a neighbourhood of b. Taking any path .γ in .U \ E which starts at b we obtain the extension of f along .γ by choosing the appropriate branches of F over the points in .γ . This gives analytic continuation of f in the complement of E in U . Suppose now that .γ intersects .E ∩ D. Without loss of generality assume that .γ terminates at a point .c ∈ E ∩ D and .γ \{c} ⊂ U \E. The set S is closed and has simply connected complement in U , hence, any two paths in the complement of S are homotopically equivalent. In particular, this means that the path .γ can be homotopically deformed avoiding the set S so that the deformation .γ˜ of .γ connects the points b and c along the path that is entirely contained in .D \ E (except the end points). Furthermore, we claim that this can be done in such a way that no curve in the deformation family intersects E (except the end point). Indeed, consider the cycle .γ ◦ γ˜ −1 which we slightly deform so that it does not intersect E near the point c. If .γ ◦ γ˜ −1 is null-homotopic in .U \ E, then the claim is trivial. If .γ ◦ γ˜ −1 is a nontrivial cycle in .U \ E, then as in the proof of simple connectivity of .U \ S, we may represent this cycle as a sum of “small” cycles around smooth points of E. We then move these small cycles along the regular locus of E until all of them are contained in D (again we used the fact that every component of E enters the domain D). As a result we conclude by the Monodromy theorem that the analytic continuation of f along .γ and .γ˜ defines the same analytic element near the point c. But since .γ˜ is contained in D, extension along .γ˜ simply gives the map f already defined at c. This gives analytic continuation of f along any path in .U \ S, which is single-valued by the Monodromy theorem. Finally, since every component of E enters the domain D at a, the set S is not a complex analytic subset of U , and hence it is a removable singularity for the extension of f in .U \S. This shows that f extends to a as a holomorphic map. 

10.4

Proof of Theorem 10.1

In this section we prove Theorem 10.1 assuming Theorem 10.11, which will be proved in the next section.

142

10 Holomorphic Correspondences

We use the notation from Sect. 9.1. Choose normal coordinates near the points .z0 , .f (z0 ) and assume .z0 = 0, .f (z0 ) = 0 . Then the defining functions .ρ and .ρ  of .D∩U , and .D  ∩U  respectively, have the form ρ(z, z¯ ) = 2xn +



ρj ( z,  z)(2yn )j , .

(10.9)

ρj ( z ,  z )(2yn )j .

(10.10)

.

j =0

ρ  (z , z¯  ) = 2xn +

 j =0



Let .λ : U → CN +1 , .λ : U  → CN +1 be the Segre maps of .∂D and .∂D  near .0 ∈ U and   .0 ∈ U respectively. It is convenient to denote their components by .λ = (λ0 , λ1 , . . . , λN ),     .λ = (λ , λ , . . . , λ ) so that in normal coordinates N 0 1 (a) λ0 (z) = zn , (b) λ0 (z ) = zn .

(10.11)

.

After rescaling of coordinates we may assume that U and .U  are polydiscs of radius 2 centered at the origin and that .F (U ) ⊂ U  , where F is the correspondence that extends the map f . In addition, we may assume that in the complexifications of the above defining  functions (Eqs. (9.6) and (9.7)), the functions .ρ(z, w), .ρj ( z,  w), .λk (w), . k λk (w)  zk and the corresponding functions in the target space are analytic in the respective polydiscs.   ∂λk | and the corresponding series in the target In particular, the series . k |λk (w)|, . k | ∂w n space converge uniformly on compact subsets of U (resp. .U  ). Further, by the choice of ∂λ

the defining functions and the coordinate systems we have .λk (0) = 0, and . ∂zk (0) = 0 for n all k. Therefore,  .

   ∂λ   k (0) = 0.  ∂z 

|λk (0)| = 0, and

k

k

(10.12)

n

By Proposition 10.8(iii), .Fn = fn = b(z)zn for some holomorphic function .b(z). In fact, we may assume that .b(z) ≡ 1. This can be achieved by an additional change of coordinates in U . Of course, these coordinates may no longer be normal. Instead we have Fn (z) = fn (z) = zn .

.



(10.13)

Denote by . a neighbourhood of .λ (0 ) in .CN +1 . We can choose the sets .U  0, .U   0 ,         .  λ (0 ) such that the mappings .f : D ∩ U → D ∩ U and .λ : U →  are proper holomorphic (the latter follows from Corollary 9.3). Consider for .M > 0 the following open sets  DM

.

⎧ ⎫ N ⎨ ⎬    λ (z )2 < 0 , = z ∈ U  : 2xn + M k ⎩ ⎭ k=0

10.4 Proof of Theorem 10.1

143



DM

N    2   λk (F (z)) < 0 . = z ∈ U : 2xn + M k=0

 , .∂D near .0 and 0 respectively, are real analytic and pseudoconvex The boundaries .∂DM M because of (10.11)(b), and do not contain nontrivial germs of complex varieties because of properness of .λ and Proposition 10.7.  ∩ U  ⊂ D ∩ U . We first prove Theorem 10.1 under an additional assumption that .DM It follows from Proposition 10.8 and (10.13) that .DM ∩ U ⊂ D ∩ U and .f : DM ∩ U →  ∩ U  is a proper holomorphic map. This implies that for DM

M = {w ∈  : 2Re w0 + M|w|2 < 0},

.

the map .λ ◦ f : DM ∩ U → M is also proper holomorphic. By Proposition 10.8(iii), the map .λ ◦ F = λ ◦ f extends holomorphically to a neighbourhood of .0 ∈ U . Let .E  ⊂ U  be the critical set of .λ : U  →  and .S ⊂ U be the branch locus of .F : U → U  . By Proposition 10.8, .F (S) ⊂ E  , moreover, .F (S) is contained in the  .(n − 1)-dimensional part of .E . By Theorem 10.11 any .(n − 1)-dimensional component     of .E enters .D ∩ U at .0 . By Proposition 10.8(ii), any irreducible component of S also enters .D ∩ U at 0, and thus f extends holomorphically to 0 by Proposition 10.9. This  ∩ U  ⊂ D  ∩ U  . However, .D  ∩ U  completes the proof of Theorem 10.1 in the case .DM M   is not necessarily a subset of .D ∩ U and the general proof of Theorem 10.1 requires an additional (mainly technical) argument. 1 Consider for .M > 1 two families of open sets depending on .ε ∈ (− M , 0]:  DMε =

.

⎧ ⎨ ⎩

⎫ ⎬



z ∈ U  : 2xn + M|zn |2 + M

N  k=1

DMε = z ∈ U : 2xn + M|zn | + M 2

N 

|λk (z )|2 < ε , ⎭

|λk

◦ F (z)| < ε . 2

k=1   , .D = DM These families are increasing for increasing .ε and .DM0 M0 = DM . The next proposition establishes some important properties of these domains.

Proposition 10.10 The following holds.  , .D (a) The sets .DMε Mε are pseudoconvex and their boundaries do not contain any nontrivial germs of complex varieties at all points in U , respectively .U  , where they are smooth real analytic. 1 1  (b) .DMε ⊂ D  ∩ U and .DMε ⊂ D ∩ U if .ε ∈ (− M , 0] is close to .− M . 1  (c) For .M > 0 sufficiently large and any .ε ∈ (− M , 0] the nonsmooth part of .∂DMε is   contained in .D ∩ U and the nonsmooth part of .∂DMε is contained in .D ∩ U .

144

10 Holomorphic Correspondences

Proof (a) In the polydiscs U and respectively .U  the opens sets are given as sublevel sets of plurisubharmonic functions, this implies pseudoconvexity. Note that .λk ◦ F (z) are single-valued by Proposition 10.8(iii).  can be rewritten as Next, observe that the defining function of .DMε   2   λ (z )2 − ε − 1/M. ρMε (z , z ) = M zn + 1/M  + M k

.

(10.14)

k  is smooth near a point .z ∈ U  and .h : D → bD  (M, ε) is a holomorphic map If .bDMε 0  , then from (10.14) it follows that .h and .λ ◦ h(t) with .h(0) = z0 and .h(D) ⊂ bDMε n k have to be constant for all .k > 0. Since the set .Az is finite for all .z , the map h itself  does not contain any nontrivial complex has to be constant which shows that .bDMε germs. The same argument applies to .DMε .  . (b) This can be seen directly from the definitions of the domains .DMε and .DMε  (c) First we show that the nonsmooth part of .bDMε is contained in .D  . Since .ε < 0, the   inclusion .z ∈ bDM,ε implies, in view of (10.14), that

  2 M zn + 1/M  + M |λk (z )|2 ≤ 1/M.

.

(10.15)

k

The following estimates follow immediately .

− 2/M ≤ xn ≤ 0, |zn |2 ≤ 4/M 2 ,



|λk (z )|2 ≤ 1/M 2 .

(10.16)

k

Since 0 is the only solution of the system zn = 0, λk (z ) = 0, for any |k| > 0,

.

 the sets .DMε shrink to the origin as .M → ∞. In particular, .z → 0 as .M → ∞.   be nonsmooth at some .z . Then .∇ρM,ε (z , z ) = 0. Therefore, Let now .bDM,ε

.

 ∂ρMε (z , z ) = 0, ∂zn

which implies 1/M + xn + Re

 ∂λ

k

.

k

∂zn

(z )λk (z ) = 0.

(10.17)

10.4 Proof of Theorem 10.1

145

By (10.16) we have .|λk (w  )| ≤ 1/M, and so       ∂λ 1   ∂λk     k   . (z )λk (z ) ≤  ∂z (z ) .  M  ∂zn n k

k

In view of (10.12) the sum on the right-hand side is .o(1) as .z → 0 uniformly in .ε ∈ (−1/M, 0]. Hence,  ∂λ

k

.

∂zn

k

(z )λk (z ) = o(1/M), as M → ∞.

(10.18)

Together with (10.17) this gives xn = −1/M + o(1/M).

(10.19)

.

Using again the inequality .|λk (z )| ≤ 1/M we also obtain  .

λk (z )zk = o(1/M).

(10.20)

k

Putting together (10.19), (10.20) into (9.8) we obtain  











ρ (z , z ) = (1 − α (z , z ))

.

2xn

+



 λk (z )zk

k

=−

2 + o(1/M) < 0, M

for sufficiently large M and uniformly in .ε. This shows that .z ∈ D  which proves the first claim. Let us now prove that the nonsmooth part of .bDMε is contained in D. Consider the defining function ρMε (z, z) = 2xn + M|zn |2 + M

.

  λ (F (z))2 − ε k

(10.21)

k

of the domain .DMε , keeping in mind that .fn (zn ) = zn . Suppose that .z ∈ bDMε is a nonsmooth boundary point. In a complete analogy to the argument for .bD  , we get 2xn + M|zn |2 + M

.

  λ (F (z))2 ≤ 0, k

(10.22)

k

and the three inequalities .

− 2/M ≤ xn ≤ 0, |zn |2 ≤ 4/M 2 ,

  λ (F (z))2 ≤ 1/M 2 . k k

(10.23)

146

10 Holomorphic Correspondences

Further, since .F (z) → {0} as .z → 0, we have  .

λk (F (z)) (F (z))k = o(1/M).

(10.24)

k ∂λ (F (z))

However, since . k∂zn does not necessarily vanish at 0, the analogue of (10.19) may not hold. Nevertheless, there exists .c > 0 such that for large M, xn ≤ −c/M.

(10.25)

.

Indeed, if (at least on a suitable subsequence) .xn = o(1/M), then we obtain from (10.22) that .|λk (F (z))| = o(1/M), and therefore,  ∂λ (F (z)) k

.

k

∂zn

λk (F (z)) = o(1/M).

But this is a contradiction to 1/M + xn + Re

 ∂λ (F (z)) k

.

k

∂zn

λk (F (z)) = 0,

which holds in analogy to (10.17). This proves (10.25). In order to show that .z ∈ D we will use the multiple-valued function  .ρ (F (z), F (z)), noting that by Proposition 10.8, for any fixed z, all of its values have the same sign, namely, D is mapped into .D  under the correspondence F , and its exterior is mapped into the exterior of .D  . Because of (9.8), we have        λk (F (z))( F (z))k , .ρ (F (z), F (z)) = 1 + α (F (z), F (z)) 2xn + k

with .α  (0, 0) = 0. Hence, we obtain from (10.24) and (10.25) that ρ  (F (z), F (z)) ≤ −2c/M + o(1/M) < 0,

.

and therefore, .z ∈ D. To finish the proof of Theorem 10.1 consider    Mε = w  ∈ CN +1 : 2u0 + M|w  |2 < ε .

.



10.5 Critical Sets of Holomorphic Maps

147

 and .λ◦f : D  If M, .ε are chosen as in Proposition 10.10, then .f : DMε → DMε Mε → Mε 1 are proper holomorphic maps. Consider the largest .ε0 ∈ (− M , 0] such that f extends to  . By Proposition 10.10, .f˜ is holomorphic a proper holomorphic map .f˜ : DMε → DMε 0 on the nonsmooth part of .∂DMε0 . Let us show that .f˜ extends holomorphically to any smooth real analytic boundary point .a ∈ U of .DMε . We only need to consider the case .a ∈ S (recall that S is the branch locus of F ). Applying, as before, Theorem 10.11 to  → Mε0 , we conclude that any irreducible .(n − 1)-dimensional the map .λ : DMε 0  at .f˜(a). By Proposition 10.8, component .Ej  f˜(a) of the critical set .E  of .λ enters .DMε 0 any irreducible component .Sj  a of S enters .DMε0 at a. Thus, by Proposition 10.9, .f˜

extends holomorphically to any such a. This means that f extends holomorphically to a neighbourhood of the closure of .DMε0 and we conclude that .ε0 = 0. This completes the proof.

10.5

Critical Sets of Holomorphic Maps

The following is a general result due to Pinchuk and Shafikov [112] that concerns the geometry of the critical set of a proper holomorphic map between real analytic hypersurfaces. This theorem is used in the proof of Theorem 10.1. Theorem 10.11 Let .D ⊂ Cn , .D  ⊂ CN , .2 ≤ n ≤ N, be bounded domains with smooth real analytic boundary. Let .f : D → D  be a proper holomorphic map that extends holomorphically to a neighbourhood .U ⊂ Cn of a point .a ∈ ∂D. Suppose that .D ∩ U ,    N is a neighbourhood of .f (a) ∈ ∂D  . Let E be .D ∩ U are pseudoconvex, where .U ⊂ C an irreducible .(n − 1)-dimensional component of the critical set of f in U with .a ∈ E. Then .E ∩ (D ∩ U ) = ∅. Without loss of generality we may assume that .a = 0, .f (0) = 0 , and .f (U ) ⊂ U  . Clearly, .f (D ∩ U ) ⊂ D  ∩ U  and .f (bD ∩ U ) ⊂ bD  ∩ U  . Since .bD  is pseudoconvex, by Theorem 1.10 for any .ε > 0 in a sufficiently small neighbourhood .U  of the origin the hypersurface .bD  ∩ U  admits a defining function .ρ  ∈ C 2 (U  ) such that .φ  := −(−ρ  )1−ε is a plurisubharmonic function on .D  ∩ U  . It follows that .φ  ◦ f is a negative plurisubharmonic function in .D ∩ U , and so by the Hopf lemma there exists a constant .C > 0 such that |φ  ◦ f (z)| ≥ Cdist (z, bD), z ∈ bD ∩ U.

.

(10.26)

We may assume that complex tangents to bD and .bD  at 0 and .0 are given respectively  = 0}. Then it follows from (10.26) that by .{zn = 0} and .{zN

148

10 Holomorphic Correspondences

.

∂fN (0) = 0. ∂zn

(10.27)

 2    2 N Indeed, if . ∂f ∂zn (0) = 0, then .fN (z) = O(|z| ), and since .ρ (z ) = 2xN + O(|z | ), we obtain

|φ  ◦ f (z)| ≤ c1 |z|2(1−ε) ,

.

which contradicts (10.26) for .ε < 1/2. In particular, we conclude that the map f extends to U as a proper holomorphic map. This can be seen as follows: (10.27) implies that  −1 (∂D ∩U  ) ⊂ ∂D. Since .∂D does not contain nontrivial .f (U \D) ⊂ U , and therefore, .f germs of complex subvarieties, the set .f −1 (0 ) is discrete, and, after shrinking if necessary the neighbourhood U , we may assume that the map f is proper in U . By Remmert’s Proper Mapping theorem .E  = f (E) ⊂ U  is an irreducible analytic set of dimension .n − 1. To illustrate the idea of the proof of the theorem consider first the simple case when E and .E  are complex manifolds. Arguing by contradiction suppose that .E ∩ (D ∩ U ) = ∅ for arbitrarily small U . Then E is tangent to .bD at the origin. Since       .E ∩ (D ∩ U ) is also empty, the manifold .E is tangent to .bD at .0 . After an additional local biholomorphic change of coordinates we may assume that .E = {zn = 0} and .E  =  = 0}. Let .z = (˜  ), and .f = (f˜, f ). Then the restriction .f | is z, zn ), .z = (˜z , zN {zN N E  ˜ given by .z˜ = f (˜z, 0). Since f is proper at the origin, .f |E is also proper at 0, and therefore ˜ the rank of the Jacobian matrix . ∂∂fz˜ (˜z, 0) is equal to .n − 1 on a dense subset .E1 ⊂ E. On ∂fn the other hand, .rank ∂f z, 0), and . ∂z (˜z, 0) = 0, .j = 1, . . . , n − 1, because ∂z < n for .z = (˜ j

N fN (˜z, 0) = 0. Therefore, . ∂f z, 0) = 0 for .(˜z, 0) ∈ E1 . By continuity, ∂zn (˜

.

.

∂fN (0) = 0, ∂zn

(10.28)

which contradicts (10.27). For the proof in the general case we will need the following technical result. We denote by .reg E the locus of regular points of a complex analytic set E, i.e., the points near which E is locally a complex manifold. Then .sing E = E \ reg E is the singular locus of E. Proposition 10.12 There exist a sequence of points .{pν } ⊂ reg E and two sequences of complex affine maps .Aν : Cn → Cn , .B ν : CN → CN such that for every .ν = 1, 2, . . . , the following holds (i) .rank(f |E ) = n − 1 at .pν , and .f (pν ) ∈ reg E  . (ii) .Aν (pν ) = pν and .B ν (f (pν )) = f (pν ).

10.5 Critical Sets of Holomorphic Maps

149

(iii) The transformations .Aν , .B ν converge to the identity maps .In : Cn → Cn and .IN : CN → CN respectively. (iv) .dAν maps .Tpν E onto .{v ∈ Tpν Cn : vn = 0} and .dB ν maps .Tf (pν ) E  onto .{v ∈ Tf (pν ) CN : vN = 0}. Theorem 10.11 can be easily deduced from Proposition 10.12. Indeed, consider the sequence of maps .f ν = B ν ◦ f ◦ (Aν )−1 . The above arguments show that .

∂fNν ν (p ) → 0 as ν → ∞, ∂zn

which yields (10.28). Again, we obtain a contradiction with the Hopf lemma. The rest of the section is devoted to the proof of Proposition 10.12. First, we will need the following Lemma 10.13 Let .U ⊂ Cn be a neighbourhood of the origin, .M  0 be a real hypersurface in U with a defining function .ρ ∈ C 1 (U ), ρ(z) = 2xn + o(|z|).

.

(10.29)

Let .A ⊂ U be an analytic set of pure dimension d, .1 ≤ d < n, such that .0 ∈ A ⊂ {z ∈ U : ρ(z) ≥ 0}. Then there exists an open subset .V ⊂ reg A with .0 ∈ V such that for any point .p ∈ V the tangent plane .Tp A is contained in a complex hyperplane Lp = {v ∈ C : vn = n

.

n−1 

ak (p)vk },

k=1

and .limV p→0 ak (p) = 0 for any .k = 1, 2, . . . , n − 1. Proof Let .C0 (A) be the tangent cone of A at 0. It is defined by .C0 (A) = limt→0 At , where At = {tz : z ∈ A}, .t ∈ R+ , are isotropic dilations of A. The set .C0 (A) is a complex cone of dimension d, i.e., it is invariant under complex dilations .z → tz, .t ∈ C\{0} (see, e.g., [38]) and .0 ∈ C0 (A) ⊂ {zn ≥ 0}. The last inclusion follows from .At ⊂ {z : tρ(z/t) ≥ 0} and .tρ(z/t) → 2xn as .t → ∞ because of (10.29). By the Maximum principle we conclude that .

C0 (A) ⊂ {zn = 0}.

.

(10.30)

150

10 Holomorphic Correspondences

Since .dim C0 (A) = d, there exists a complex plane .L  0, .dim L = n − d, such that L ∩ C0 (A) = {0}. Without loss of generality we assume that

.

L = {z ∈ Cn : z1 = 0, . . . , zd = 0}.

.

(10.31)

Let .z˜ = (z1 , . . . , zd ), .z˜˜ = (zd+1 , . . . , zn−1 ) so that .z = (˜z, z˜˜ , zn ). It follows from (10.30) that .|zn | = o(|˜z| + |z˜˜ |) on A, i.e., there exists a continuous function .α(t) ≥ 0 for .t ≥ 0 such that |zn | ≤ α(|˜z| + |z˜˜ |)(|˜z| + |z˜˜ |), z ∈ A.

.

(10.32)

We also have the following estimate for some .c1 > 0 and all .z ∈ C0 (A): |zn | + |z˜˜ | ≤ c1 |˜z|,

.

(10.33)

which follows from .L∩C0 (A) = {0} and (10.31). This implies that the origin is an isolated point of .L ∩ A. Hence, (10.33) also holds for .z ∈ A, possibly with a different .c1 . Now we can choose U = U˜ × U˜˜ × Un ⊂ Cd × Cn−d−1 × C

.

such that .π : A ∩ U → U˜ is a branched analytic covering of some multiplicity .m ≥ 1. Its discriminant set .σ˜ ⊂ U˜ and the tangent cone .C0 (σ˜ ) ⊂ Cd are analytic sets of dimension at most .d − 1. Therefore, there exists a complex line .l˜ ⊂ Cd such that .C0 (σ˜ ) ∩ l˜ = {0}. We may assume that .l˜ = {(z1 , 0, . . . , 0) ∈ Cd : z1 ∈ C}. Since .C0 (σ˜ ) is a closed cone, there exists .δ > 0 such that {˜z ∈ Cd : |zj | < δ|z1 |, j = 2, . . . , d} ∩ C0 (σ˜ ) = ∅.

.

(10.34)

With possibly smaller .δ > 0 we also have {˜z ∈ U˜ : |zj | < δ|z1 |, j = 2, . . . , d} ∩ σ˜ = ∅.

.

(10.35)

The set V˜δ := {˜z ∈ U˜ : |zj | < δ|z1 |, j = 2, . . . , d} ∩ {˜z ∈ U˜ : Re z1 > 0}

.

is simply connected, open in .U˜ and contains the origin in its closure. Since .V˜δ ∩ σ˜ = ∅, the set .A∩(V˜δ × U˜˜ ×Un ) is the union of the graphs of m holomorphic mappings .V˜δ → U˜˜ ×Un .

10.5 Critical Sets of Holomorphic Maps

151

˜˜ h ), and let .A = A ∩ (V˜ × U˜˜ × U ) be its graph. For Consider one of them, .H = (h, n δ δ n ˜˜ pn ) ∈ Aδ the tangent plane .Tp A is contained in the tangent plane at p to ˜ p, any .p = (p, the hypersurface in .V˜δ × U˜˜ × Un defined by one equation .zn = hn (˜z), which is given by .

v ∈ C : vn = n

d 

ak (p)v ˜ k , ak (p) ˜ =

k=1

∂hn (p). ˜ ∂zk

Thus, to finish the proof of the lemma, it remains to show that .

∂hn (p) ˜ = 0, k = 1, . . . , d. ∂zk ˜ V˜δ p→0 lim

(10.36)

Using (10.32)–(10.35) we successively obtain for certain constants .cj > 0 and all .p˜ ∈ Vδ , with .δ 1, and let .f : D → D be a proper holomorphic map that extends continuously to .D. Then f extends holomorphically to a neighbourhood of .D. The theorem was proved by Diederich and Pinchuk in [53].

11.1

Examples and General Setup

Note that the assumptions on the domains in Theorem 11.1 implies that bD and .bD  do not contain any nontrivial complex germs (Theorem 9.2). The following examples show that these assumptions cannot be dropped in general.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_11

155

156

11 Extension of Proper Holomorphic Mappings

(i) Let D = {z ∈ C2 : Re z2 + |z1 |2 < 0},

.

(11.1)

and .D  = {z ∈ C2 : Re (z2 )2 + |z1 |2 < 0}. Note that D is biholomorphic to the unit ball, and that .D  has two connected components, none of which has smooth boundary near the origin. However, .bD  is real analytic. Since .z2 = 0 in D, it √ is possible to choose a well-defined branch of . z2 in D, and with this choice, let √ .f (z1 , z2 ) = (z1 , z2 ). Then f is a biholomorphic map between D and a connected component, say, .D1 of .D  . The graph of f extends as a complex analytic set to a neighbourhood of .(0, 0 ) ∈ bD ×bD1 but the map f itself extends only continuously, but not holomorphically to .0 ∈ bD. (ii) With D defined as in (11.1), let .D  = {z ∈ C2 : Re z2 + |z1 |2 |z2 |2 < 0}. Then the boundaries bD and .bD  are smooth and real analytic, but .bD  contains a complex line .{z2 = 0}. The map .f = (z1 /z2 , z2 ) is a biholomorphism between D and .D  . Its graph extends to a neighbourhood of .(0, 0 ) ∈ bD × bD  as a complex analytic set but f does not extend to 0 even continuously. (iii) Let .g(z) be an inner function on .Bn (.n > 1). Then .g(z) has unimodular boundary values almost everywhere on .bBn by Fatou’s theorem. However it is known (see Proposition 19.1.3 in [125] for example) that there is a dense .Gδ -subset of .bBn such that the cluster set of any point from it must intersect the unit disc .D ⊂ C. In fact the image of any radius in .Bn that ends at a point on this .Gδ under .g(z) is dense in .D. Therefore, .f (z) = (g(z), 0, . . . , 0) is a (nonproper) holomorphic self map of .Bn for which there does not exist a neighbourhood U of any given boundary point with the property that the cluster set of .U ∩ bBn does not intersect .Bn . Evidently f fails to admit even a continuous extension to any .p ∈ bBn . In the remaining part of this section we outline the general setup for the proof of Theorem 11.1. Let .M = bD and .M  = bD  . We may assume that .0 ∈ M, .0 ∈ M  , and .f (0) = 0 . The goal is to show that f extends holomorphically to a neighbourhood U of the origin. Denote by .ρ(z, z) (resp. .ρ  (z, z)) real analytic defining functions of M (resp.   .M ) near 0 (resp. .0 ). By choosing an appropriate sign of .ρ we may assume that f is a map that is holomorphic in .U2− and extends continuously to M with .f (M) ⊂ M  . Further, we may assume that we have chosen normal coordinates z and .z such that ρ(z, z) = 2xn +

∞ 

.

ρν ( z,  z)(2yn )ν ,

(11.2)

ν=0

with .ρν ( z,  0) ≡ 0 for all .ν, and similarly for .M  . Denote by .U 1 ⊂ U2 the standard neighbourhoods of 0 (resp. .U1 and .U2 ). Then, for any .w ∈ U1 , the Segre variety Qw = {z ∈ U2 : ρ(z, w) = 0}

.

11.2 Extending the Map to a Dense Subset of M

157

is a well-defined closed smooth complex hypersurface in .U2 . For .w ∈ U1 ⊂ M, we define the symmetric point .s w as the unique intersection between the complex line through w normal to M and .Qw . For .w ∈ M we put .s w := w. We also use the notation Uj± := {z ∈ Uj : ±ρ(z, z) > 0}, j = 1, 2.

.

For .w ∈ U1+ , we define the canonical component .Qcw of .Qw as the component of .Qw ∩U2− containing the symmetric point .s w. For any point .ζ ∈ Qw , we denote by .s Qw the germ of .Qw at .ζ . We denote by .M + the set of strictly pseudoconvex points on M and by .M − the set of strictly pseudoconcave points. By .M ± we mean the set of all points where the Levi form has eigenvalues of both signs on .H (M), and by .M 0 we denote the set of points where the Levi form has at least one zero eigenvalue on .H (M). Note that .M 0 is a closed real analytic subset of M of dimension at most .2n − 2. We have M = M + ∪ M − ∪ M ± ∪ M 0.

.

Finally, we denote by . the set of all points in M such that f extends holomorphically to a neighbourhood of ..

11.2

Extending the Map to a Dense Subset of M

In this section we show that under the assumptions of Theorem 11.1, the set . is dense in M. To begin with, observe that properness of the map f implies that the differential .Jf is generically of full rank. −  − ± First note that .(M − ∪ M ± ) ∩ U2 ⊂ U 2 ; hence, f is holomorphic on .M ∪ M . Since .M 0 is nowhere dense in M, it remains to show that f extends holomorphically to a neighbourhood of a dense subset of .M + . Assume that .0 ∈ M + . In what follows we may shrink the neighbourhoods .Uj and .Uj if needed. Several cases can occur. (1) If .0 = f (0) ∈ M  + , then by Theorem 5.7 the map f is a homeomorphism of class 1/2 on M near the origin, and thus by Theorem 4.8 it extends holomorphically to a .C neighbourhood of the origin. (2) Suppose that .0 ∈ M  ± ∪ M  − . Then the point .0 belongs to the envelope of holomorphy of .D  , and so the map f extends holomorphically by Lemma 10.3. But, in fact, in this case f is necessarily a constant map. If .0 ∈ M  − this clearly follows by the invariance of the signature of the Levi form. For the proof when .0 ∈ M  ± , assume that on the contrary f is not constant. Then there exists a point .a ∈ U1− arbitrarily close to M such that .a  := f (a) belongs to .U1 − or .U2 + . Indeed, at a point .p ∈ U1− where Df has maximal rank, the image of a small neighbourhood of p is a complex

158

11 Extension of Proper Holomorphic Mappings

analytic set of positive dimension. Since .M  does not contain any nontrivial complex germs, the assertion follows. Note that such a point a can be chosen arbitrarily close to the origin. For an arbitrary .a ∈ U1− with .a  ∈ U1 − , there exists a (closed) complex analytic subset .Xa  ⊂ U1 of dimension 1 such that .a  ∈ Xa  and .Xa  ∩ M  is empty. This follows from the assumption .0 ∈ M ± and the Kontinuitätssatz. Then .a ∈ Xa := f −1 (Xa  ) ⊂ U2− . However, since .0 ∈ M + , the complex analytic set .Xa must have limit points on .M ∩ U2 . But then, since .f (M) ⊂ M  , the same holds for .Xa  , which is a contradiction. (3) The remaining case is when .0 = f (0) ∈ M  0 . We will show that in an arbitrarily small neighbourhood of the origin in M there exist points that belong to ., which will prove that . is dense in M. For that we first establish the following. Lemma 11.2 Let .0 ∈ N  ⊂ M  be a real .C 2 -smooth generic manifold, with .dim N  ≤ 2n − 2, and let U be any neighbourhood of .0 ∈ M + . Then .f (M ∩ U ) is not contained in  .N . Proof Without loss of generality assume that .dim N = 2n − 2. There exists a complex plane .L containing .0 such that .L ∩ N  is a totally real manifold of dimension 2 near   n   .0 . For .a ∈ C denote by .La  the plane parallel to .L and passing through .a . For small     enough .Uj , the intersections .Sa  := La  ∩ N ∩ U2 are totally real and of real dimension 2. Let .φa  be a strictly plurisubharmonic function in .U2 such that: (i) .φa  ≥ 0; (ii) .φa  (z) = 0 ⇐⇒ z ∈ Sa  . We may assume that .U1 ∩ M ⊂ M + , .f (U1− ) ⊂ U1 , and that ρ(z, z) = 2xn + | z|2 + o(|z|2 ).

.

(11.3)

For .a = (a1 , . . . , an ) ∈ U1− , put .a = {z ∈ U1− : zn = an } and Xa = a ∩ f −1 (Lf (a) ∩ U1 ).

.

Note that .a is a complex manifold of dimension .n − 1 which is compactly contained in U1 , and .f (ba ) ⊂ M  . If the lemma were false, we would have .f (ba ) ⊂ N  . Since −  .dimC L f (a) = 2, it follows that .Xa is an analytic set in .U1 of dimension at least 1 and .a ∈ Xa . The function .ψa = φf (a) ◦ f is plurisubharmonic and nonnegative on .Xa , and  .ψa |bXa = 0 because .f (bXa ) ⊂ N ∩ Lf (a) . Hence, .ψa |Xa ≡ 0. But .φf (a) is strictly plurisubharmonic, and therefore, .f |Xa is constant with the image contained in .N  ⊂ M  . Since .a ∈ U1− was chosen arbitrarily, it follows that .f (U1− ) ⊂ M  , but this contradicts case (2).   .

11.3 Correspondences that Extend the Map

159

To complete the argument for case (3), observe that the real analytic set .M  0 can be stratified into smooth generic submanifolds of dimension .≤ 2n−2. Applying Lemma 11.2 to each of them, we see that .f (M + ) is not contained in .M  0 . Hence, there exist points on + arbitrarily close to 0 that are mapped by f into .M  + ∪ M  − ∪ M  ± . According to case .M (2) described previously, such point cannot be mapped to .M  ± because f is nonconstant. Therefore, by case (1) these points must be in .. Combining all cases together we obtain Proposition 11.3 The set . is dense in M.

11.3

Correspondences that Extend the Map

We assume that the standard neighbourhoods are chosen so that .f (U2− ) is compactly contained in .U1 . The following set plays an important role for the holomorphic extension of f to a neighbourhood of the origin: F + := {(w, w  ) ∈ U1+ × U1 : f (Qcw ) ⊂ Qw }.

.

(11.4)

Consider the canonical projections .π : F + → U1+ given by .π(w, w  ) = w and .π  : F + → U1 given by .π  (w, w  ) = w  . Lemma 11.4 The set .F + is complex analytic in .U1+ × U1 . Proof From the definition of .F + in (11.4), we have .(w, w  ) ∈ F + if and only if   c .ρ (f (z), w ) = 0 for all .z ∈ Qw . Furthermore, .z ∈ Qw if and only if .ρ(z, w) = 0. The last equality holds if and only if .zn = h( z, w) where the function .h( z, w) = hw ( z) is defined as in (9.2). Hence, near any pair .(w 0 , w  0 ) ∈ U1+ × U1 , the set .F + is defined by ρ  (f ( z, h( z, w)), w  ) = 0,

.

for . z close to .s  w0 . This is a family of (anti) holomorphic equations in w, .w  .

 

Now consider the set α := {w ∈ U1+ : Jf = 0 on Qcw }.

.

Clearly, .α must be discrete. We set U + := π(F + ) ⊂ U1+ .

.

Lemma 11.5 The map .π |(F + \π −1 (α)) is locally proper.

(11.5)

160

11 Extension of Proper Holomorphic Mappings

Proof We have to show that for any .w 0 ∈ U + \ α, the set π −1 (w 0 ) ∩ F + = {(w 0 , w  ) ∈ F + }

(11.6)

.

is discrete. Let .b ∈ Qcw0 be such that .Jf (b) = 0. Then .f (b Qcw0 ) is a complex analytic germ of dimension .n − 1, and therefore, the inclusion .f (Qcw0 ) ⊂ Qw completely determines the Segre variety .Qw . The lemma now follows from the fact that .Aw is a finite set for any   .w ∈ U .   2 Corollary 11.6 The dimension of .F + is n. Proof By Lemma 11.5, the set .F + has dimension at most n at all points not lying in −1 (α). But since .α is discrete, the dimension of .F + must be at most n everywhere. .π However, the dimension of .F + cannot be less than n because .F + contains the graph of f   near all points of .. Next we modify the set .F + by excluding from it all irreducible components of dimension less than n. Furthermore, we choose .U1 so small that the Segre map λ : U1 → λ (U1 ) ⊂ S 

.

is proper. Lemma 11.7 The map .π : F + → U + is proper and hence .U + ⊂ U1+ is open. Proof The proof consists of two steps. First, we show that the restriction .π : F + \ π −1 (α) → U + \ α is proper. It suffices to show that .F + \ π −1 (α) has no limit points on  + ν ν + −1 (α) be a sequence such that .w ν → w 0 ∈ U + \α .(U \α)×bU . Let .(w , w ) ∈ F \π 1 and .w  ν → w  0 ∈ U  . Since .w 0 ∈ U + \ α, there exists a point .(w 0 , w˜ 0 ) ∈ F + \ π −1 (α) 1

with .w˜ 0 ∈ U1 . Furthermore, there is a sequence .(w ν , w˜ ν ) ∈ F + \ π −1 (α) with ν .w˜ → w˜ 0 ; this follows because the map .π is locally proper away from .π −1 (α). By / α, we conclude that .Qw ν = Q ˜ ν . (11.4) we have .f (Qcwν ) ⊂ Qw ν ∩ Q ˜ ν . Since .w ν ∈ w w Now the properties of the Segre map .λ together with the fact that .w˜ 0 ∈ U  imply that 1

also .w  0 ∈ U1 . For the second step, first observe that for any point .w ∈ α, the set (11.6) has dimension at most .n − 1. Indeed, .f (Qcw ) contains at least one point, and therefore, the set of .w  such that .Qw pass through that point has dimension at most .n − 1. This means that after eliminating the lower-dimensional components from .F + , we have .F + = F + \ π −1 (α). Now, by the previous step, the set .F + is (over .U + \ α) contained in a set of points whose .w  -coordinates are given by a system of monic polynomials in .wk , .k = 1, . . . , n,

11.3 Correspondences that Extend the Map

161

with coefficients holomorphic in .w ∈ U + \ α. Since all of these coefficients extend holomorphically across .α, and since .F + is (over .α) contained in the set defined by these extended polynomials, the lemma follows.   Next, we have the following Lemma 11.8 The map .π  : F + → U1 is locally proper. Proof We need to prove that any point .(w 0 , w  0 ) ∈ F + is isolated in π

.

−1

(w  ) = {(w, w  ) ∈ U1+ × U1 : f (Qcw ) ⊂ Qw 0 }. 0

0

This set is complex analytic. If it is not discrete, then its dimension is at least 1 and hence the set  .

Qcw

(w,w 0 )∈π −1 (w 0 )

contains an open subset of .U1− . This would imply that .f (U1− ) ⊂ Q  0 . This is a w   contradiction since .Jf does not vanish identically. The two lemmas above imply that .F + , equipped with the projections .π and .π  , can be viewed as a holomorphic correspondence .F + = π  ◦ π −1 : U + → U1 . By construction, for .w ∈ U + we have F + (w) = {w  ∈ U1 : f (Qcw ) ⊂ Qw }.

.

Now we define another holomorphic correspondence .F − : U1− → U1 by F − (w) := {w  ∈ U1 : Qw = Qf (w) }.

.

We put U := U1− ∪ U + ∪ ( ∩ U1 ).

.

By the invariance property of the Segre varieties, the correspondences .F + and .F − coincide near any point from . ∩ U1 . Therefore, together they give a holomorphic correspondence ˆ : U → U  with .Fˆ |U + = F + and .Fˆ | − = F − . .F U1 1 Let F := {(w, w  ) ∈ U × U1 : w  ∈ Fˆ (w)}

.

(11.7)

162

11 Extension of Proper Holomorphic Mappings

be the “graph” of .Fˆ . It is an analytic set in .U × U1 of pure dimension n with proper projection .π : F → U , and .Fˆ = π  ◦ π −1 . It follows by the definition of .Fˆ that all values .w 1 , . . . , w m ∈ Fˆ (w) have the same is well-defined for all .w ∈ U . Segre varieties. Therefore, .Qˆ F (w)

Lemma 11.9 Assume that for all .w ∈ U there exists .w  ∈ U1 such that .w k ∈ Qw for some value .w k ∈ Fˆ (w). Then .w l ∈ Qw , .l = 1, . . . , m. Therefore, in this case we can write .Fˆ (w) ⊂ Qw . Proof Note that .w k ∈ Qw if and only if .w  ∈ Qwk . But we also have .Qwk = Qwl . This   means that .w l ∈ Qw .

11.4

Extension as an Analytic Set Implies Holomorphic Extension

In Sect. 10.1 we defined what it means for a holomorphic map f to extend as a correspondence. An even weaker notion is that f extends as a (closed) complex analytic set .A ⊂ U1 × U1 , i.e., without imposing any conditions on the projections .π : A → U1 and .π  : A → U1 . In this section we prove the following intermediate result. Theorem 11.10 In the assumptions of Theorem 11.1, if f extends near .(0, 0 ) as an analytic set .A ⊂ U1 × U1 of pure dimension n, then f extends holomorphically across 0. Proof Without loss of generality we may assume that A is irreducible. We define S = {(z, z ) ∈ A : π :→ U is not locally biholomorphic near (z, z )}.

.

Let .U + , U be defined as in the previous section, and let F be given as in (11.7). Step 1.

We claim that (A \ S) ∩ (U × U1 ) ⊂ F.

.

To see this first note that by definition, the set F contains a component of .A ∩ (U × U1 ). Choose some .(p, p ) ∈ F ∩ (A \ S) and let .(q, q  ) ∈ A \ S be an arbitrary point. One can show that .dim S < n which implies that .dim π −1 (π(S)) < n, and consequently the set .A \ π −1 (π(S)) is path-connected. Therefore, the chosen points can be connected by a path γ : [0, 1] → (A \ π −1 π(S)) ∪ {(p, p ), (q, q  )}

.

11.4 Extension as an Analytic Set Implies Holomorphic Extension

163

with .γ (0) = (p, p ) and .γ (1) = (q, q  ). We show that .π(γ ) ⊂ U . Indeed, let [0, t0 ) ⊂ {t ∈ [0, 1] : π(t) ∈ U˜ }

.

be a connected component containing the origin. Then .γ (t) ∈ F for .0 ≤ t < t0 , and π(γ (t0 )) ∈ U1 ∩ bU . Since .M \  ⊂ π(S), it follows that .π(γ ) does not intersect + .M \ , and so the only possibility is that .π(γ (t0 )) ∈ bU ∩ U . But in this situation 1 +   .γ (t0 ) ∈ U 1 × bU , which is clearly a contradiction as .γ lies in .U × U . This shows that .t0 = 1, and therefore, .π(γ ) can be lifted to a path in F starting at .(p, p ). By the uniqueness theorem for analytic sets it follows that .(A \ S) ⊂ F . By construction of the set F it follows that for .(w, w  ) ∈ (A \ S) ∩ (U + × U1 ) the invariance property .

f (Qcw ) ⊂ Qw

.

(11.8)

holds. But further, since S is nowhere dense in A, it follows that (11.8) holds for all (w, w  ) ∈ A ∩ (U + × U1 ).

.

Step 2. We show that .π  : A → U1 is proper for a suitable choice of neighbourhoods  −1 (0 ) is discrete. We .U1 and .U . For this it suffices to show that the analytic set .σ = π 1 may regard .σ as an analytic set in .U1 , and if it is not discrete, then it has an irreducible component .σ1 of positive dimension. Since M does not contain any complex germs of positive dimension, it follows that no open subset of .σ1 is contained in M. Further, if −  .σ1 ⊂ U 1 × U1 , then by the Strong Disc Theorem (see [155]) we may conclude that −   .(0, 0 ) is in the envelope of holomorphy of .U 1 × U , which implies that f extends holomorphically across 0. To see the latter note that given an arbitrary .g ∈ O(U1− ), we may regard .g ∈ O(U1− × U1 ), i.e., it is independent of variable .z . Then g extends holomorphically to a neighbourhood of .(0, 0 ) and by the unique continuation, the extension of g is also independent of .z . Hence, g is holomorphic near 0. Assume now that .σ1 ∩ U1+ = ∅. Then by (11.8) we have f (Qcw ) ⊂ Q0

.

for all .w ∈ σ1 ∩ U1+ . Moreover, the union of the canonical components of .Qw for + contains an open set in .U − . Since .J ≡ 0 and the Segre map .λ : U → S .w ∈ σ1 ∩ U f 2 1 is finite, it follows that .σ1 ∩ U + is a finite set. Thus, .σ is discrete, which allows us to shrink neighbourhoods .U1 and .U1 so that the projection .π  : A → U1 is proper. Step 3. An argument similar to that of Sect. 10.2 shows that the projection .π : A → U1 is also locally proper near .(0, 0 ). In order to apply Theorem 10.1 it remains to show that .f (U1− ) ⊂ U1 − . Denote by . = π  ◦ π −1 the correspondence associated with the set A. Then from Proposition 10.4 we conclude that for all .(w, w  ) ∈ A ∩ (U1 × U1 ), the inclusion

164

11 Extension of Proper Holomorphic Mappings

(Qw ) ⊂ Qw

(11.9)

.

holds. We now claim that the standard coordinates can be chosen so that .(U1 ∩ M) ⊂ U1 ∩M  and .−1 (U2 ∩M  ) ⊂ U2 ∩M. To show the first inclusion, choose .w0 ∈ U1 ∩M and .w0 ∈ (w0 ). Then (11.9) implies .(Qw0 ) ⊂ Qw = Qf (w0 ) . However, .f (w0 ) ⊂ 0

M  , and therefore, .w0 ∈ U1 ∩ M  . For the second inclusion of the claim, observe that since the projection .π  : A → U1 is proper, the set .f (M ∩ U1 ) contains an open subset of .0 in .M  . By shrinking all the neighbourhoods we can ensure that for all .z ∈ U2 ∩M  there exists .z ∈ −1 (z ) such that .z ∈ U2 ∩M. For any other .z˜ ∈ −1 (z ) the invariance of Segre varieties for the inverse correspondence implies .−1 (Qz ) ⊂ Qz˜ . By the argument above there exists .z ∈ −1 (z ) ∩ M, and so .−1 (Qz ) ⊂ Qz . This shows that .Qz = Qz˜ , and so by Lemma 9.1(d) we conclude that .z ˜ ∈ M. The claim above now shows that .f (U1− )∩M  = ∅, and after possibly interchanging the roles of .U1 + and .U − we conclude that .f (U1− ) ⊂ U1 − , and we may apply Theorem 10.1 to conclude that f extends holomorphically to a neighbourhood of the origin.  

11.5

Pairs of Reflection

In what follows it will be convenient to use the following terminology. We say that a pair (w 0 , z0 ) ∈ U × (Qw0 ∩ U ) is a pair of reflection if there are open neighbourhoods .(w 0 ) of .w 0 and .(z0 ) of .z0 such that, for all .w ∈ (w 0 ), one has

.

Fˆ (Qw ∩ (z0 )) ⊂ Qˆ

.

F (w)

.

A typical example of a pair of reflection is the situation when .w 0 ∈ U + and .z0 ∈ Qcw0 . Another simple example is the pair .(w 0 , w 0 ) with .w 0 ∈ . However, a pair .(w 0 , z0 ) in general is not a pair of reflection if we just have .w 0 ∈ U + and .z0 ∈ Qw0 ∩ U , since .Qw 0 ∩ U may be disconnected. Lemma 11.11 If .(w 0 , z0 ) is a pair of reflection, then also .(z0 , w 0 ) is a pair of reflection. Proof We take .z ∈ (z0 ) and .w ∈ Qz ∩(w 0 ). Then we have .z ∈ Qw ∩(z0 ) and hence ˆ (z) ⊂ Q ˆ . From this it follows that .Fˆ (w) ⊂ Q ˆ and hence .Fˆ (Qz ∩ (w 0 )) ⊂ .F F (w) F (z)   Qˆ . This proves the lemma. F (z)

Recall that .C(F, p) denotes the cluster set of the map F at a point p.

11.5 Pairs of Reflection

165

Lemma 11.12 We have: (i) (ii) (iii) (iv)

C(Fˆ , w 0 ) ⊂ bU1 for any .w 0 ∈ bU ∩ U1+ ; ˆ , 0) ⊂ Q  ; .C(F 0 if .C(Fˆ , 0) = {0 }, then .0 ∈ ; F is a closed analytic subset of .[U1 \ (M \ )] × U1 . .

Proof (i) Let .(w ν , w  ν ) ∈ F with .(w ν , w  ν ) → (w 0 , w  0 ) ∈ (bU ∩ U1+ ) × U1 as .ν → ∞. For any .ν = 1, 2, . . . we have .f (Qcwν ) ⊂ Qw ν ; If .w  0 ∈ U1 , we may pass to the limit and obtain f (Qcw0 ) ⊂ Qw 0 .

.

But this means that .(w 0 , w  0 ) ∈ F and hence .w 0 ∈ U . This is a contradiction that proves .w  0 ∈ bU1 . (ii) Let .w ν ∈ U , .w ν → 0. It is enough to consider the following two cases: (a) .w ν ∈ U1− ∪ ( ∩ U1 ) for all .ν; (b) .w ν ∈ U + for all .ν. Since f is continuous up to M, in the first case .f (w ν ) → 0 and, for any .w  ν ∈ Fˆ (w ν ) we have .Qw ν = Qf (wν ) . Since we may assume that equality .Qw = Q0 holds only

for .w  = 0, this means that .w  0 → 0 . Therefore, it only remains to consider the case when all .w  ν ∈ U + . We have .f (Qcwν ) ⊂ Qcw ν for any .w  ν ∈ Fˆ (w ν ). Suppose that  ν → w  0 ∈ U  ; then .Q .w → Q  0 . Since .w ν → 0, also .dist(Qcwν , 0) → 0. Hence 1 w ν w

dist(Qw ν , 0 ) → 0, implying .0 ∈ Q  0 . We conclude that .w  0 ∈ Q0 . w (iii) If .C(Fˆ , 0) = {0}, then by (i) we have .0 ∈ U , and hence .0 ∈  or .

dist(0, bU ∩ U1+ ) > 0.

.

Thus, the remaining case is when .dist(0, bU ∩ U1+ ) > 0. We choose a small open neighbourhood .U˜ 1 of 0, compactly contained in .U1 , such that .U˜ 1+ ∩ bU = ∅. Then .U˜ 1+ ⊂ U ; hence .U˜ 1 \ (M \ ) is contained in U . We now replace .U1 by .U˜ 1 . The correspondence .F |U1+ is a component of the zero set of a system of pseudopolynomials with bounded holomorphic functions on .U1+ as coefficients. By Trépreau’s theorem (Theorem 1.3), these coefficients extend holomorphically to .U1 . The zero set of the extended system of pseudopolynomials contains a component

166

11 Extension of Proper Holomorphic Mappings

that is an extension of .F |U + to .U1 . But . is dense in M. Hence, this component 1

must coincide with .F + over .U1− , giving an extension of F over all of .U1 ; we still denote this extension by .Fˆ . The projection .π : F → U1 is again proper. Hence, by Theorem 10.1 it follows that .0 ∈ . (iv) This is a consequence of (i).  

11.6

Extension Along Segre Varieties

In this section we study the possible extension of f along Segre varieties, similar to the technique discussed in Sect. 9.3. To begin with, we observe the following. For any .w 0 ∈ U one can find a neighbourhood . =   × n of .w 0 , compactly contained in U , and a neighbourhood .V ⊂ U1 of .Qw0 ∩ U1 such that the intersection .Qz ∩  is not empty for .z ∈ V . Using our previous notation, for such a pair .(, V ) we define F˜ := F˜ (w 0 , , V ) := {(z, z ) ∈ V × U1 : Fˆ (Qz ∩ ) ⊂ z }.

.

(11.10)

Lemma 11.13 The set .F˜ is analytic in .V × U1 , and .dim F˜ is at most n. Proof The statement clearly holds if .Fˆ is a (single-valued) holomorphic map. Indeed, Eq. (11.10) can be written in the form   F˜ = {(z, z ) ∈ V × U1 : ρ Fˆ ( w, h( w, z)), z = 0, ∀  w ∈  },

.

(11.11)

where .wn = h( w, z) is the equation for .Qz . This is a family of (anti)holomorphic equations for .z, z . For the general case when .Fˆ :  → U1 is a holomorphic correspondence with the sheet number .m ≥ 2, there exists an analytic set .σ1 ⊂  of dimension at most .n − 1 such that for any .w 0 ∈  \ σ1 , there exists a neighbourhood .U (w 0 ) =  U (w 0 ) × Un (w 0 ) of .w 0 in which .Fˆ consists exactly of m separate holomorphic branches .F 1 , . . . , F m . If .(z, z ) ∈ F˜ and (Qz ∩ U (w 0 ))  w = ( w, wn ),

.

then the functions αj = αj ( w, z, z ) := ρ  (F j ( w, h( w, z)), z ), j = 1, . . . , m,

.

11.6 Extension Along Segre Varieties

167

satisfy the equations αj = 0, whenever (z, z ) ∈ F˜ and Qz ∩ U (w 0 )  w.

(11.12)

.

Conversely, if (11.12) holds for all .j = 1, . . . , m, . w ∈  U (w 0 ), then by analyticity ρ  (Fˆ ( w, h( w, z)), z ) = 0

.

for all . w ∈   and all branches of .Fˆ . Consider now the polynomial P ( w, z, t) = t m + a1 ( w, z)t m−1 + · · · + am ( w, z),

.

whose roots are precisely .αj , .j = 1, . . . , m. As symmetric polynomials in .αj , the coefficients .ak = ak ( w, z, z ), .k = 1, . . . , m, are the functions that are well-defined on   .(  \ σ1 ) × (V \ σ2 ) × U , where .σ2 = {z ∈ V : Qz ∩  ⊂ σ1 }. Note that these functions 1  are holomorphic in . w and antiholomorphic in .z, z . Furthermore, the set .σ2 is discrete. Thus, the coefficients .ak may be continued to .  × V × U1 is such a way that they remain to be holomorphic in . w and antiholomorphic in .z, z . The set .F˜ is now defined as {(z, z ) ∈ V × U1 : ak ( w, z, z ) = 0, ∀ k = 1, . . . , m, ∀ w  ∈  U }.

.

This shows that .F˜ is analytic.

 

We now assume additionally that .s w 0 ∈ U . Then, by Lemma 11.11, the sets F and ˜ coincide near the points of the form .(s w 0 , w  0 ) with .w  0 ∈ Fˆ (s w 0 ). We remove from .F ˜ those components that do not contain at least one of these points and denote the new .F analytic set again by .F˜ . Then .F˜ is of pure dimension n. By the uniqueness theorem we have the following lemma. Lemma 11.14 If .(w 0 , z0 ) is a pair of reflection, then .F˜ = F˜ (w 0 , , V ) contains F near every point .(z0 , z 0 ) ∈ F . Proof Let .(z, z ) ∈ F with .z ∈ (z0 ); take also .w ∈ Qz ∩ (w 0 ). Then .z ∈ Qw ∩ (z0 ) and .Fˆ (Qw ∩(z0 )) ⊂ Q . By Lemma 11.11 we have .Fˆ (Qz ∩(w 0 )) ⊂ Q = Q  . Hence, .(z, z )

∈ F˜ .

Fˆ (w)

As an immediate consequence we obtain the following.

Fˆ (z)

z

 

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11 Extension of Proper Holomorphic Mappings

Corollary 11.15 In the situation of Lemma 11.14, after removing low-dimensional components from .F˜ , the set .F˜ is of pure dimension n, and .F ∩ (V × U1 ) ⊂ F˜ . More precisely,  ˜ ∩ (V × U  ). .F ∩ (V × U ) is the union of suitable irreducible components of .F 1 1 Let .w 0 , .0 , V , .F˜ be as before and suppose that .(w 0 , z0 ) is a pair of reflection with ∈  = M ∩U . Denote by .S(w 0 , z0 ) the irreducible component of .F˜ ∩[(Qw0 ∩U1 )×U1 ] containing the germ of the graph of f at .(z0 , f (z0 )). Obviously, .S(w 0 , z0 ) is independent of the choice of . or V and is an analytic set of dimension .n − 1 in .(Qw0 ∩ U1 ) × U1 .

0 .z

Lemma 11.16 Let .(w 0 , z0 ) be a pair of reflection with .z0 ∈ . Then: (i) .S(w 0 , z0 ) ⊂ [(U1 ∩ Qw0 ) × (U1 × Qˆ

F (w0 )

)] ∩ F ;

(ii) .S(w 0 , z0 ) is an analytic set in .(U1 ∩ Qw0 ) × (U1 ∩ Qˆ

F (w0 )

) and

π(S(w 0 , z0 )) ∩ (M \ ) = ∅;

.

(iii) the projection .π : S(w 0 , z0 ) → π(S(w 0 , z0 )) ⊂ U ∩ Qw0 is proper. Note that we do not claim that the projection .π : S(w 0 , z0 ) → U1 ∩ Qw0 is proper. Proof Part (i) follows from the uniqueness theorem, Lemma 11.14 and the inclusion Fˆ (Qw0 ∩ (z0 )) ⊂ Qˆ

.

F (w0 )

.

Part (ii) follows from (i), Corollary 11.15 and Lemma 11.12 (iv). Part (iii) follows from   the properness of .π : F → U . The set .S(w 0 , z0 ) may be considered as the maximal analytic continuation of the germ of the graph of f at .(z0 , f (z0 )) along .Qw0 ∩ U1 . From Lemma 11.11 and the definition of ˜ and .S(w 0 , z0 ), the next lemma follows immediately. .F Lemma 11.17 For any .z ∈ π(S(w 0 , z0 )), the point .(w 0 , z) is a pair of reflection. We use the following notation. Let .(Aν ) be a sequence of (closed) subsets of a domain D in .Cn . Define its cluster set .C(Aν ) as C(Aν ) := {z ∈ D : ∃ zν ∈ Aν : z is a point of accumulation of (zν ) }.

.

Our next goal is to show the holomorphic extendability of .f : M → M  at certain points in M by studying the cluster sets of certain sequences .S(w ν , zν ).

11.6 Extension Along Segre Varieties

169

Proposition 11.18 Let .(w ν , zν ) ∈ U ×  be a sequence of pairs of reflection and choose  ν ∈ Fˆ (w ν ). Assume that .(w ν , zν ) → (0, 0) and .w  ν → w  0 ∈ U  . Suppose that the .w 1 cluster set .S := C(S(w ν , zν )) contains a point .(ζ 0 , ζ  0 ) ∈ U1 × U1 with .ζ0 ∈ U . Then .0 ∈ . Proof Let .(ζ ν , ζ  ν ) ∈ S(w ν , zν ) be chosen such that .ζ νk → ζ 0 and .ζ  νk → ζ  0 for a certain subsequence .(νk ). By Lemmas 11.11 and 11.17, .(ζ νk , w νk ) is a pair of reflection for any k. Let . ⊂ U and .V ⊂ U1 be connected open neighbourhoods of 0 and .Q .ζ ζ 0 ∩ U1 respectively, such that for all .w ∈ V , the intersection .Qw ∩  is connected and nonempty. Then .F˜ (ζ 0 , , V ) is an analytic set in .V × U1 . After shrinking ν .U1 we have .ζ k ∈  and .Qζ νk ∩ U1 ⊂ V for k big enough. Therefore, it follows by (11.10) that .F˜ (ζ 0 , , V ) = F˜ (ζ νk , , V ). By Lemma 11.14, the set .F˜ (ζ νk , , V ) contains the graph of f near .(zνk , f (zνk )) and hence .F˜ (ζ 0 , , V ) contains the point  ν ν .(0, 0 ) = limk (z k , f (z k )). This means that the graph of f extends as an analytic set   to a neighbourhood of .(0, 0 ). It follows now from Theorem 11.10 that .0 ∈ . Unfortunately, the situation of Proposition 11.18 cannot always be achieved, because the set .π(S(w ν , zν )) may not be analytic for lack of properness of the projection .π on ν ν .S(w , z ). Therefore, we begin with the construction of some new sequences of analytic sets. This will allow us finally to overcome this difficulty. Lemma 11.19 There exist sequences .(w ν , zν ) ∈ U ×  and .w  ν ∈ Fˆ (w ν ), and analytic sets .σν ⊂ U1 such that (1) (2) (3) (4) (5)

(w ν , zν ) is a pair of reflection for any .ν; ν ν .(w , z ) → (0, 0); ν  .w → w  0 ∈ U1 ; there is an integer .p ≥ 1 such that .σν are complex analytic sets of pure dimension p; ν ν ν .z ∈ σν ⊂ π(S(w , z )), for all .ν. .

Proof Choose an arbitrary sequence .zν ∈ , .zν → 0. Assume that there is a radius .r > 0 such that, for any .ν big enough, the set .π(S(zν , zν )) contains .Qzν ∩ B(zν , r). Then (after shrinking .U1 ) properties (1)–(5) of the lemma are satisfied for .w ν = zν , .w  ν = f (zν ), and .σν = Qzν ∩ U1 . Thus we may now assume that there is no such radius r. This means that, for any sufficiently small .r  > 0, there exists a sequence .(w ν , w  ν ) ∈ S(zν , zν ) such that ν  ν → w  0 with .|w  0 | = r  . The conditions .w ν = zν and .w  ν = f (zν ) of .w → 0 and .w course no longer hold; moreover, .w ν ∈ U + . By Lemma 11.12 (ii) we have .w  0 ∈ Q0 ∩ U1 . Since .r  > 0 is arbitrary and .λ is a finite map, we may assume that .Q  0 = Q0 . By w Lemmas 11.11 and 11.17, .(w ν , zν ) is a pair of reflection for every .ν. Put .Sν := S(w ν , zν ).

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11 Extension of Proper Holomorphic Mappings

It remains to show that .π(Sν ) contains and analytic set .σν ⊂ U1 , .zν ∈ σν , of some fixed pure dimension .p ≥ 1. Since .w  0 ∈ Q0 , we have .0 ∈ Qw 0 . Since .Q  0 = Q0 , there w exists a normal coordinate system in the target space such that .0 is an isolated point of     .Q 0 ∩ {z : z = . . . = zn = 0}. Hence, there exists an .ε > 0 such that, after shrinking 2 w  .U , the intersection 1  q  := Qw 0 ∩ {z ∈ U1 : z2 = . . . = zn−1 = 0, |zn | < ε} 0

.

has no limit points on .bU1 . Note that .q  0  0 is an analytic set of dimension 1 in .U1 ∩ {|zn | < ε}. Thus, for .ζ  ν := f (zν ) and .ν big enough, the sets q  := Qw ν ∩ {z ∈ U1 : zk = ζ  k , k = 2, . . . , n − 1, |zn | < ε}

.

ν

ν

contain .ζ  ν and are analytic sets of dimension 1 in .U1 ∩ {|zn | < ε} without limit points on    ν ν .bU . Since .Sν ⊂ (U1 ∩ Qw ν ) × (U ∩ Q ν ) and .Sν  (z , f (z )), the intersections 1 1 w sν := Sν ∩ {(z, z ) : zk = ζ  k for k = 2, 3, . . . , n − 1}

.

ν

are analytic sets of dimension at least 1 in .U1 × (U1 ∩ {|zn | < ε}). Since the sets .q  ν have no limit points on .bU1 , the sets .sν have no limit points on .U1 × (bU1 ∩ {|zn | < ε}). By Lemma 11.12, we have .C(Fˆ , 0) ⊂ Q0 = {zn = 0}. Thus, after shrinking .U1  0, the projections .π : sν → U1 are proper and the images .σν := π(sν ) are analytic sets of   dimension at least 1 in .U1 with .zν ∈ σν . The following lemma will play a crucial role in the forthcoming argument. Lemma 11.20 Let .w ν , .w  ν , .σν be sequences with all the properties stated in Lemma 11.19. Assume that .0 ∈ M \ . Then .C(σν ) ⊂ M ⊂ . Proof Suppose there is a point .ζ 0 ∈ C(σν ) ∩ (U1 \ (M \ )). By Lemma 11.19 there is a sequence .(w ν , zν ) ∈ U ×  with .(w ν , zν ) → (0, 0) and a sequence .(ζ ν , ζ  ν ) ∈ Sν = S(w ν , zν ) with .ζ ν ∈ σν such that .ζ ν → ζ 0 , .ζ  ν → ζ  0 ∈ U1 . Since .ζ ν ∈ U , we / M \ , Lemma 11.12 implies .ζ  0 ∈ bU1 , which is a have .ζ 0 ∈ U ∩ U1 . But since .ζ 0 ∈ contradiction.  

11.7

Final Steps in the Proof

The main role in the final part of our proof is played by the following Main Conjecture Let N be a real analytic CR manifold in a domain .W ⊂ Cn . Suppose that N does not contain any germs of complex varieties of positive dimension. Let .(Aν ) be

11.7 Final Steps in the Proof

171

a sequence of (closed) complex analytic sets of pure dimension .p ≥ 1 in W . Then .C(Aν ) cannot be contained in N . This important conjecture is open in general. It amounts to having a kind of more global uniform Łojasiewicz inequality. One of the difficulties that arises in the proof comes from the fact that .C(Aν ) need not contain any even 1-dimensional complex-analytic germs, as examples in [161] and [137] show. (If this were the case then Lemma 11.20, together with the fact that N is variety free, would give a contradiction.) We will prove this conjecture under some additional hypothesis, and this will conclude the proof of the main results of this chapter. Let U be an open set in .Cn , .a ∈ U , and let .B ⊂ U be a closed subset, .a ∈ B. Suppose that .Aν ⊂ U are closed complex analytic sets of pure dimension .p ≥ 1. We say that the sequence .(Aν ) clusters along B at a if .a ∈ C(Aν ) ⊂ B. Lemma 11.21 Suppose that there exists a continuous, plurisubharmonic in U , function .φ such that .φ(a) > φ(z) for all .z ∈ B \ {a}. Then the sequence .(Aν ) does not cluster along B at a. Proof Suppose that .(Aν ) were to cluster along B at a. We fix an open neighbourhood W of a compactly contained in U . Then .Aν ∩bX is not empty for all .ν and hence .C(Aν )∩bW is not empty. Therefore, also .B ∩ bW is not empty. One has .

sup {φ(z) : z ∈ B ∩ bW } < φ(a).

Hence, by continuity of .φ, there exists a sufficiently small open neighbourhood V of .B ∩ bW so that c := sup {φ(z) : z ∈ V } < φ(a).

.

Next we can choose an open neighbourhood .V1 of a such that .φ(z) > c on .V1 . Note, however, that for .ν sufficiently large, .Aν ∩ bW ⊂ V and .Aν ∩ V1 is not empty. But this is a contradiction to the Maximum principle applied to the plurisubharmonic function .φ|Aν .   In some important cases peak plurisubharmonic functions required by this lemma are known to exist. For example, this is the case when B is a strictly pseudoconvex hypersurface, totally real manifold, or if B is a pseudoconvex real analytic hypersurface of finite type. The latter follows by a result of Diederich and Fornæss [50]. Now we explain how the Main Conjecture implies holomorphic extendability of the map f . From Lemmas 11.20 and 11.21 with .Aν = σν and .B = M we obtain the following. Proposition 11.22 Let .a ∈ B = (M \ ). Then there is no open neighbourhood of a with a continuous plurisubharmonic function .φ on U such that .φ(a) > φ(z) for all .z ∈ B \ {a}.

172

11 Extension of Proper Holomorphic Mappings

Proof It follows by Lemma 11.20 that necessarily .a ∈ C(σν ) ⊂ B. According to Lem  ma 11.21, however, this is impossible. Proof of Theorem 11.1 Note that the set .E = M \  ⊂ M is compact. Assume that it is not empty. For a point .p ∈ E, let .R > 0 be the smallest number such that .B(p, R) contains all of E. Let .a ∈ E ∩ bB(p, R). Then there exists a sphere such that it touches from the outside the ball .B(p, R), and therefore, the set E, precisely at the point a. Without loss of generality we may assume that this sphere is centred at the origin. By construction the function .φ(z) = |z|2 is a plurisubharmonic peak function for .a ∈ E. But this contradicts   Proposition 11.22, which proves that .a ∈ E, and this gives the proof of the theorem.

11.8

Comments and Further Results

When the domains D and .D  in Theorem 11.1 are pseudoconvex, the condition of continuity of f up to the boundary can be dropped. This result was obtained independently by Baouendi and Rothschild [4] and Diederich and Fornæss [50]. In this case case Condition R holds and f extends smoothly up to the boundary. This considerably simplifies the Reflection principle. We also note that in the pseudoconvex case f extends continuously up to the boundary according to Diederich and Fornæss [47]; their proof is based on the estimates for the Kobayashi-Royden metric on bounded domains with real analytic pseudoconvex boundary. Hence, Theorem 11.1 can be applied directly which provides a purely geometric proof without the use of the Bergman projection techniques. A stronger version of Theorem 11.1 is the following result due to Diederich and Pinchuk [53]. Theorem 11.23 Let .U, U  be domains in .Cn , .n > 1, and let .M ⊂ U and .M  ⊂ U  be smooth real analytic closed real hypersurfaces that contain no nontrivial germs of complex curves. Then any continuous CR map .f : M → M  extends holomorphically to a neighbourhood of M. This result can be considered as a local version of Theorem 11.1. Its proof is similar to that of Theorem 11.1, but one needs to prove the Main Conjecture in Sect. 11.7 in a more general situation. This is done by using volume estimates of the varieties in the sequence with a consequent application of Bishop’s theorem, and a special stratification of the limiting CR submanifold due to Tumanov [146]. Theorem 11.23 is a generalization of many previously known results. For example, Pinchuk-Tsyganov’s theorem (Theorem 5.7) assumes that M and .M  are strictly pseudoconvex, and the results of Diederich and Webster [55] and Baouendi et al. [6] assume that the map f is a .C ∞ -smooth CR diffeomorphism; the latter also holds for

11.8 Comments and Further Results

173

CR manifolds of higher codimension. A simple proof of this in the hypersurface case was also obtained by Bedford and Pinchuk [18]. Finally, the special case of Theorem 11.23 when .n = 2 was studied by Diederich and Pinchuk [52] and Huang [82]. The result in [52] establishes analytic continuation of proper holomorphic maps without any a priori boundary regularity, the proof of this is presented in the next chapter. There are some partial results for the case when .dim M  > dim M, see Forstneriˇc [65] and Pinchuk and Sukhov [113].

Extension in C2 .

12

In this chapter we give the proof of holomorphic extension past the boundary without the assumption of continuous extension to the boundary. Theorem 12.1 Let .D, D  be bounded domains in .C2 with smooth real analytic boundaries and .f : D → D  be a biholomorphic map. Then f extends holomorphically to an open neighbourhood of .D. This is a special case of a general result due to Diederich and Pinchuk [52] that holds for proper holomorphic maps. To simplify the argument we only consider the case of a biholomorphic f . We emphasize that in the above theorem we do not assume that the map f extends continuously to the boundary of D, as otherwise the result would immediately follow from Theorem 11.1. Lack of initial regularity of the map f on the boundary makes this problem particularly difficult. It fact, it remains open for dimensions .n ≥ 3. And this is another illustration of the importance of boundary regularity of holomorphic maps as discussed in Chap. 2.

12.1

Possible Extension as a Correspondence

In view of Theorem 10.1 it suffices to show that the map f extends as a proper holomorphic correspondence. This correspondence will be constructed using Segre varieties. For a given .z0 ∈ bD in a standard pair of neighbourhoods .U1  U2 and a suitably chosen neighbourhood .U  of .bD  define V = {(w, w  ) ∈ (U1 \ D) × (U  \ D  ) : f (Qw ∩ D) ⊃ s w Qw }.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3_12

(12.1)

175

12 Extension in .C2

176

We call V the Segre correspondence for f on .U1 \ D. Further, let E = {z ∈ D ∩ U2 : z ∈ f −1 (s w  ) ∩ Qw for some (w, w  ) ∈ V }.

.

(12.2)

These definitions make sense for .U  that can be covered by a finite collection of standard coordinate neighbourhoods. By the invariance of Segre varieties, these definitions are independent of the choice of these neighbourhoods. Lemma 12.2 If E is relatively compact in .U2 , then 

(a) V is a complex analytic subset of .(U1 \ D) × (U  \ D ). (b) There are no limit points of V on .(U1 \ D) × bU  . (c) If .  ⊂ bD  is the set of points where the inverse map .f −1 extends as a proper holomorphic correspondence, then V does not have any limit points on .(U1 \ D) ×   . Proof (a) Let .(w 0 , w 0 ) ∈ V . In view of Lemma 10.6, for a sufficiently small neighbourhood   4 0 0 .W ⊂ C of .(w , w ), a suitable standard pair of neighbourhoods .U 1  U2 with  0  .w ∈ U1 and any .(w, w ) ∈ W , the condition f (Qw ∩ D) ⊃ s w Qw

.

is equivalent to f (Qw ∩ D) ⊃ w˜  Qw ,

.

(12.3)

where .w˜  is a unique point on .Qw with .w˜ 1 = s w10 . It is easy to see that the condition (12.3) is equivalent to a system of holomorphic equations. This shows that  V is locally complex-analytic. To show that V is closed in .(U1 \ D) × (U  \ D ), let  ν ν 0 0  .(w , w ) ∈ V be a sequence converging to a point .(w , w ) ∈ (U1 \ D) × (U \ D ). Then, since .w ν → w 0 , we also have .s w ν → s w 0 ∈ D  , and therefore, by definition of V there exist points .ζ ν ∈ f −1 (s w ν ) ∩ Qwν such that .ζ ν → ζ 0 ∈ f −1 (s w 0 ) ⊂ D. By the continuity of the family of Segre varieties and the compactness of E, we conclude that .ζ 0 ∈ Qw0 ∩ U2 , which implies that f (Qw0 ∩ D) ⊃ s w0 Qw0 .

.

This proves (a). (b) We set .δ = dist(bU  , bD  ) > 0. From (9.1) and elementary considerations we have .dist(w, bD) ∼ = dist(s w, bD) as .dist(w, bD) → 0. From this it follows that there exists

12.1 Possible Extension as a Correspondence

177



δ1 > 0 such that for any .w  ∈ U  \ D with .dist(s w  , bD  ) < δ1 , the inequality

.

dist(w , bD  ) < δ/2

.

holds. Since .f : D → D  is biholomorphic, there exists .δ2 > 0 such that f maps the set .D δ2 = {z ∈ D : dist(z, bD) < δ2 } to the set .D δ1 = {z ∈ D  : dist(z , bD  ) < δ1 }. Therefore, we may assume that the standard pair of neighbourhoods is chosen such that .dist(z, bD) < δ2 for an .z ∈ D ∩ U2 . From this (b) follows. (c) Assume that there exists a sequence .(w ν , w ν ) ∈ V converging to a point .(w 0 , w 0 ) ∈ (U1 \ D) ×   , and consider a sequence .ζ ν ∈ Qwν such that .f (ζ ν ) = s w ν . Since ν → w 0 ∈ bD  , we also have .s w ν → s w 0 = w 0 and .dist(ζ ν , bD) → 0. We may .w suppose, after passing to a subsequence if necessary, that the sequence .ζ ν converges to a point .ζ 0 ∈ bD. By compactness of the set E, .ζ 0 ∈ U2 , and in view of .w 0 ∈   we have .ζ 0 ∈ . We immediately conclude that .f (ζ 0 ) = w 0 . Therefore, by Theorem 10.1 there are neighbourhoods .U˜ of .ζ0 and .U˜  of .w 0 such that f extends holomorphically to .D ∪ U˜ and .f |U˜ : U˜ → U˜  is proper. Set .g˜ := (f |U˜ )−1 . Then we can choose .zν ∈ g(w ˜ ν ) such that .zν → ζ 0 . We have f (Qzν ∩ U˜ ) ⊂ Qwν ∩ U2 .

.

Therefore, Q z ν = Qw ν

.

and so by continuity, Q w 0 = Qζ 0 .

.

But since .ζ 0 ∈ bD, by Lemma 9.1(d), the set .Aζ 0 is contained in bD. This is a contradiction, since .w 0 ∈ / bD.  The decisive hypothesis in Lemma 12.2 is compactness of the set E. The following lemma gives a simple geometric criterion for this condition. Lemma 12.3 Let .z0 ∈ bD be a point such that .Qz0 ∩ D = {z0 }. Then, for a suitable standard pair of neighbourhoods .U1  U2 of .z0 the set E given by (12.2) is relatively compact in .U2 . In particular, this is the case when .z0 is a strictly pseudoconvex boundary point of D. Proof It follows from (12.2) that

12 Extension in .C2

178

E⊂



.

Qw ∩ D.

(12.4)

w∈U1 \D

We choose a neighbourhood .U  U2 of .z0 . Since .Qz0 ∩ D = {z0 }, for a suitable constant .c > 0 we have dist(Qz0 ∩ (U2 \ U ), D) ≥ 2c.

.

Therefore, if .U1  U is sufficiently small, then also dist(Qw ∩ (U2 \ U ), D) ≥ c,

.

for all .w ∈ U1 such that  .

Qw ∩ D  U  U2 .

w∈U1 \D

In view of (12.4), the lemma now follows.



Unfortunately, the hypothesis of Lemma 12.3 is not satisfied for all .z ∈ bD. But even when V is already known to be complex analytic we still need to show that the set .

Vˆ = V ∩ ((U1 \ D) × U  ) ⊂ ((U1 \ D) × U  )

is complex analytic and the projection .Vˆ → U1 \ D is proper.

12.2

The Structure of the Boundary

In this section we stratify the boundary of the domain D and identify a certain exceptional subset of .bD, which is the cause of the greatest difficulties for the extension. Our goal is to show that this set is pluripolar. For an arbitrary .z0 ∈ bD we choose a neighbourhood U of .z0 and a real analytic vector field X on .bD ∩ U with values in .H (bD) that does not vanish on .bD ∩ U . Let r be a real analytic defining function for D on U . Then we write LbD (z) = Lr (z; X)

.

for the Levi form of .bD. Since D is bounded, there are always strictly pseudoconvex points on .bD, which we denote by .bDs+ . Let .bDs− denote the set of points in .bD, where .LbD has only negative

12.2 The Structure of the Boundary

179

eigenvalues. Define ◦

bD + = bDs+

.

the pseudoconvex region of .bD (the interior is taken in the relative topology on bD). Similarly, let ◦

bD − = Ds−

.

 the envelope of be the pseudoconcave part of .bD. It is well-known that .bD − ⊂ D, holomorphy of D. In some sense, the most interesting part of bD is the remaining part of the boundary: M = bD \ (bD + ∪ bD − ),

.

(12.5)

which we call the border of bD. We note that in .bD, the border M separates the  may pseudoconvex and pseudoconcave regions. Also, the envelope of holomorphy .D −  contain other parts of .bD besides .bD . In fact, most of M is contained in .D. We define T = {z ∈ bD : LbD (z) = 0}.

.

(12.6)

Then T is a compact real analytic subset of .bD of dimension at most 2, and .M ⊂ T . Furthermore, it is easy to see that M is always a closed semianalytic set. We also set .T + = T ∩bD + and call it the set of weakly pseudoconvex points of .bD, and .T − = T ∩bD − —the set of weakly pseudoconcave points of bD. The following holds. Lemma 12.4 The set T admits the following stratification into semianalytic sets T = T2 ∪ T1 ∪ T0 ,

.

where .Tk is a locally finite union of smooth real analytic submanifolds of dimension k, .k = 0, 1, 2. The set .T1 ∪ T0 is real analytic of dimension at most 1. The proof is immediate from real analyticity of .bD and elementary properties of semianalytic sets. We now further refine this stratification to suit our purposes. First observe that T h = {z ∈ T2 : dim Hz (T2 ) = 1}

.

(12.7)

12 Extension in .C2

180

is semianalytic of dimension at most 1 in .T2 , since otherwise .T2 would contain a piece of a complex analytic curve, which contradicts Theorem 9.2. Note that the set .T2 \ T h is maximally totally real. Denote by .oL (z) the order of vanishing of the Levi form .LbD at a point .z ∈ bD. Then the set T d = {z ∈ T : lim inf oL (ζ ) < oL (z)}

.

T ζ →z

(12.8)

is real analytic of dimension at most 1. Lemma 12.5 The set .T1 ∪ T0 ∪ T h ∪ T d can be stratified in the form .Tˆ1 ∪ Tˆ0 , where .Tˆ1 is a locally finite family of smooth curves and .Tˆ0 is a locally finite union of points. In particular, the set .Tˆ1 ∪ Tˆ0 is pluripolar. The set .

Tˆ2 = T2 \ (Tˆ1 ∪ Tˆ0 )

is a locally finite family of maximally totally real submanifolds. To simplify the notation, from this point we write .Tk instead of .Tˆk , .k = 0, 1, 2, and set = Tk ∩ bD + . The following holds.

+ .T k

 Proposition 12.6 The set .Me = M ∩ (T1 ∪ T0 ) is semianalytic and .M \ Me ⊂ D. Proof Our proof follows [46]. Semianalyticity of .Me follows from elementary considerations. To prove the second assertion note that the problem is local, and so we may restrict our attention to a connected component of .Me (which we still denote by .Me ). We choose local coordinates near a point .p ∈ Me such that .p = 0 and near 0 we have bD = {y2 +

∞ 

.

j

aj (x1 , x2 )y1 = 0}, Me = {y1 = y2 = 0}.

(12.9)

j =3

Since bD does not contain any nontrivial germs of complex analytic curves, for each .q ∈ Me there exists .j ≥ 3 such that .aj (·, x2 (q)) = 0, since otherwise, the variety .{y2 = 0, x2 = x2 (q)} would be locally contained in bD. For .q ∈ Me define j (q) = min{j : aj (·, x2 (q)) ≡ 0} < ∞; j0 = min{j (q) : q ∈ Me }.

.

These definitions make sense due to the following Lemma 12.7 The integers .j (q) and .j0 are independent of the choice of holomorphic coordinates, and therefore, .j0 is constant on each component of .Me . Further, each .j (q) is odd.

12.2 The Structure of the Boundary

181

Proof For each .q ∈ Me , the number .j  (q) = min{j : aj (x1 (q), x2 (q) = 0} is equal to the highest possible order of contact between bD and 1-dimensional complex manifolds passing through q whenever .j  (q) = j (q). This is a biholomorphic invariant. Further, if .j (q) = 2k, we can choose the origin in .Me such that .j (0) = 2k and .a2k (0, 0) = 0. The curve ⎛ ⎞ ∞  .λ(t) = ⎝it, −i aj (0)t j ⎠ j =2k

is contained in .bD and is transverse to .Me at the origin. Therefore, there are values of t arbitrarily close to 0 such that .λ(t) ∈ bDs+ and .λ(t) ∈ bDs− . On the other hand, a direct computation shows that the Levi form of bD at .λ(t) is LbD (λ(t)) =

.

2k(k − 1) a2k (0)t 2k−2 + O(t 2k−1 ). 16

This, however, does not change the sign.



To continue with the proof of the proposition, let .j (p) = 2l + 1 and rewrite (12.9) in the form ∞ 

ρ = y2 +

j

bj (x1 )y1 + x 2 y13 B(x1 , x2 , y1 ), b2l+1 (0) > 0,

.

(12.10)

j =2l+1

with .D = {ρ < ∞} near the origin. For .ε > 0 define a rectangular .Rε in the .z1 -plane by Rε = {x1 + iy1 : |x1 | ≤ ε, −ε2 ≤ y1 ≤ ε3(l+1) }.

.

Furthermore, let Xε = {(z1 , z2 ) ∈ C2 : z2 = −ε6l+5 z1 (1 + iz1 )}.

.

 if the following holds: Note that .0 ∈ Xε ∩ bD. By Kontinuitätssatz, the origin will be in .D for any .ε > 0 sufficiently small we have {(z1 , z2 ) ∈ Xε : z1 ∈ bRε } ⊂ D.

.

We have to show that for all small .ε > 0 and .z1 ∈ bRε , σε (z1 ) := ρ(z1 , −ε6l+5 z1 (1 + iz1 )) < 0.

.

We have

12 Extension in .C2

182 ∞ 

bj (x1 )y1 .

j

(12.11)

+ 2ε6l+5 x1 y1 ).

(12.12)

σε (z1 ) = −ε6l+5 y1 + ε6l+5 (y12 − x12 ) +

.

2l+1

+(−ε

6l+5

x1 + 2ε

6l+5

x1 y1 )y13 B(x1 , y1 , −ε6l+5 x1

There exists a constant .K > 0 such that on the line segment .{y1 = ε3(l+1) , |x1 | ≤ ε} ⊂ bRε we have 1 σε (x1 + iy1 ) ≤ − ε9l+8 (1 + εK) < 0, 2

.

provided that .ε > 0 is small enough. On the two line segments .{x1 = ±ε, 0 ≤ y1 ≤ ε3l+3 } ⊂ bRε we obtain σε (z1 ) ≤ ε6l+7 (−1 + O(ε)) < 0.

.

Next we consider .{x1 = ±ε, −ε3 ≤ y1 ≤ 0} ⊂ bRε . Since now .y12l+1 ≤ 0, we have the following estimate for small .ε > 0, σε (z1 ) ≤ ε6l+7 (−1 + ε + ε4 + Kε8 ) < 0.

.

Finally, for .{x1 = ±ε, y1 = −ε3 } ⊂ bRε we get

1 σε (z1 ) = ε6l+3 − b2l+1 (0) + ε5 + ε8 + Kε12 < 0. 2

.

This completes the proof of the proposition.

12.3



Extension Across a Dense Subset of the Boundary

As a starting point for extending the given map .f : D → D  we need to know that it extends holomorphically across an open dense subset of .bD. Proposition 12.8 There exists a dense open subset . ⊂ bD such that f extends holomorphically to a neighbourhood of .D ∪ . The set . can be chosen in such a way that this extension is locally biholomorphic at every point .a ∈ .  the map f extends across .bDs− , and the extension is locally Proof Since .bDs− ⊂ D, biholomorphic almost everywhere in .bDs− . Therefore, we only need to consider .bDs+ . Suppose that there exists a point .a ∈ bDs+ with a relatively open neighbourhood .S ⊂ bDs+ such that f does not extend holomorphically to any point .z ∈ S. We claim that for all .z ∈ S the cluster set .CD (f ; z) ⊂ Me , where .Me is the exceptional set as in Proposition 12.6.

12.4 Extension Across Strictly Pseudoconvex Points

183

 , then .z ∈ D  according To prove the claim assume that .z ∈ CD (f, z). Then if .z ∈ D  + to Lemma 10.3. If, on the other hand, .CD (f ; z) contains a point .z ∈ bD , i.e., in the pseudoconvex region of .bD  , then by Theorem 3.10, f extends continuously to .bD ∪ V for a certain neighbourhood V of z. The image of .S ∪ V under the extension necessarily lies in .bD + and the result follows from Theorem 11.1. This proves the claim. We now use the fact that .Me is pluripolar. There exists a function .φ = −∞ plurisubharmonic on .C2 and such that .φ|Me ≡ −∞. The function .ψ = φ ◦ f is plurisubharmonic on D and .D  z → z0 ∈ S, then .ψ → −∞. However, since .S ⊂ bD is relatively open, it follows that .ψ ≡ −∞, which implies that .ψ ≡ −∞ on D–a contradiction.  Now that we know that the given map .f : D → D  extends biholomorphically across an open dense subset of bD, we can make the following change in the definition of the variety V in (12.1): Suppose that in the context of Eq. (12.1) we choose a point .a ∈ bD∩U1 of holomorphic extendability of f , and furthermore, assume that .E  U2 . Then we denote by V the irreducible component in .(U1 \ D) × (U  \ D  ) of the variety given by (12.1), which contains the graph of the extended map f near a. Then it follows from Lemma 12.2 that V is a closed complex analytic variety. We will see later that V does not depend on a.

12.4

Extension Across Strictly Pseudoconvex Points

In this section we prove the following. Lemma 12.9 Let .z0 ∈ bDs+ . Then .z0 is a point of local biholomorphic extendability of f . Proof In view of Theorem 10.1 we only need to show that f extends as a proper holomorphic correspondence. We choose neighbourhoods .U1  U2 of .z0 and .U  of .bD  as in (12.1). Then the Segre correspondence V is an irreducible closed complex analytic subvariety of .(U1 \ D) × (U  \ D  ). Our goal is to show that V is contained in a complex analytic subset .Vˆ ⊂ (U1 \ D) × U  so that the projection .π : Vˆ → U1 \ D is proper. First we claim that V does not have any limit points on (U1 \ D) × (bU  ∪ (bD  \ Me )).

.

To see this note that by Lemma 12.2 the set V does not have limit points on .(U1 \D)×bU  and on  ). (U1 \ D) × (bD  ∩ D

.

(12.13)

12 Extension in .C2

184

So suppose that V has a limit point .(w 0 , w 0 ) ∈ (U1 \ D) × bD + . Then there is a sequence k k k k 0 0 k k .(w , w ) ∈ V such that .(w , w ) → (w , w ). Choose .z ∈ Qw k such that .f (z ) = s w k . Since .s w k → w 0 ∈ bD + and from the definition of V , we may assume (by passing to a subsequence if necessary) that the sequence .(zk ) converges to a point .z˜ ∈ bD ∩ U2 . However, by Lemma 12.3, .z˜ cannot be contained in .bU2 , and so .z˜ ∈ bDs+ ∩ U2 . Therefore, we have shown that the set .CD (f, z˜ ) contains the point .w 0 ∈ bD + . By Theorem 3.10, the map f extends as a Hölder continuous map to .D ∩ U for some neighbourhood U of .z˜ , and therefore, f extends holomorphically by Theorem 11.1. By continuity of the family of Segre varieties it follows that .z˜ Qw0 = z˜ Qz˜ , but this contradicts Lemma 9.1. This proves the claim. Next we eliminate the possibility that V has limit points on .(U1 \ D) × Me . We may assume that V is closed in .((U1 \ D) × U  ) \ E  , where E  = (U1 \ D) × Me ⊂ (U1 \ D) × U  .

.

According to Lemma 12.5 and Proposition 12.6, the set .E  is pluripolar. Therefore, there exists a plurisubharmonic function .φ = −∞ on .(U1 \ D) × U  such that E  ⊂ {(z, z ) ∈ (U1 \ D) × U  : φ(z, z ) = −∞}.

.

Let E˜  = {(z, z ) ∈ (U1 \ D) × U  : φ(z, z ) = −∞},

.

and .

V˜ = V ∩ ((U1 \ D) × U  ) \ E˜  .

Then .V˜ is a closed complex analytic subset of .((U1 \ D) × U  ) \ E˜  . Further, .V˜ = ∅ and .dim V˜ = 2. Nevertheless, there are points .z ∈ U1 ∩ bD such that the map f extends biholomorphically to a neighbourhood .U˜ of z. Therefore, on .(U˜ \ D) × U  , the Segre correspondence V is just the graph of a holomorphic mapping. Indeed, let .(w, w  ) ∈ V with .w ∈ U˜ \ D. After shrinking .U˜ , if necessary, we may assume that .Qw ∩ D and s  .f (Qw ∩ D) are small and close to z and .f (z) respectively. Therefore, . w ∈ f (Qw ∩ D) +  is also close to .f (z). Since .f (z) ∈ bDs , the Segre map .λ is locally one-to-one near .f (z) and we conclude that .w  is uniquely defined, i.e., .w  = f (w). The above argument shows that we are in the situation of Bishop’s lemma (see, e.g., [37]): Let E be a closed complete pluripolar subset of a bounded domain .U = U  × U  ⊂ Cp × Ck , and A be a complex analytic subset of of .U \ E of pure dimension p with no limit points on .U  × bU  . Suppose that .U  contains a nonempty subdomain .V  such that the set .A ∩ (V  × U  ) is complex analytic. Then the set .A ∩ U is complex analytic in

12.5 Proof of Theorem 12.1

185

U . This implies that .Vˆ := V˜ is a closed complex analytic subset of .(U1 \ D) × U  of pure dimension 2 without limit points on .(U1 \ D) × bU  . In particular, the projection ˆ → U1 \ D is proper. We also have .π : V V ∩ (U1 \ D) × (U  \ D  )) = V .

.

The map .π : Vˆ → U1 \ D is proper holomorphic, hence, is a finite branched covering. Let m be the sheet number of .π. From this we obtain the extension of f as a proper holomorphic correspondence in a neighbourhood of .z0 . Indeed, it follows that there exist two pseudopolynomials of the form Pk (z, zk ) = zkl + ak1 (z)zkl−1 + · · · + akl (z) = 0, l ≤ m,

.

(12.14)

where .k = 1, 2, and the holomorphic coefficients .akj (z) ∈ O(U1 \ D) are such that .

Vˆ ⊂ F˜1 := {(z, z ) ∈ (U1 \ D) × U  : Pk (z, zk ) = 0}.

(12.15)

If U is a sufficiently small neighbourhood of .z0 , then all coefficients .akj extend as holomorphic functions to U so that .

Vˆ ⊂ F˜ := {(z, z ) ∈ U × U  : Pk (z, zk ) = 0, k = 1, 2},

(12.16)

where .F˜ is a closed complex analytic subset of .U ×U  containing the graph of .f |D∩U . The irreducible component F of .F˜ containing this graph gives the required proper holomorphic correspondence extending f to U . 

12.5

Proof of Theorem 12.1

It remains the extend the map f to the set .T + ∩ Me . We will do that in a step-by-step approach by constructing local extensions as proper holomorphic correspondences and applying Theorem 10.1. We begin by establishing the following. Lemma 12.10 For any point .z0 ∈ T2+ there exists a standard pair of neighbourhoods .U1  U2 such that (Qz0 ∩ bD) \ {z0 } ⊂ bDs+ .

.

Proof After an appropriate change of coordinates we may assume that .z0 = 0 and that the 2-dimensional totally real submanifold .T2+ is just the real plane .iR2 near .z0 . Then a local real analytic defining function r of D can be chosen near zero to be of the form

12 Extension in .C2

186

r(z) = 2x2 + (2x1 )2m a(z1 , y2 ),

.

(12.17)

where .m > 1 and the real analytic function .a(z1 , y2 ) is positive near 0. The complexification of r has the form r(z, w) = z2 + w2 + (z1 + w1 )2m a(z, ˜ w),

.

(12.18)

with .a(0, ˜ 0) > 0. Therefore, .Q0 is locally given by the equation z2 + (z1 )2m a(z, ˜ 0) = 0.

.

(12.19)

Its restriction to .T2+ is iy2 + (−1)m (y1 )2m a(iy, ˜ 0) = 0.

.

Since .Re[a(iy, ˜ 0)] > 0, the Segre variety .Q0 intersects .T2+ only at the origin.



In the next lemma we consider another situation in which the set V , as defined by (12.1) can be used for the construction of a proper holomorphic correspondence extending f . Lemma 12.11 Let .z0 ∈ bD, .U  be a suitable neighbourhood of .bD  , .U1  U2 be a suitable standard pair of neighbourhoods of .z0 and .V ⊂ (U1 \ D) × (U  \ D  ) be given as in (12.1). If .

  (Qz0 ∩ bD) \ {z0 } ⊂ ,

where . is the set of points of holomorphic extendability of f , then V is a closed complex analytic subset of .(U1 \ D) × (U  \ D  ) without limit points on .(U1 \ D) × (bU  \ D  ). Proof Shrinking .U2 if necessary, we may assume that .Qz0 ∩ bD ∩ bU2 ⊂  and that there exists a radius .τ1 > 0 such that for all .w ∈ Qz0 ∩ bU2 ∩ D, the map f extends holomorphically to the ball of radius .t1 centred at w. Let .R  (z0 ) be the set of all points   .w ∈ U \ D  for which f (Qz0 ∩ D) ⊃ s w Qw .

.

Then .R  (z0 ) is discrete in .U  \ D  because the Segre map .λ is finite. Therefore, the set R(z0 ) := {w ∈ Qz0 ∩ D ∩ U2 : f (w) = s w  for some w  ∈ R  (z0 )}

.

is also discrete in D. Shrinking .U2 and .U1 again, we may assume that

(12.20)

12.5 Proof of Theorem 12.1

187

R(z0 ) ∩ bU2 = ∅,

.

and, further, there exists a radius .τ2 > 0 such that for all .w ∈ U1 \D, the set .R(w), defined as in (12.20), does not intersect the neighbourhood of size .τ2 /2 of the set .Qw ∩ D ∩ bU2 (we assume that such a neighbourhood of the empty set is empty). Hence, the hypothesis .E  U2 of Lemma 12.2 is satisfied, and hence, V is a closed complex analytic subset of   .(U1 \ D) × (U \ D  ) without limit points on .(U1 \ D) × (bU \ D  ).  Once we prove that the projection .π : V → (U1 \ D) is proper, the same argument as in the previous section shows that f extends as a correspondence, and therefore as a holomorphic map. To achieve this goal we first prove the following. Lemma 12.12 One has .T2+ ⊂ . Proof Let .z0 ∈ T2+ and make all choices as in Lemma 12.10. Then we have (Qz0 ∩ bD) \ {z0 } ⊂ bDs+ .

.

By Lemma 12.9, we obtain (Qz0 ∩ bD) \ {z0 } ⊂ .

.

(12.21)

Note that in this case .CD (f, z0 ) ∩   = ∅, because by Theorem 11.1 we have .z0 ∈ . In view of Theorem 3.10, this implies that we are done if .CD (f, z0 ) ∩ bD  = ∅. Since − ⊂   , we only need to consider the case when .bD CD (f, z0 ) ⊂ (T1 ∪ T0 ) \ bD − .

.

But according to Lemma 12.5 the set .T1 ∪ T2 is pluripolar, which allows us to use Bishop’s lemma provided that we prove the following claim: For a suitable choice of .U1 there exists a nonempty open set .U˜ ⊂ U1 \D such that .V ∩(U˜ ×(U  \D  )) is a closed complex analytic subset of .U˜ × U  . For the proof of the claim we have to make a careful selection of .U1  U2 . Since .Qz0 is a 1 dimensional complex manifold and .λ is an open map, for any sufficiently small .δ > 0 and .U0 := {z : |z − z0 | < δ}, the set W :=



.

Qw

w∈U0 + is an arbitrarily small neighbourhood of .Qz0 . Denote for a moment by .T2δ the .δ/2+ + neighbourhood of .T2 . We then choose .U1 such that .U1 ∩ T2δ ⊂ U0 , but there also exists a

12 Extension in .C2

188

nonempty open set .U˜  U1 \ (W ∪ D). It then necessarily follows that for .z ∈ U˜ , the set .Qz ∩ U0 is empty. Hence, for all .z ˜ ∈ U we have Qz ∩ bD ⊂ bDs+ ⊂ .

.

This implies that f is well-defined on .Qz ∩ bD and f (Qz ∩ bD) ⊂ bDs+ ⊂   .

.

We now conclude the proof as follows. The set .V ∩ [U˜ × (U  \ D  )] has no limit points on ˜ ×bD  , as otherwise for a point .(z0 , z0 ) ∈ V ∩(U˜ ×bD  ) we would obtain, analogously to .U Lemma 12.2(c), that .z0 ∈ f (Qz0 )∩bD  ⊂   , and this is a contradiction to Lemma 12.11. This proves the claim. Now Bishop’s lemma implies that .V is an analytic set in .(U1 \ D) × U  . Since it has no limit points on .(U1 \ D) × bU  , the map f can be extended as a proper holomorphic correspondence to a neighbourhood of .z0 exactly as in the second half of the proof of Lemma 12.9. This completes the proof of the lemma.  To complete the proof of Theorem 12.1 we still need to consider the points in .(T \ T2 ) \ bD − . For this we establish the following. Lemma 12.13 Let .z0 ∈ (T \ T2 ) \ bD − , and suppose that (Qz0 ∩ bD) \ {z0 } ⊂ .

.

(12.22)

Then .z0 ∈ . Proof Since we already know that .bD  \ (T1 ∪ T0 ) ⊂   , we conclude from Lemma 12.11 that the set V , as considered here, does not have any limit points on   (U1 \ D) × bD  \ (T1 ∪ T0 ) ,

.

nor on .(U1 \ D) × bU  . The same argument as before allows us the apply Bishop’s lemma to prove that .V is an analytic set in .(U1 \ D) × U  . And this implies that .z0 ∈ .  The remaining case is when .z0 ∈ (T \ T2 ) \ bD − , and (Qz0 ∩ (T1 ∪ T0 )) \ {z0 } = ∅.

.

(12.23)

12.6 Boundary Regularity: Some Historic Remarks

189

If in this situation the set .Qz0 ∩ (T1 ∪ T0 ) is discrete, we can simply shrink the neighbourhoods .U1 and .U2 and we are in the situation of Lemma 12.13. Therefore, it suffices to show the following. Lemma 12.14 Let .z0 ∈ (T \ T2 ) \ bD − , and suppose that .Qz0 ∩ (T1 ∪ T0 ) contains a real analytic set of dimension 1 near .z0 . Then .z0 ∈ . Proof If .U1 is chosen suitably small, then any of the finitely many smooth curves contained in the set .(T1 ∪ T0 ) ∩ U1 has a complexification .Tˆ . Then the following holds: Whenever for a point .z ∈ U1 × (T \ T2 ) the set .z Qz ∩ (T2 ∪ T0 ) is the germ of a smooth real curve, then there is a component .Tˆ , as defined above, such that .z Tˆ = z Qz (i.e., the germs at z agree). But since the sets .Az are discrete by Corollary 9.3, the hypothesis in the above statement can only be satisfied for a finite number of points .z ∈ U1 ∩ (T \ T2 ). Therefore, we have shown that f is holomorphically extendable across all points of .U1 \ bD with a possible exception of a finite number of points. After shrinking .U1 and .U2 again, we may therefore assume that .(Qz0 ∩ bD) \ {z0 } ⊂ . Hence, Lemma 12.13 applies, and 0 .z ∈ .  This sequence of lemmas now completely proves our main result of this chapter.

12.6

Boundary Regularity: Some Historic Remarks

The problem of boundary regularity of holomorphic mappings between smoothly bounded domains in .Cn has a long history. It prompted the development of many sophisticated and powerful techniques in several complex variables, and this in itself can (and should!) be a topic of a separate monograph. We refer the reader to several textbooks and survey papers for a broader overview of the subject and connections to other areas of complex analysis, see [9, 11, 67, 116, 138]. In a short discourse below we mention some important milestones of the subject relevant for the case of domains with smooth real analytic boundaries. To a certain extend this summarizes the core results of the book. Among the first works on boundary regularity in higher dimensions are the papers by Margulis [96], Henkin [76] and Vormoor [156], where it was proved that a biholomorphic map .f : D → D  between bounded domains in .Cn , .n > 1, with smooth strictly pseudoconvex boundaries extends as a Hölder continuous map to .D. The Carathéodory metric is introduced, which lays ground for further research. For proper holomorphic maps the result was proved by Pinchuk [105]. The first account of holomorphic extension is the Reflection Principle of Lewy and Pinchuk (Theorem 2.1); the boundaries are assumed to be strictly pseudoconvex and the map is assumed to be .C 1 -smooth up to the boundary. Combining this with Fefferman’s theorem [60] (in which the boundaries are assumed to be .C ∞ -smooth) gives holomorphic

190

12 Extension in .C2

extension of biholomorphic maps between strictly pseudoconvex domains with real analytic boundary without any a priori regularity on the boundary. For proper holomorphic mappings this result can be proved using the technique of Chap. 5, see Corollary 5.6. The local version of this is the theorem of Pinchuk and Tsyganov [114] (Theorem 5.7). The next step was to prove the result for domains with pseudoconvex real analytic boundaries. Presence of weakly pseudoconvex points poses difficulty already for continuous extension. While partial results were obtained by pushing the technique of using Carathéodory metric, the break-through came in the paper by Diederich and Fornæss [47] with the use of Kobayashi-Royden metric. The required estimates for the Kobayashi-Royden metric are obtained using a subtle construction of peak plurisubharmonic functions. As a result, Diederich and Fornæss [47] proved that if .f : D → D  is a proper holomorphic map between bounded pseudoconvex domains, .bD ∈ C 2 , and .bD  is smooth real analytic, then the map f extends to .D as an .ε-Hölder continuous map for some .ε > 0. As already discussed in Sect. 5.4, Bell and Catlin [21] and Diederich and Fornæss [48] proved the following result about smooth extension: a proper holomorphic map .f : D → D  between smoothly bounded pseudoconvex domains in .Cn extends smoothly to .D if D satisfies Condition R. Note that this result does not require any a priori regularity of f on the boundary. Since it was already proved by Diederich and Fornæss [45] that bounded pseudoconvex domains with real analytic boundary satisfy Condition R, it follows that holomorphic maps between such domains admit smooth extension to the boundary. For maps between domains with real analytic boundaries further generalization is possible by showing holomorphic extension past the boundary. First results in this direction were obtained by Diederich and Webster [55] and Baouendi et al. [6] under the assumption that the extension of f to the boundary is a .C ∞ -smooth CR diffeomorphism. For a proper holomorphic map .f : D → D  between bounded pseudoconvex domains in n .C with real analytic boundaries, the smooth extension of f to bD is a smooth CR map, but not necessarily a diffeomorphism, and so a different argument was needed to conclude holomorphic extension. For dimension .n = 2 this was done by Baouendi et al. [7]. The result for a general n was obtained in two papers: Diederich and Fornæss [50] proved this by using the Reflection principle and properties of Segre varieties to construct an ndimensional complex analytic set that extends the graph of the map f , an idea that was later successfully used by many others, and that lies at the heart of the geometric constructions in the last chapters of this book. At the same time Baouendi and Rothschild [4] proved the result using a more algebraic version of the Reflection Principle. The next improvement to the result would be the removal of pseudoconvexity from the assumptions. Theorem 12.1, which is due to Diederich and Pinchuk [52], completely settles the case .n = 2 (the case when f is a priori known to extend to bD as a homeomorphism was proved by Diederich et al. [56]). Analogous result for proper holomorphic correspondences was proved later by Verma [150], and a local version of Theorem 12.1 was given by Shafikov and Verma [132]. Finally, Theorem 11.1 and its local version, Theorem 11.23, both proved by Diederich and Pinchuk [53], give the holomorphic

12.6 Boundary Regularity: Some Historic Remarks

191

extension in all dimensions once it is known that the map f extends continuously up to the boundary. A somewhat weaker assumption is that the graph of the map f locally extends as an n-dimensional complex analytic set. This also gives holomorphic extension as proved by Diederich and Pinchuk [54]. The remaining obstacle is to prove continuous extension of the map to the boundary of D for .n > 2. This problem remains open to this day.

A

Appendix

In this chapter we give some more technical proofs of the results discussed in Chap. 1.

A.1

Edge-of-the-Wedge Theorem

We present the proof of a version of the edge-of-the-wedge theorem which we have used several times. Our approach is based on the gluing holomorphic discs construction already used in Sect. 4.3. We need the following approximation theorem, which is due to Baouendi and Trèves [5]. Theorem A.1 Let M be a smooth generic manifold in .Cn and .E ⊂ M be a smooth totally real manifold of dimension n. Then in a neighbourhood of any point .p ∈ E, any CR function f of class .C s , .s ≥ 0, on M can be approximated in the .C s norm on M by the sequence of holomorphic functions ⎛ (1E f ) ∗ exp ⎝−k

n 

.

⎞ zj2 ⎠ , k = 1, 2, . . . ,

j =1

where .1E denotes the characteristic function of E and the asterisk .∗ denotes the convolution operator. Let E be a generic submanifold in a domain . ⊂ Cn defined by .{ρ = (ρ1 , . . . , ρd ) = 0}. Recall that the wedge .W (, E) in . with the edge E is the domain W (, E) = {z ∈  : ρj (z) < 0, j = 1, . . . , d}.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3

193

194

A Appendix

For .δ > 0 we also consider a .δ-“truncated” wedge Wδ (, E) =

.

⎧ ⎨ ⎩

z ∈  : ρj (z) − δ



ρk < 0, j = 1, . . . , d

k=j

⎫ ⎬ ⎭

⊂ W (, E).

We describe the construction of attaching analytic discs adapted to our case. Without loss of generality, we may assume that in a neighbourhood . of the origin a smooth totally real manifold E is defined by the equation .x = r(x, y), where a smooth vector function .r = (r1 , . . . , rn ) satisfies the conditions .rj (0) = 0, .∇rj (0) = 0. Fix a positive noninteger s and consider for a real function .u ∈ C s (∂D) the Hilbert transform .H : u → H (u). It is uniquely defined by the conditions that the function .u + iH (u) is the trace of a function holomorphic on .D and the integral average of .H (u) over the circle is equal to 0. This is a classical singular linear integral operator; it is bounded on the space .C s (∂D). Let + = {eiθ : θ ∈ [0, π ]} and .S − = {eiθ : θ ∈ (π, 2π )} be the semicircles. Fix a .C ∞ .S smooth real functions .ψj on .∂D such that .ψj |S + = 0 and .ψj |S − < 0, .j = 1, . . . , n. Set .ψ = (ψ1 , . . . , ψn ). Consider the generalized Bishop equation u(ζ ) = r(u(ζ ), H (u)(ζ ) + c) + tψ(ζ ), ζ ∈ ∂D,

.

(A.1)

where .c ∈ Rn and .t = (t1 , . . . , tn ), .tj ≥ 0, are real parameters. It follows by the Implicit function theorem that this equation admits a unique solution .u(c, t) ∈ C s (∂D) depending smoothly on the parameters .(c, t). Consider now the analytic discs f (c, t)(ζ ) = PD (u(c, t)(ζ ) + iH (u(c, t))(ζ )),

.

where .PD denotes the Poisson operator of harmonic extension to .D. The map .(c, t) → f (c, t)(0) (the centres of discs) is of class .C s . Every disc is attached to E along the upper semicircle. It is easy to see that this family of discs fills the wedge .Wδ (, E) when .δ > 0 and a neighbourhood . of the origin are chosen small enough. Indeed, this is immediate when the function r vanishes identically (i.e., .E = iRn ), while the general case follows by a small perturbation argument. We now can easily prove the edge-of-the-wedge theorem following Airapetyan [1] and Tumanov [144]. Consider the generic manifolds .Ej = {z ∈  : ρk (z) = 0, k = j, k = 1, . . . , n} of dimension .n+1. On the unit circle we consider the open arcs .Sj , .j = 1, . . . , n, bounded by the points {e

.

2πj n

i

, j = 0, . . . , n − 1}.

Let .ψj be .C ∞ -smooth functions on .∂D such that .ψj |Sj < 0 and .ψj |∂D\Sj = 0, s .j = 1, . . . , n. The Eq. (A.1) admits a solution in .C (∂D) smoothly depending on the parameters .(c, t) in a neighbourhood of the origin in .R2n (note that .tj are not assumed

A Appendix

195

to be positive here). Every analytic disc from the family .f (c, t)(ζ ) obtained as above has the boundary attached to the union .∪j Ej . Furthermore, their centres .f (c, t)(0) fill a neighbourhood of the origin in .Cn . Indeed the map .(c, t) → f (c, t)(0) has the maximal rank 2n in a neighbourhood of the origin (this is obvious when .r = 0 and hence remains true under small perturbations). In combination with the approximation result (Theorem A.1) we obtain Proposition A.2 Let f be a continuous CR function on .∪j Ej . Then f extends holomorphically to a neighbourhood of E in .Cn . Indeed, by the Maximum principle (applied along every analytic disc) the approximating family of holomorphic functions converges in a neighbourhood of the origin. As a corollary we obtain the edge-of-the-wedge theorem. Define the domains ± = {z ∈  : ±ρj > 0, j = 1, . . . , n}.

.

Then we have the following Corollary A.3 Let .f + and .f − be functions holomorphic on the wedges .+ and .− respectively and continuous up to the edge E. If .f + and .f − coincide on E, then they extend to a holomorphic function in a neighbourhood of E.

A.2

PSH Defining Functions

In this section we give the proof of Theorem 1.10, which we formulate below in a more precise form. Theorem A.4 Let . be a bounded pseudoconvex domain with .C 2 -smooth boundary. Then there exist a defining function .r(z) of ., constants .K > 0 and .0 < η0 < 1 such that for .0 < η ≤ η0 the function ρ(z) = −(−r(z) e−K|z| )η 2

.

(A.2)

is a bounded strictly plurisubharmonic exhaustion function on .. Proof Our proof follows Range [118]. Near .b the function .r(z) can be chosen to be the “signed” distance function: δ (z) =

.

⎧ ⎨−dist(z, b),

if z ∈ ,

⎩dist(z, b),

if z ∈ / .

(A.3)

196

A Appendix

It is well-known that in a sufficiently small neighbourhood of .b the function .δ is .C 2 smooth. We then may smoothly cutoff .δ at some small negative level to obtain a global smooth function on .. For any .z ∈ b we have a decomposition .Tz Cn = Hz b ⊕ Tz , where .Hz b is the holomorphic tangent and .Tz is the orthogonal complement. This decomposition can be naturally extended to a tubular neighbourhood of .b, say, .U = b × (−ε, ε) for some small .ε > 0. So for .z ∈ U and .t ∈ Tz Cn , we have .t = t  ⊕ t  ∈ Hz ⊕ Tz . From pseudoconvexity of . and the fact that the level sets of .δ are locally translations, it follows that for .z ∈ U , Lδ (z, t  ) ≥ 0 for t ∈ Tz Cn .

.

For .t = t  ⊕ t  we have Lδ (z, t) = Lδ (z, t  ) + 2 Re Lδ (z, t  , t  ) + Lδ (z, t  ) = Lδ (z, t  ) + O(|t||t  |),

.

where the first term on the right is the 2-form corresponding to the Levi-form of .δ . Since  .∂δ (z) defines the direction .Tz , we have |t  | = O(|∂δ (z), t|).

.

Combining the three formulas above we conclude that there exists a constant .C > 0 such that Lδ (z, t) ≥ −C |t| · |∂δ (z), t| for z ∈  ∩ U, w ∈ Tz Cn .

.

(A.4)

Now let .ρ be given as in (A.2) for some .0 < η < 1, .K > 0. Using the properties of the Levi form one can show that for .t ∈ Cn , Lρ (z, t) = η(−δ )η−2 e−ηK|z| · D(z, t), 2

.

(A.5)

where

 D(z, t) = K(δ )2 |t|2 − ηK|z, t|2   + (−δ ) Lδ (z, t) − 2ηK Re∂δ , tz, t + (1 − η)|∂δ , t|2 .

.

(A.6)

The form .Lρ (z, t) in (A.5) will be positive-definite for all .z ∈ U ∩  and .t ∈ Tz Cn if .D(z, t) > 0 there. Using estimate (A.4) for the second term in .D(z, t) we see that there exists .η0 > 0 and .M = M(η0 ) > 0 such that for .0 < η < η0 , we have for .z ∈ U ∩ ,

A Appendix

197

1 K (δ )2 |t|2 − M(−δ )|t||∂δ , t| + |∂δ , t|2 . 2 2

D(z, t) ≥

.

Clearly, constants K and .η can now be chosen so that .ρ is strictly plurisubharmonic in U ∩ .  

.

A.3

Hörmander-Wermer Lemma

In this section we prove Lemma 1.8 of the Introduction. For convenience we state it below. Lemma A.5 Let .M ⊂ Cn be a maximally totally real manifold of class .C k , .k ≥ 1, .k ∈ R, with .0 ∈ M and .T0 M = Rn . For any function .f ∈ C k (M) there exists an extension .f˜ in a neighbourhood of 0, which is of class .C k on M and infinitely differentiable outside M and which has the property that the coefficients of .∂ f˜ vanish on M to order .k − 1 in a neighbourhood of the origin. Proof Our proof follows Chirka [36]. Without loss of generality we may assume that in a neighbourhood U of the origin, M is given by the equations yν = φν (x), φν ∈ C k (U ), ν = 1, . . . , n.

.

For a fixed .a  ∈ Rn , let .a = a  + iφ(a  ) ∈ M, .φ = (φ1 , . . . , φn ). Then for .z = x + iy ∈ M the following identity in the vector form holds x − a  = (z − a) − i(φ(x) − φ(a  )) = (z − a) − iDx φ(a  )(x − a  ) + r0 (x, a  ).

.

From this it follows that on M  −1 x − a  = In + iDx φ(a  ) ((z − a) + r0 (x, a  )) = (a  )(z − a) + r1 (x, a  ),

.

(A.7)

where .In is the identity matrix, the components of .(a  ) are functions of class .C k−1 in      .a , and .r1 (x, a ) = (a )r0 (x, a ) = o(|x − a |). In a similar fashion we can express any  linear function in .x − a . A polynomial .Q(x − a  ) of degree 2 can be represented in the form .Q1 + Q02 , where 0  .Q is a degree 2 homogeneous polynomial in .x − a . The linear term .Q1 we may represent 2 ˜ 1 (z − a) + Q1 (r1 ), and if .k ≥ 2, then by the Taylor representation, on M as .Q1 = Q r1 (x, a  ) = (a  )

.

 1 j Dx φ(a  ) (x − a  )j + r2 (x, a  ), j!

|j |=2

198

A Appendix

where .r2 up to the factor .(a  ) is the remainder of the vector-function .φ at the point .a  (of order higher than 2). Combining the order two terms in .Q1 (r1 ) and .Q02 , we obtain that on M, ˜ 1 (z − a) + P 0 (x − a  ) + Q1 (r2 ), Q=Q 2

.

where .P20 is a homogeneous polynomial of degree two with coefficients of class .C k−2 in   .a . After substituting .x − a with its representation (A.7) we obtain ˜ 0 (z − a) + Q0 (x − a  , r2 ), P20 = Q 2 2

.

˜ 0 , .Q0 are homogeneous quadratic polynomials. Note that every monomial of .Q0 where .Q 2 2 2 contains terms of .r2 of positive degree and that the coefficients of .Q02 are functions in .a  of class .C k−1 . This can be continued by induction: from the representation of monomials in .x − a  of lower degree as functions of .z − a we obtain polynomials in .x − a  and the Taylor remainders of .φ at .a  . Their initial representations we add to monomials of the next order, and represent these as functions of .z − a again. We may assume that f is a function of x. We represent f using Taylor’s expansion centred at a point .a  : f (x) =

.

 1 j  1 j j Dx (a  )(x − a  )j + (Dx f (ξ ) − Dx f (a  ))(x − a  )j , j! j!

(A.8)

|j |=l

|j |≤l

where .ξ is some point on the interval .[a  , x] and .l = [k]. We perform the substitution discussed above in the first sum in (A.8). As a result we obtain on M an expression of the form .Pa (z) + Qa (x, ra ), where .Pa is a polynomial in z, and .Qa is a homogeneous polynomial of degree l in x, and .ra is the Taylor remainder of order .> l with coefficients that are functions of .a  and are Hölder continuous of order .α = k − [k] (or simply continuous if .α = 0). Further, every monomial in .Qa contains some terms of .ra of positive degree. From the Taylor expansion for .φ we see that raν (x, a  ) =

.

 1 j j (Dx φν (ξ ) − Dx φν (a  ))(x − a  )j . j!

|j |=l

Denote by .rf (z, a) the Taylor remainder of f (the second sum in (A.8)). Then on M we have Pa (z) − Pb (z) = Qa (z, ra ) − Qb (z, rb ) + rf (z, a) − rf (z, b).

.

The right-hand side, defined on .Cn (with .a, b ∈ M), can be represented in the form

A Appendix

199

λ(z, a, b) × (|z − a|l + |z − b|l ),

.

where .λ(z, a, b) → 0 as .(a − b) → 0, uniformly in .z ∈ Cn , and if, in addition, .α > 0, then .λ(z, a, b)| ≤ C|a − b|α with some constant .C > 0. This follows from the Taylor remainders of f and .φ, since .f, φ ∈ C k . Thus, the polynomials .Pa (z) satisfy Whitney’s conditions and therefore, there exists a .C k -smooth function .f˜ in a neighbourhood of the origin, and of class .C ∞ away from M such that its Taylor polynomials centred at any .a ∈ M close to 0 equal .Pa (z). In particular, .f˜|M = f . Since .Pa (z) is a polynomial in z, then .∂Pa (z) ≡ 0, and so .∂ f˜ = ∂(f˜ − Pa ) = o(|z − a|k−1 ), if k is an integer, and k−1 , if k is a noninteger. Since .a ∈ M was arbitrary, this implies that .∂ f˜ .|∂ f˜| ≤ C|z − a| vanishes on M to order .k − 1.  

A.4

Separate Algebraicity

The material in this section is adapted from Bochner and Martin [26]. Recall that a holomorphic function .f (z), .z ∈ Cn , .n ≥ 1, is called rational if there exist polynomials .P (z) and .Q(z) such that .P (z)f (z) + Q(z) ≡ 0. We say that a holomorphic function .f (z) is algebraic of degree .s > 0, if there exist polynomials .P0 (z), . . . , .Ps (z), such  that . sk=0 Pk (z)(f (z))k ≡ 0. In particular, a rational function is an algebraic function of degree .s = 1. If every component of a holomorphic map from a domain in .Cn into m n+m of .C is algebraic, then the graph of the map is contained in an algebraic variety in .C dimension n. To formulate our main result we first establish the following. Lemma A.6 Let .D ⊂ Cz and .D  ⊂ Cw be domains, .z = (z1 , . . . , zp ), .w = (w1 , . . . , wq ), and .p, q ∈ N. Let .F1 (z, w), . . . , FN (z, w) be holomorphic functions in  .D × D , not all .≡ 0. If every .Fj (z, w) is a rational function in w for every fixed .z ∈ D, and if they satisfy a relation p

q

c1 (w)F1 (z, w) + · · · + cN (w)FN (z, w) ≡ 0,

.

(A.9)

for some functions .cj (w) satisfying |c1 (w)|2 + · · · + |cN (w)|2 > 0,

.

(A.10)

then there exist polynomials .C1 (w), . . . , CN (w), not all .≡ 0, such that C1 (w)F1 (z, w) + · · · + Cn (w)FN (z, w) ≡ 0.

.

(A.11)

Proof We rewrite (A.9) for some N values in D, call them .z(1) , . . . , z(N ) . This gives a system of N linear homogeneous equations in .cj (w), .j = 1, . . . , N. In view of condition (A.10), its determinant

200

A Appendix

   F (z(1) , w) . . . F (z(1) , w)  N   1   .  ... ... ...   F1 (z(N ) , w) . . . FN (z(N ) , w)

(A.12)

vanishes identically for .w ∈ D  and .z(j ) ∈ D, .j = 1, . . . , N . Replacing .z(N ) with the variable z, and developing the determinant in terms of the last row, we obtain the following relation N  .

 Cj z(1) , . . . , z(N −1) , w Fj (z, w) ≡ 0,

(A.13)

j =1

  where the functions .Cj z(1) , . . . , z(N −1) , w up to the sign are the .(N − 1)-dimensional determinants of the matrix ⎡ ⎤ F1 (z(1) , w) . . . FN (z(1) , w) ⎢ ⎥ .⎣ ... ... ... ⎦. (N −1) (N −1) , w) . . . FN (z , w) F1 (z   First assume that not all .Cj z(1) , . . . , z(N −1) , w vanish identically in all variables. Then there exist values .z(j ) = aj , .j = 1, . . . , N − 1, such that the functions Cj (w) = Cj (a1 , . . . , aN −1 , w)

.

(A.14)

do not vanish identically in w. For this choice of .z(j ) , Eq. (A.13) reduce to (A.11), because the functions .Cj (w) being rational combinations of the functions .Fj (ak , w) are rational functions in w by the hypothesis of the lemma.   Secondly, if all .Cj z(1) , . . . , z(N −1) , w vanish identically, there exists a minor    F (z(α1 ) , w) . . . F (z(α1 ) , w)  αm   α1   .  ... ... ...   Fα1 (z(αm ) , w) . . . Fαm (z(αm ) , w) of the determinant in (A.12) which vanishes identically but has at least one subdeterminant of order .(m − 1) which does not vanish identically. By the previous argument there exists a relation m  .

Cαk (w)Fαk (z, w) ≡ 0,

k=1

and by inserting the vanishing coefficients .Cj (w) we obtain (A.9).

 

A Appendix

201

Theorem A.7 Let .D ⊂ Cz and .D  ⊂ Cw be domains. If a function .f (z, w), holomorphic on .D × D  , is a rational function in z for each w and a rational function in w for each z, then .f (z, w) is rational in .(z, w). p

q

Proof Let .p( z), p2 (z), . . . be a sequence of all monomials in z arranged in any order. Then for each .w ∈ D  we have ⎞ ⎛ m n   .⎝ aj (w)pj (z)⎠ f (z, w) + bk (w)pk (z) ≡ 0, (A.15) j =1

k=1

with m  .

|aj (w)|2 +

j =1

n 

|bk (w)|2 > 0.

(A.16)

|bk (w)|2 = 1.

(A.17)

k=1

Assume the normalization m  .

|aj (w)|2 +

j =1

n  k=1

Let .{wν }, .ν = 1, 2, . . . , be any sequence of points in .D  for which the relation (A.15) is available for some fixed indices .(m, n), and let the sequence be convergent to a point  .w0 ∈ D . In view of (A.17), by passing to a subsequence we obtain .

lim aj (wν ) = aj ,

ν→∞

lim bk (wν ) = bk ,

ν→∞

for some .aj and .bk . By continuity of .f (z, w) the relation (A.15) with those limit values  is the set of those points in .D  for which a is also valid for .w0 . This means that if .D(m,n)  is closed in .D  . Since relation (A.15) holds for the given .(m, n), then .D(m,n) .

  ∪∞ m,n=1 D(m,n) = D ,

 and .D  is open, some .D(m,n) must have an nonempty interior, which we denote by .D0 . Replacing .D  with .D0 we have the relation (A.15) to hold for .m, n that are independent of w. Putting .N = m+n, .Fj (z, w) = pj (z)f (z, w), .j = 1, . . . , m and .Fm+k (z, w) = pk (z), .k = 1, . . . , n, we conclude from Lemma A.6 that there exist polynomials .aj (w), .bk (w) such that (A.15) and (A.15) hold, and so .f (z, w) is rational in .(z, w).  

202

A Appendix

Theorem A.8 Let .f ∈ O(D × D  ), where .D × D  ⊂ Cz × Cw . If a function .f (z, w) is an algebraic in z for each w and algebraic in w for each z, then .f (z, w) is algebraic in .(z, w). p

q

Proof The argument is similar to that of Theorem A.7. For every .w ∈ D  we have a relation s m   .

aj k (w)pj (z)(f (z, w))k ≡ 0,

(A.18)

j =1 k=0

and for some nonempty subdomain .D0 ⊂ D  this relation holds for a fixed .(m, s). Assume that for each .a ∈ D, the function .f (w) = f (a, w) is algebraic in w, i.e., satisfies C0 (w)f l + C1 (w)f l−1 + · · · + Cl (w) ≡ 0,

.

(A.19)

where the coefficients .Ci are polynomials in w, .i = 1, . . . , l. We apply Lemma A.6 to the functions Fi (z, w) = Hj,k (z, w) = pj (z)(f (z, w))k .

.

(A.20)

It follows from the lemma, or rather from its proof, that there exists a fixed pair of indices .(m, s) such that relation (A.18) holds for .aj k (w), each of which is a polynomial in terms that arise from (A.20) by specifying the value of z. In other words, each .aj k (w) is a polynomial of terms that are algebraic functions of w, and therefore, .aj k are themselves algebraic functions in w. Thus we obtained (A.18) with coefficients .pj (z) being polynomials, and .aj k (w) being algebraic functions. But from this we may obtain a   relation (A.18) in which both .pj (z) and .aj k (w) are polynomials in their variables.

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Index

A Alexander’s theorem, 50 Algebraic function, 199 Algebraic hypersurface, 128 Algebroid function, 48 Analytic continuation along CR-curves, 94 Anisotropic dilation, 22 Automorphism group, 3

B Baouendi-Trèves approximation theorem, 193 Bedford-Pinchuk theorem, 59 Border of the boundary of a domain, 179 Boundary regularity of holomorphic discs, 41 Bounded exhaustion function, 3 Bounded PSH exhaustion functions, 13 Burns-Shnider example, 86

C Canonical component of a Segre variety, 157 Carathéodory (pseudo) distance, 30 Cartan’s theorem, 3 Cauchy transform, 42 Cayley’s transform, 2 Chain, 81 Cluster set, 3 Complete hyperbolic domain, 27 Complex Hessian, 4 Complexification, 76 Condition R, 56 Continuous extension of correspondences, 49

Continuous extension to the boundary, 31, 33 CR-manifold, 6 CR-curve, 93 CR-dimension, 6 CR function, 7 Current of integration, 7

D Defining function, 6 Diederich-Pinchuk theorem, 133, 155, 175 Domain, 2

E Edge, 7, 8, 10 Edge-of-the-wedge theorem, 8, 194 Essentially finite hypersurface, 121 Exhaustion function, 3 Extension along Segre varieties, 123, 166 Extension as holomorphic correspondence, 134

F Fefferman’s metric, 83, 95

G Generalized Bishop equation, 45 Generic manifold, 6 Gluing discs argument, 44

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Pinchuk et al., Geometry of Holomorphic Mappings, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-37149-3

211

212 H Hörmander-Wermer lemma, 11, 197 Hardy-Littlewood lemma, 32 Hilbert transform, 45 Holomorphic tangent space, 6 Holomorphic tangent vector field, 66 Hopf lemma, 14 Hyperbolic domain, 27 Hyperquadric, 132

Index

I Indeterminacy locus, 20 Infinitesimal Lie symmetry, 78 Invariance of Segre varieties, 76, 77

Parabolic subgroup, 65 Peak function, 5 Pinchuk extension theorem, 87, 95 Pinchuk-Tsyganov theorem, 52 Pluriharmonic functions, 4 Pluripolar set, 5 Plurisubharmonic function, 4 Poincaré distance, 25 Poincaré-Alexander theorem, 18 Polar set, 4 Projectivization, 12 Proper holomorphic correspondence, 47, 133 Proper map, 3 Pseudoconvex domain, 12 Psh-barrier property, 5

K Khurumov’s example, 46 Kobayashi (pseudo) distance, 26 Kobayashi-Royden (pseudo) metric, 26 Kobayashi-Royden metric estimates, 28

R Rado’s theorem, 4 Rational function, 199 Reflection principle, 10, 17, 88 Removable singularity, 5

L Levi form, 11 Lewy extension theorem, 7 Lie symmetry, 78 Local equivalence problem, 75

S Scaling along a sequence, 38 Scaling method, 23, 38 Schwarz reflection principle, 17 Segre map, 118, 121 Segre variety, 19, 76, 117 Separate algebraicity, 130, 202 Sibony metric, 30 Smooth extension to the boundary, 37 Space of Segre varieties, 121 Spherical hypersurface, 87 Square distance function, 10 Standard complex structure, 6 Standard coordinate system, 118 Standard pair of neighbourhoods, 118 Strictly plurisubharmonic function, 4 Strictly pseudoconvex domain, 12 Subharmonic function, 3 Sussmann’s theorem, 93 Symmetric point, 137, 157 Symmetry group, 78

M Maximally totally real manifold, 8 Maximum principle, 4 Mean value inequality, 3 Meromorphic map, 87 Moser normal form, 81

N Nonspherical hypersurface, 94 Nonvanishing of Jacobian, 20 Normal form of a defining function, 12

P Pair of reflection, 164 Parabolic group, 65 Parabolic point, 65

T Tangential Cauchy-Riemann equations, 7

Index Totally real manifold, 8 Trépreau’s theorem, 7

U Uniformization of domains, 110, 112

213 W Weakly pseudoconvex domain, 13 Webster’s theorem, 128 Wedge, 7, 8, 10 Weighted homogeneous polynomial, 67 Wong-Rosay theorem, 22