A Course on Holomorphic Discs 303136063X, 9783031360633

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Birkhäuser Advanced Texts Basler Lehrbücher

Hansjörg Geiges Kai Zehmisch

A Course on Holomorphic Discs

Birkhäuser Advanced Texts Basler Lehrbücher A series of Advanced Textbooks in Mathematics

Series Editors Steven G. Krantz, Washington University, St. Louis, USA Shrawan Kumar, University of North Carolina at Chapel Hill, Chapel Hill, USA

This series presents, at an advanced level, introductions to some of the fields of current interest in mathematics. Starting with basic concepts, fundamental results and techniques are covered, and important applications and new developments discussed. The textbooks are suitable as an introduction for students and non–specialists, and they can also be used as background material for advanced courses and seminars.

The Virtual Series on Symplectic Geometry Series Editors Alberto Abbondandolo Helmut Hofer Tara Suzanne Holm Dusa McDuff Claude Viterbo Associate Editors Dan Cristofaro-Gardiner Umberto Hryniewicz Emmy Murphy Yaron Ostrover Silvia Sabatini Sobhan Seyfaddini Jake Solomon Tony Yue Yu

Hansjörg Geiges • Kai Zehmisch

A Course on Holomorphic Discs

Hansjörg Geiges Mathematisches Institut Universität zu Köln Köln, Germany

Kai Zehmisch Fakultät für Mathematik Ruhr-Universität Bochum Bochum, Germany

ISSN 2296-4894 (electronic) ISSN 1019-6242 Birkhäuser Advanced Texts Basler Lehrbücher ISBN 978-3-031-36063-3 ISBN 978-3-031-36064-0 (eBook) https://doi.org/10.1007/978-3-031-36064-0 Mathematics Subject Classification (2020): 32Q65, 53D35, 58D15, 58J05, 57R17, 46T10 © 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

Contents Preface I

Gromov’s Nonsqueezing Theorem I.1 Liouville’s theorem in Hamiltonian mechanics . . . . . . . . . . I.2 Symplectomorphisms and symplectic embeddings . . . . . . . . I.3 Gromov’s nonsqueezing theorem . . . . . . . . . . . . . . . . . I.3.1 Statement of the theorem . . . . . . . . . . . . . . . . . I.3.2 Almost complex structures . . . . . . . . . . . . . . . . I.3.3 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . I.4 The monotonicity lemma . . . . . . . . . . . . . . . . . . . . . I.4.1 The strong maximum principle . . . . . . . . . . . . . . I.4.2 Symplectic energy of holomorphic curves . . . . . . . . . I.4.3 Proof of the monotonicity lemma . . . . . . . . . . . . . I.5 Isoperimetric inequalities . . . . . . . . . . . . . . . . . . . . . I.5.1 Proof of Lemma I.4.25 . . . . . . . . . . . . . . . . . . . I.5.2 Symplectic action and energy of loops . . . . . . . . . . I.6 Holomorphic curves are minimal surfaces . . . . . . . . . . . . . I.7 The space of almost complex structures . . . . . . . . . . . . . I.7.1 Linear algebra of complex structures . . . . . . . . . . . obius transformation I.7.2 The Cayley transformation as a M¨ I.7.3 The Cayley transformation on matrices . . . . . . . . . I.7.4 Interpolating almost complex structures . . . . . . . . . I.8 A moduli space of J-holomorphic discs . . . . . . . . . . . . . . I.8.1 The Schwarz lemma . . . . . . . . . . . . . . . . . . . . I.8.2 Automorphisms of the unit disc . . . . . . . . . . . . . . I.8.3 The moduli space M . . . . . . . . . . . . . . . . . . . . I.8.4 Symplectic energy of J-holomorphic discs . . . . . . . . I.8.5 Flat discs . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 4 6 7 8 9 13 14 18 19 21 22 23 26 27 28 29 31 32 33 33 34 34 36 38

II Compactness II.1 Geometric a priori bounds . . . . . . . . . . . . . . . . . . . . . . II.1.1 Schwarz reflection in the unit circle . . . . . . . . . . . . . . II.1.2 A bound on boundaries . . . . . . . . . . . . . . . . . . . .

41 42 42 44

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CONTENTS II.1.3 A bound on nonstandard discs . II.1.4 The boundary lemma of E. Hopf II.1.5 Boundaries are embedded . . . . II.1.6 The evaluation map . . . . . . . II.2 Uniform gradient bounds . . . . . . . . II.2.1 The Fr´echet metric . . . . . . . . II.2.2 The Arzel` a–Ascoli theorem . . . II.2.3 Examples of bubbling . . . . . . II.2.4 The bubbling-off argument . . . II.3 An asymptotic isoperimetric inequality . II.4 Boundary singularities . . . . . . . . . .

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III Bounds of Higher Order III.1 The a priori estimate . . . . . . . . . . . . . . . . . . . III.1.1 Sobolev norms . . . . . . . . . . . . . . . . . . III.1.2 The Poincar´e inequality . . . . . . . . . . . . . III.1.3 The inhomogeneous Cauchy–Riemann equation III.1.4 The Calder´ on–Zygmund inequality . . . . . . . III.1.5 The a priori estimate . . . . . . . . . . . . . . III.1.6 Why do we need Sobolev norms? . . . . . . . . III.2 Various Sobolev estimates . . . . . . . . . . . . . . . . III.3 The C k -norm of nonstandard discs is bounded . . . .

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IV Elliptic Regularity IV.1 The linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.1.1 Notions of differentiability . . . . . . . . . . . . . . . . . . IV.1.2 The linearisation of C k . . . . . . . . . . . . . . . . . . . . IV.1.3 Spaces of continuous maps are smooth Banach manifolds IV.1.4 The linearisation of ∂ J . . . . . . . . . . . . . . . . . . . . IV.2 The Sobolev completion . . . . . . . . . . . . . . . . . . . . . . . IV.2.1 The definition of Sobolev spaces . . . . . . . . . . . . . . IV.2.2 The Sobolev embedding theorem . . . . . . . . . . . . . . IV.2.3 Rules of differentiation in Sobolev spaces . . . . . . . . . IV.2.4 Sobolev estimates . . . . . . . . . . . . . . . . . . . . . . IV.2.5 The completion B of C is a Banach manifold . . . . . . . IV.2.6 Differentiability of ∂ J . . . . . . . . . . . . . . . . . . . . IV.3 Elliptic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.3.1 The topology on M . . . . . . . . . . . . . . . . . . . . . IV.3.2 A local estimate . . . . . . . . . . . . . . . . . . . . . . . IV.3.3 The shift operator . . . . . . . . . . . . . . . . . . . . . . IV.3.4 Proof of Theorem IV.3.1 . . . . . . . . . . . . . . . . . . . IV.3.5 Proof of the deceptively simple lemma . . . . . . . . . . . IV.3.6 The need for the Sobolev completion . . . . . . . . . . . .

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89 90 91 94 95 99 100 101 103 104 106 107 108 113 113 114 115 117 119 123

CONTENTS V Transversality V.1 Fredholm theory . . . . . . . . . . . . . . . . . . . . . . V.1.1 Fredholm operators . . . . . . . . . . . . . . . . V.1.2 The Fredholm index of ∂ J . . . . . . . . . . . . . V.1.3 Transformation to a perturbed ∂-operator . . . . V.1.4 Fredholm plus compact is Fredholm . . . . . . . V.1.5 The components of ∂ . . . . . . . . . . . . . . . . V.2 Regular values and the Sard–Smale theorem . . . . . . . V.2.1 The implicit function theorem . . . . . . . . . . . V.2.2 Regular values . . . . . . . . . . . . . . . . . . . V.2.3 The Sard–Smale theorem . . . . . . . . . . . . . V.2.4 How to prove that M is a manifold . . . . . . . . V.3 The Carleman similarity principle . . . . . . . . . . . . . V.3.1 Behaviour near the boundary . . . . . . . . . . . V.3.2 From a nonlinear to a linear equation . . . . . . V.3.3 The similarity principle and its corollaries . . . . V.3.4 Proof of the Carleman similarity principle . . . . V.4 Injective points . . . . . . . . . . . . . . . . . . . . . . . V.4.1 Inj(u) is open and dense . . . . . . . . . . . . . . V.4.2 (Self-)Intersections of holomorphic discs . . . . . V.5 The Floer space of almost complex structures . . . . . . V.5.1 The Floer norm . . . . . . . . . . . . . . . . . . . V.5.2 A separable Banach space . . . . . . . . . . . . . V.5.3 A Banach manifold of almost complex structures V.6 The universal moduli space . . . . . . . . . . . . . . . . V.6.1 The tangent space TJ J0 . . . . . . . . . . . . . . V.6.2 The linearisation of ∂ • . . . . . . . . . . . . . . . V.6.3 The Weyl lemma . . . . . . . . . . . . . . . . . . V.6.4 The universal moduli space is a Banach manifold V.7 The moduli space M and the evaluation map . . . . . . V.7.1 The moduli space MJ is a manifold . . . . . . . V.7.2 The mod 2 degree of a smooth map . . . . . . . V.7.3 The degree of the evaluation map . . . . . . . . .

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125 126 126 127 128 131 135 140 140 141 142 144 144 144 145 145 147 150 151 152 155 155 157 159 160 161 163 166 167 168 168 169 172

Bibliography

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Index

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Preface ¨ beganDie Lehrer der verschiedenen mathematischen Ubungen nen ihren Kursus, mit wenigen Ausnahmen, durch einige magere Worte u ¨ber den Sinn des Titels und begannen dann unaufhaltartsschreitend ohne umzusehen, ob sam die Sache selbst, vorw¨ uckbleibe oder nicht. andnis zur¨ einer mit dem Verst¨ une Heinrich Gottfried Keller, Der gr¨

The holomorphic discs in the title of this book refer to a mathematical object that is, strictly speaking, neither holomorphic nor a disc. This is in accordance with the principle, as laid down by [Hirsch 1976], that “in mathematics a red herring does not have to be either red or a herring”. By discs we mean smooth maps defined on the unit disc in the complex plane C, taking values in some manifold. We shall be speaking of curves as smooth maps defined on a domain (or its closure) in the complex plane1 . More generally, the domain of definition may be a Riemann surface (possibly with boundary); such curves will not be considered in the present text, though. The usage of the word holomorphic requires a more detailed explication. In the strict sense of the word, this refers to curves where the target space is a complex manifold, and the map satisfies the Cauchy–Riemann equations. Equivalently, these are maps whose differential at any point of the domain of definition is a complex linear map. In the main body of the text, we shall adhere to this narrow interpretation of the word ‘holomorphic’. The holomorphic discs of the title in point of fact refer to the more general pseudoholomorphic or J-holomorphic maps. The idea behind this concept is quite simple. In order to speak sensibly of maps with complex linear differential, there is no need for (local) complex coordinates on the target space. All that is required is a notion of complex multiplication on each tangent space, which is precisely what an almost complex structure J on a manifold amounts to. (In two real dimensions, any almost complex structure comes from complex coordinates, thanks to the existence of isothermal coordinates2 for any conformal structure; for this reason we need not consider a wider class of domains of definition.) Beware that many authors nowadays use ‘holomorphic curve’ with the latter, wider meaning. To keep the title of this book snappy, we have done the same there, but there only. The ground-breaking observation in [Gromov 1985] was that pseudoholomorphic curves share many desirable features with holomorphic curves, provided that the almost complex structure J is compatible, in a sense to be defined, with a symplectic form ω (that is, a closed nondegenerate differential 2-form). This includes 1 Except when, as will be clear from the context, we are speaking of real curves defined on intervals in the real line. 2 This is a deep result; see [Bers 1957/8] or [Donaldson 2011].

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them being minimal surfaces, and energy estimates that guarantee the compactness of certain ‘moduli spaces’ of pseudoholomorphic curves. Such compactness results pave the way for cobordism- or degree-theoretic topological arguments. Gromov’s first application of the theory of pseudoholomorphic curves in symplectic manifolds was the nonsqueezing theorem, which may be regarded as the big bang for the new field of symplectic topology. Symplectic geometry is a time-honoured subject, as it provides the mathematical framework for classical mechanics; see [Arnol’d 1989]. Liouville’s theorem on the preservation of volume in phase space is in fact a statement about symplectic mappings. The nonsqueezing theorem demonstrates that symplectic mappings are considerably more restrained than volume-preserving ones. It says that one cannot squeeze a ball (in the standard symplectic space R2n ) into a cylinder of smaller radius by a symplectic embedding.

Outline of the text The book you hold in your hands (if you are an old-fashioned reader) presents a complete proof of the nonsqueezing theorem. In Chapter I we expand on the classical context just mentioned and introduce all the concepts required to give a precise formulation of the theorem and a sketch of its proof. The later chapters successively fill in the details of this proof. Given a symplectic embedding of a ball of radius r into a cylinder of radius R, the aim is to show that r ≤ R. Ultimately, the proof boils down to establishing the existence of a J-holomorphic disc passing through the centre of the embedded ball, and whose boundary is mapped to the boundary of the cylinder. The almost complex structure J in question is adapted to the embedding of the ball. Energy estimates and an isoperimetric inequality for holomorphic discs then combine to yield the desired inequality on the radii. The existence of such a disc rests crucially on the properties of the moduli space of all pseudoholomorphic discs with boundary behaviour as described. Broadly speaking, chapters II and III are devoted to establishing the compactness of this space; chapter V deals with the manifold property of this space. A major complication in the proof stems from the fact that one cannot work solely within the framework of smooth discs, as spaces of smooth maps fail to be Banach manifolds. Chapter IV shows how to resolve this issue by considering spaces of weakly differentiable pseudoholomorphic curves and then establishing regularity results for such curves. The nonsqueezing theorem is very easy to state, and the formidable mathematical effort we need to expend on its proof may seem surprising and daunting. In order to assuage the intellectual strain, at several places in the text we pause to explain why one cannot do without the intricate, mostly analytic tools that we introduce.

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The rationale for this text Beautiful and unexpected as the nonsqueezing theorem may be, its proof should not be regarded as an end in itself. We principally take Gromov’s theorem as a motivation to explore the powerful tools it calls for. As the title of the book suggests, the true aim of this text is to provide a self-contained, comprehensive, and yet concise introduction to the theory of pseudoholomorphic curves. This theory has become indispensable in symplectic topology, and we develop its basics in sufficient generality for the method to be widely applicable. The special emphasis on discs leads us to analyse boundary behaviour in considerable detail. In our education of students we have, for a long time, felt a dearth of texts on pseudoholomorphic curves to which we could unhesitatingly direct them for instruction and edification. The exhaustive monograph [McDuff & Salamon 2012] contains everything a seasoned researcher may be looking for, but it is a steep climb3 for a beginner. The books [Wendl 2018] and [Wendl 2020] stem from minicourses or lectures for a relatively mature audience, focus on specific themes, and are brief on the mathematical basics. The beautiful and impressive Diplomarbeit [Hummel 1997] contains a proof of the compactness theorem (for closed curves), and we have used parts of it in this text, but it never pretends to cover all fundamentals of pseudoholomorphic curves theory. Hummel’s thesis takes Gromov’s differential geometric approach via hyperbolic surfaces, whereas we follow a more analytical line of argument, based on ideas developed by Helmut Hofer. The present text began its life as a set of notes prepared by the second-named alische Wilhelms-Universit¨ at author for a one-semester course taught at the Westf¨ M¨ unster in 2015 to an audience of Master and doctoral students. Since then, we have taught from and — with the assistance of our students — improved upon at Gießen, Ruhrthese notes in courses and seminars at the Justus-Liebig-Universit¨ Universit¨at Bochum, and the Universit¨ at zu K¨ oln. During the conversion of the handwritten notes into a typed text, while preserving the overall structure of the course, we added some background material (e.g. on Fredholm operators), shed light into some black boxes (such as the Banach manifold property of spaces of maps), and polished the arguments in various places. This book is meant to serve two purposes. On the one hand, the text is structured so as to be suitable for self-study. On the other, we hope that this text will inspire colleagues to teach their own students from it. For this reason, we have tried to stay true to the spirit of the course from which this text originates. The scores of exercises throughout the text, some from our teaching, some added as the notes expanded, should be helpful for either purpose.

3 Our text, by comparison (and with apologies to Felix Schlenk), is intended to feel like a gentle hike on Uetliberg; cf. reference [13] in [Cieliebak et al. 2007].

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Discs versus spheres This section is for readers who already have some familiarity with symplectic and contact topology. Gromov’s original proof of the nonsqueezing theorem works with a partial 2 2 × R2n−2 , and then utilises pseucompactification SR/2 × R2n−2 of the cylinder BR doholomorphic spheres. As mentioned earlier, our proof instead studies pseudoholomorphic discs whose boundary lies on the boundary of the cylinder. Arguably this is a more natural approach. Pseudoholomorphic discs made their first appearance in applications to contact topology, where one works with the symplectisation of a contact manifold. This symplectisation carries an exact symplectic form and therefore does not admit nonconstant closed pseudoholomorphic curves such as spheres. A comprehensive monograph on the use of pseudoholomorphic discs in contact topology is [Abbas & Hofer 2019]. The ideas for filling cylinders with discs, again in the context of contact topology (and Reeb dynamics), go back to [Eliashberg & Hofer 1994]. That paper mostly dealt with 3-dimensional contact topology; in [Geiges & Zehmisch 2016] we extended the theory to higher-dimensional contact manifolds. There, we also systematically began to explore degree-theoretic arguments for pseudoholomorphic discs — here, the content of Chapter V — that originate (for spheres) from [McDuff 1991]. Again for spheres, this was developed further in [Barth, Geiges & Zehmisch 2019]. In [Eliashberg & Hofer 1994] and [Geiges & Zehmisch 2016] the use of discs was necessitated by the contact-geometric setting. In the symplectic setting, pseudoholomorphic discs were used in [Gromov 1985] to deal with Lagrangian submanifolds. The present text might also serve to encourage more mature readers to revisit some of the results in this area, and to find other applications of pseudoholomorphic discs in symplectic topology. An interesting variation on the theme of pseudoholomorphic discs can be found in [Sukhov & Tumanov 2014].4 There, the authors establish the existence of pseudoholomorphic discs — and use this to give a proof of Gromov’s nonsqueezing theorem — by solving a vector analogue of the classical Beltrami equation. For analytical reasons in dealing with the boundary problem, the authors replace the round cylinder with a cylinder having a triangular base. For further reading on applications of pseudoholomorphic discs we modestly suggest [Geiges & Zehmisch 2010], where we fleshed out an idea by Eliashberg to give a contact-geometric proof of Cerf’s theorem that every diffeomorphism of the 3-sphere S 3 extends to a diffeomorphism of the 4-ball. This proof relies on a ‘filling’ of S 3 ⊂ C2 by holomorphic discs, i.e. discs in the 4-ball with boundary on S 3 , adapted to a contactomorphism of the standard contact structure5 on S 3 . In [Geiges & Zehmisch 2013], more generally, we studied the filling of symplec4 We 5 We

thank Felix Schlenk for this reference. shall not pursue contact-geometric themes in this text, but see Exercise I.4.13.

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tic cobordisms with pseudoholomorphic discs. This allowed us to give a unified treatment of some ‘classical’ results in symplectic and contact topology (including Reeb dynamics). A number of arguments developed in these papers have found their way into the presentation here.

Mathematical prerequisites The theory of pseudoholomorphic curves draws on methods and results, some of them quite deep, from diverse areas of mathematics. We have tried to develop the theory from first principles wherever feasible, but it is impossible to include everything without the book becoming encyclopaedic. The first two chapters can probably be understood with little more than a background in complex analysis. In fact, we have read this part profitably (for all involved) with students in the third and final year of their Bachelor studies. Advanced calculus, which German students take in their second and third semesters, is taken for granted. This includes concepts such as (sub-)manifolds, tangent spaces, differential forms, and Stokes’s theorem (especially in the divergence version known as the theorem of Gauß). anich 1996]: Everything we require from point-set topology can be found in [J¨ compactness vs. sequential compactness, continuity vs. sequential continuity, first and second countability axioms for topological spaces. Concerning analysis on manifolds, we freely (but sparingly) use concepts such as the flow of a vector field and the Lie derivative (including Cartan’s formula). Where possible, we provide alternative and more elementary arguments. The remaining chapters require some basic concepts and results from linear functional analysis, such as Banach spaces, bounded (a.k.a. continuous) linear operators, and the open mapping theorem. In a few places, and in passing, we quote the Hahn–Banach theorem. Everything we use can be found in the undergraduate text [Bollob´as 1999] or the slightly more advanced [Hirzebruch & Scharlau 1971]. We need much less than these books cover. For instance, all the prerequisites from Fredholm theory are introduced in Chapter V, even though they can be found in the text by Hirzebruch and Scharlau. As concerns Sobolev spaces, some acquaintance with the fundamental concepts in [Adams & Fournier 2003] is certainly a boon, but not required. We quote without proof some Sobolev inequalities and embedding theorems, but the theory of weakly differentiable functions is developed from scratch in Chapter IV. From differential geometry, we use the concept of geodesics without explanation, but we do discuss the exponential map — the only reason we need to talk about geodesics — in detail. We also use some concepts from elementary differential geometry such as the curvature of plane curves. Concerning differential topological methods, we occasionally apply them without being overly loquacious. Books such as [Br¨ ocker & J¨ anich 1973], [Hirsch 1976], or [Wall 2016] cover much more than what is needed here, but a certain acquaintance with one of these books certainly helps to smooth some corners in the text.

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The mapping degree theory in Chapter V is developed in detail, following a differential topological route. The alternative approach, using homological methods, is sketched for readers familiar with the relevant algebraic topology. No previous knowledge of symplectic geometry is assumed.

A guide for the reader and the lecturer (and a little rant) In the German Diplomstudiengang of lore — before it was scrapped by EU bureaucrats and an overambitious federal minister of education, misappropriating the name of a renowned Italian university town6 for a process of intellectual vandalism7 — a course such as this would have, we believe, been welcomed by students as a challenging treat in their final year, even if few of them would have elected it as ufung. In this day and age of credit points, international part of their oral Diplompr¨ ‘harmonisation’ of educational systems, and curricula geared towards ‘employability’, the situation is a little more vexatious. (We hear an inner voice quoting from Christian Morgenstern’s Der Ginganz : “Der Knecht wirft beide Arm’ empor, als wollt’ er sagen: Laß doch, laß!”) On the positive side, even if occasionally we may have been carried away by romanticised conceptions of what students ‘in the old times’ were like (including an idealised view of our own student days), we have been rewarded by students who have put in to some extent ‘voluntary’ work in reading parts of this text in various seminars and Arbeitsgemeinschaften, the latter our own humble version of the seminar concept designed by Felix Klein. That being said, our text can be used in a variety of settings. First and foremost it can naturally be the basis for a one-semester course of four hours per week, like the one from which these notes originate. The book now contains slightly more material than can reasonably be covered in such a course, and the selection of parts to exclude can be made dependent on the students’ background. Section I.6 is optional, as it will not be used anywhere else. Some of the more gruesome analytical details in Chapters III and IV, such as in Section IV.3.5, should probably be skimmed. A few parts of the material in Section IV.1, notably the discussion of manifolds of maps, may safely be put back into a black box or left as vocational reading. The selection of topics in Chapter V may depend on whether the students have been educated more thoroughly in either functional analysis or differential topology. Alternatively, one can cover selected parts of the book in student seminars. As already mentioned, the first two chapters lend themselves to a seminar following a first course on complex analysis. Other parts can be used for seminars on functional analysis or differential topology. We have placed well over a hundred exercises throughout the text rather than at the end of sections or chapters. They form an integral part of the course. 6 The

university of Bologna, founded in 1088, is the oldest university in Europe. declaration that specifies objectives to be reached “within the first decade of the third millennium” is, in our view, more than just a little grandiose. 7A

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Indeed, many exercises ask for filling in smaller details of proofs, and in this way provide an important self-assessment for the lone reader.

The sources we consulted With a course that has developed over several years, it has become shrouded by the mists of time exactly which sources were of help in writing certain parts of the text. We make only modest claims to originality. The initial design of the course and the approach to the nonsqueezing theorem via discs have been inspired by unpublished notes of Helmut Hofer, some of which made their way into [Abbas & Hofer 2019]. A copy of [McDuff & Salamon 2012] was of course always at hand during the writing. The proof of the monotonicity lemma in Section I.4.3 closely follows [Hummel 1997]. For writing Section V.5 on the Floer space of almost complex structures we reread the thesis [Schwarz 1995]. Course notes by Kai Cieliebak on nonlinear functional analysis were of great help in writing Section IV.1. We also used lecture notes by Joa Weber on Sobolev spaces to streamline parts of the discussion in Section IV.2. For cultural reasons, in the bibliography we have not replaced the German texts we consulted with English ones. The reader will easily find equivalent sources in his favourite language.

Acknowledgements For an uncountable number of inspiring mathematical conversations that have contributed to the writing of this book we thank, especially, Alberto Abbondandolo, Peter Albers, Helmut Hofer, Leonid Polterovich and Felix Schlenk, and we apologise to all the friends and colleagues whose input we may have absorbed unconsciously. Murat Sa˘ glam taught examples classes for a course based on a draft of this text, worked through many of the exercises, and made innumerable constructive suggestions. One of our lengthier discussions led to the writing of [Geiges, Sa˘glam & Zehmisch 2023]. Dirk Horstmann and Guido Sweers were our sounding boards for some of the more analytical parts of the text. The students who followed the original course and the later repeats, as this text began to take shape, helped us to clarify our arguments and improve the exposition. We should like to single out Tilman Becker for his perceptive reading of the draft. We thank the editors of the Virtual Series on Symplectic Geometry and of the Basler Lehrb¨ ucher, respectively, for accepting this text into their series. The two anonymous referees did a tremendous job, and we have gratefully taken up most of their suggestions for improving the text. We also thank Christopher Tominich at Birkh¨auser for his efficient handling of this book project, and our copy editor Paula Francis for her eagle-eyed reading of the manuscript. Part of the final writing was done at the Lorentz Center at Leiden University during the workshop on ‘Symplectic Dynamics Beyond Periodic Orbits’ in August

xviii

Preface

2022, organised by Federica Pasquotto and Viktor Ginzburg, jointly with the firstnamed author. We thank the Lorentz Center and its staff, especially the scientific coordinator Federica Burla and the workshop coordinator Tanja Uitbeijerse, for providing an excellent working environment. During the writing of this book and the research that, in parts, found its way into this text, the authors were supported by the DFG (Deutsche Forschungsgemeinschaft) through the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’ (Project ID 281071066 – TRR 191). Cologne & Bochum, May 2023

org Geiges & Kai Zehmisch Hansj¨

Chapter I

Gromov’s Nonsqueezing Theorem And now, if e’er by chance I put My fingers into glue, Or madly squeeze a right-hand foot Into a left-hand shoe. Lewis Carroll, Through the Looking-Glass

The theorem in the title is the protagonist of this text. We state it in Section I.3 and present a sketch proof, where (pseudo-)holomorphic discs make their first appearance. This is preceded by a section on Liouville’s theorem in classical mechanics, which serves as a motivation to study properties of symplectic maps; these we define in Section I.2. Geometrically, the proof of the nonsqueezing theorem rests crucially on two estimates. One is a lower bound on the area of certain holomorphic curves. This bound is derived from a monotonicity formula (Section I.4) and an isoperimetric inequality (Section I.5). The monotonicity lemma is an expression of the fact that holomorphic curves are minimal surfaces. Although our argument does not rely explicitly on this minimality property, in Section I.6 we explain this connection. The second geometric input is a computation of the symplectic energy of certain J-holomorphic curves, that is, curves that are ‘holomorphic’ with respect to an almost complex structure J. Section I.7 is devoted to the relevant space of almost complex structures, which is foundational for the study of the ‘moduli space’ of J-holomorphic discs. The proof of the nonsqueezing theorem rests on an abundance of such discs. The first basic properties of this moduli space are discussed in Section I.8; this section includes the energy calculation mentioned above. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Geiges, K. Zehmisch, A Course on Holomorphic Discs, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-36064-0_1

1

2

I.1

Chapter I. Gromov’s Nonsqueezing Theorem

Liouville’s theorem in Hamiltonian mechanics

The time evolution of many classical mechanical systems, such as a simple pendulum or, more interestingly, a number of celestial bodies under mutual gravitational attraction, is governed by a set of differential equations known as Hamilton’s equations. For an elementary derivation of these equations from a variational principle, see [Geiges 2016, Chapter 9]. The geometric ideas behind Hamiltonian mechanics are beautifully elucidated in [Levi 2014]. The bare-bones description of Hamilton’s equations that follows merely serves to provide the terminology for formulating Liouville’s theorem — in its simplest version — on the preservation of volume in phase space. We are considering R2n = Rn ⊕ Rn with Cartesian coordinates (x, y). In the physical interpretation, x = (x1 , . . . , xn ) is regarded as a point in the configuration space of the mechanical system (e.g., describing the positions of the celestial bodies); y = (y1 , . . . , yn ) describes the corresponding momenta (or velocities, if you will). The full space R2n is the phase space (rather, its local description) of the system. The Hamilton equations corresponding to a smooth ‘Hamiltonian’ function1 H : R2n → R are ∂H  x˙ = ,   ∂y (I.1) ∂H  y˙ = − ,  ∂x where we write time derivatives with dots in the Newtonian fashion. Denote by ∇H =

∂H ∂(x, y)

the gradient of H. Then, (I.1) becomes ˙ x) ˙ = ∇H. (−y,

(I.2)

Now we identify Rn ⊕Rn with Cn via2 (x, y) 7→ x+iy =: z. Let J0 : Cn → Cn be the isomorphism corresponding to multiplication by the imaginary unit i, but thought of on the level of tangent spaces, that is, J0 (∂xj ) = ∂yj ,

J0 (∂yj ) = −∂xj ,

j = 1, . . . , n.

Here, ∂xj , ∂yj denote unit vectors in the coordinate directions; the notation comes from the interpretation of these tangent vectors as directional derivatives. Notice 1 The attribute ‘Hamiltonian’ does not refer to any special properties of the function, but simply to its role in giving rise to a dynamical system in the Hamiltonian formalism. As explained in [Geiges 2016], the letter H actually stands for Huygens. 2 Later on, when we speak of the identification of Cn with R2n , our convention is to order the real coordinates as (x1 , y1 , . . . , xn , yn ); see Section I.7.1, for instance.

I.1. Liouville’s theorem in Hamiltonian mechanics

3

˙ Hence, (I.2) can be rewritten even more ˙ = −y˙ + ix. that J02 = −id and J0 (x˙ + iy) succinctly as z˙ = −J0 ∇H(z) =: XH (z), (I.3) and XH is called the Hamiltonian vector field corresponding to H. The flow of this vector field describes the evolution of the mechanical system, and Liouville’s theorem says that this flow preserves the standard volume form volR2n := dx1 ∧ dy1 ∧ . . . ∧ dxn ∧ dyn on phase space. Theorem I.1.1 (Liouville). The flow of the Hamiltonian vector field XH on phase space is volume-preserving. Proof. Thanks to the divergence theorem of Gauß, it suffices to observe that the vector field XH is divergence-free (or see the subsequent elementary exercise):     n  X ∂ ∂H ∂ ∂H div(XH ) = + − = 0. ∂xj ∂yj ∂yj ∂xj j=1



Exercise I.1.2. This exercise is meant to elucidate the interpretation of divergence as ‘source strength’, without appealing to the theorem of Gauß. Let v : Rm → Rm be a vector field on Rm . Write e1 , . . . , em for the unit vectors in the coordinate directions. For a small ε > 0 we consider the little cube spanned by the vectors εei , i = 1, . . . , m at some point a ∈ Rm . Now we study how this cube moves under the flow defined by the vector field v. In a small time t, the corner a of the cube moves, in first approximation, to the point a + tv(a); the corners a + εei move to a + εei + tv(a + εei ). These corners define a parallelepiped spanned by the vectors
ui (ε, t) := εei + t v(a + εei ) − v(a) , i = 1, . . . , m. Write Vε (t) for the volume of this parallelepiped, that is,
Vε (t) = det u1 (ε, t), . . . , um (ε, t) , where we interpret the ui (ε, t) as column vectors of an (m × m)-matrix. Compute V˙ ε (0) and show that V˙ ε (0) = div v(a). ε→0 Vε (0) lim

♦

We shall see presently that one can avoid referring to the geometric interpretation of divergence, and prove something considerably stronger to boot. Enter symplectic geometry.

4

I.2

Chapter I. Gromov’s Nonsqueezing Theorem

Symplectomorphisms and symplectic embeddings

We now consider the 2-form ω :=

n X

dxj ∧ dyj

(I.4)

j=1

on R2n . Notice that in terms of the standard inner product h . , . i, for arbitrary vector fields v, w on R2n we have ω(v, w) = hJ0 v, wi. Exercise I.2.1. Verify this identity. Further, show that J0 is an element of the orthogonal group O(2n), and that it preserves the 2-form ω, in the sense that ω(J0 v, J0 w) = ω(v, w). So, we also have ω(v, J0 w) = hv, wi.

♦

The 2-form ω is obviously closed, i.e. dω = 0, and it is nondegenerate in the sense that for any nonzero tangent vector vp ∈ Tp R2n one can find a tangent vector wp ∈ Tp R2n such that ωp (vp , wp ) 6= 0; indeed, one may take wp := J0 vp . Definition I.2.2. A closed and nondegenerate 2-form Ω on a smooth manifold M is called a symplectic form. The pair (M, Ω) is then called a symplectic manifold. The 2-form ω in (I.4) is the standard symplectic form on R2n . Note I.2.3. The nondegeneracy condition can be interpreted as saying that the linear map Tp M −→ Tp∗ M 7−→ Ωp (vp , . ) vp from the tangent space to its dual is injective, and hence an isomorphism, at all points p ∈ M . The nondegeneracy is a pointwise condition, and some simple linear algebra shows that it forces the manifold to be of even dimension 2n, and that the nondegeneracy of Ω can equivalently be phrased as the condition that the nth exterior power Ωn of Ω be nowhere zero, i.e. a volume form. In our concrete example, we have ω n = n! volR2n . Exercise I.2.4. Prove the claims in the preceding paragraph.

♦

Proposition I.2.5. With the help of ω, the defining equation (I.3) for the Hamiltonian vector field XH can be written as iXH ω = dH, where i denotes the interior product.

(I.5)

I.2. Symplectomorphisms and symplectic embeddings

5

Proof. By Note I.2.3, equation (I.5) defines a unique vector field XH for any given smooth function H. Therefore, it suffices to verify that the Hamiltonian vector field XH defined in (I.3) also satisfies (I.5): iXH ω = hJ0 XH , . i = h∇H, . i = dH.



Here is the promised improvement on Liouville’s theorem. For the basic facts about Lie derivatives we are using in the proof and the subsequent discussion, the reader may wish to refer to [Warner 1983]. Cartan’s formula is Proposition 2.25 (d) in that text. Theorem I.2.6. The flow of the Hamiltonian vector field XH preserves the symplectic form ω on phase space. Proof. This follows from the vanishing of the Lie derivative: LXH ω = (d ◦ iXH + iXH ◦ d)(ω) = d2 H = 0, where we have used Cartan’s formula LX = d ◦ iX + iX ◦ d.



If we write φt for the (local) flow of XH , then d |t=t0 φ∗t ω = φ∗t0 LXH ω, dt so the vanishing of the Lie derivative LXH ω is indeed equivalent to φ∗t ω = ω for all times t (when and wherever φt is defined). Alternatively, one can perform a direct and elementary computation. Write φt : (x01 , y10 , . . . , x0n , yn0 ) 7−→ (xt1 , y1t , . . . , xtn , ynt ). Then φ∗t ω =

n X

dxtj ∧ dyjt

j=1

and, suppressing the superscript t from the notation, d ∗ φ ω dt t

=

X

dx˙ j ∧ dyj + dxj ∧ dy˙ j




j

=

"  X X  ∂2H ∂2H dxk ∧ dyj + dyk ∧ dyj ∂xk ∂yj ∂yk ∂yj j k # X  ∂2H ∂2H − dxj ∧ dxk + dxj ∧ dyk ∂xk ∂xj ∂yk ∂xj k

= 0, where we have used (I.1) in the second line and symmetry of second derivatives throughout.

6

Chapter I. Gromov’s Nonsqueezing Theorem

Definition I.2.7. A diffeomorphism Φ of a symplectic manifold (M, ΩM ) is called symplectic or a symplectomorphism if Φ∗ ΩM = ΩM . An embedding Φ : (M, ΩM ) ,→ (N, ΩN ) of symplectic manifolds is symplectic if Φ∗ ΩN = ΩM . (Here it is not assumed that dim M = dim N .) With this terminology in place, Theorem I.2.6 becomes: The flow map φt of the Hamiltonian vector field XH is a symplectomorphism of phase space. A symplectomorphism Φ of (M, ΩM ) preserves, a fortiori, the volume form ΩnM (where n = (Φ∗ ΩM )n = ΩnM . So the classical Liouville theorem dim M = 2n), since Φ∗ ΩM is indeed a direct consequence of Theorem I.2.6. For completeness we mention that a diffeomorphism that arises as the time-1 map of a Hamiltonian flow is called a Hamiltonian diffeomorphism. On a compact symplectic manifold (M, ΩM ) without boundary, a Hamiltonian function H : M → R necessarily has singular points (a minimum and a maximum), and thus XH has zeros by (I.5). Thus, the shift (ϕ, θ) 7→ (ϕ + ϕ0 , θ) on the 2-torus T 2 = S 1 × S 1 , with circular coordinates ϕ, θ and symplectic form dϕ ∧ dθ, would be a simple example of a non-Hamiltonian symplectomorphism. A concise textbook on symplectic geometry is [Cannas da Silva 2001]. For a considerably more comprehensive treatment, including more topological aspects, see [McDuff & Salamon 2017]. The use of symplectic geometry in Hamiltonian mechanics is amply explained in [Abraham & Marsden 1978], [Arnol’d 1989] and [Libermann & Marle 1987].

I.3

Gromov’s nonsqueezing theorem

The emancipation of symplectic geometry from its servant role to classical mechanics, and its subsequent metamorphosis into the field of symplectic topology, can be dated quite precisely. The seminal event was the publication of M. Gromov’s paper Pseudoholomorphic curves in symplectic manifolds [Gromov 1985]. The methods developed there opened vistas of thought that have led to spectacular advances over the course of several decades. One of the key results of that paper, and the protagonist of these lecture notes, is the celebrated3 nonsqueezing theorem. As we have seen, Liouville’s theorem is usually formulated as a result about the preservation of volume in phase space under Hamiltonian flows, although these flows actually preserve the symplectic form ω, not only the volume form ω n . Thus, it seems natural to ask: how much more restrictive is the condition for a map (a diffeomorphism, an embedding, . . .) to be symplectic rather than merely volumepreserving? The nonsqueezing theorem provides an instance where one can establish the nonexistence of a symplectic embedding, although a volume-preserving embedding can be found. This proves the existence of ‘symplectic obstructions’. 3 We

were too young to be invited to the party.

I.3. Gromov’s nonsqueezing theorem

7

I.3.1 Statement of the theorem In order to formulate the theorem, we introduce the following notation. Write  Br2n := z ∈ Cn : |z| < r for the open ball4 of radius r > 0 in Cn , and B 2n := B12n for the open unit ball. We drop the superscript 2n when the dimension is understood. The open unit cylinder in Cn is the set  Z := Z 2n := z = (z1 , . . . , zn ) ∈ Cn : |z1 | < 1 = B 2 × Cn−1 ⊂ C × Cn−1 . For n ≥ 2 and any s > 0, the complex linear diffeomorphism Φ = Φs : (z1 , z2 , . . . , zn ) 7−→


z1 , sz2 , . . . , zn s

of Cn is volume-preserving for the volume form ω n (since det Φ = 1), but it is not symplectic unless s = 1: Φ∗ ω

= =

Φ∗ (dx1 ∧ dy1 + dx2 ∧ dy2 + · · · + dxn ∧ dyn ) 1 dx1 ∧ dy1 + s2 dx2 ∧ dy2 + · · · + dxn ∧ dyn . s2

For 0 < r ≤ 1, the ball Br embeds into Z simply by inclusion. For r > 1, if we choose s ≥ r, the restriction of Φs to Br defines a volume-preserving embedding of Br as a ‘cigar’ into Z, that is, Φs (Br ) ⊂ Z for s ≥ r; see Figure I.1. By contrast, in the symplectic realm we have the following theorem. Theorem I.3.1 (Gromov’s Nonsqueezing Theorem). The ball (Br , ω) embeds symplectically into (Z, ω) if and only if r ≤ 1. Remark I.3.2. This theorem permits a physical interpretation as a classical (in the sense of non-quantum) version of the uncertainty principle in statistical mechanics. We regard the ball Br in phase space as an initial collection of states of a physical system. The nonsqueezing theorem then says that, under the Hamiltonian flow, this collection of states never evolves into one where position and momentum in the (x1 , y1 )-direction are compressed into a disc of radius smaller than r. For a more precise formulation in terms of the so-called covariance ellipsoid, and for a comprehensive discussion of the physical context, see [de Gosson 2009]. 4 For

n = 1, we usually speak of the disc Br2 ⊂ C.

8

Chapter I. Gromov’s Nonsqueezing Theorem

Figure I.1: The volume-preserving embedding Φs : Br ,→ Z for s ≥ r > 1.

I.3.2 Almost complex structures Before we sketch the proof of the nonsqueezing theorem, we introduce a little more terminology. Definition I.3.3. An almost complex structure on a smooth manifold M is a bundle isomorphism J : T M → T M satisfying J 2 = −id. An almost complex structure J turns each tangent space Tp M , p ∈ M , into a complex vector space by defining (a + ib)v = av + bJv. In particular, M must be even-dimensional. Remark I.3.4. The dimension of M being even is not sufficient for the existence of an almost complex structure. For instance, in dimension 4 there are obstructions coming from the signature theorem. This allows one to show that the 4-sphere S 4 does not carry an almost complex structure. In fact, S 2 and S 6 are the only spheres admitting an almost complex structure. Definition I.3.5. An almost complex structure J on M is called compatible with a given symplectic form Ω on M if hv, wiJ := Ω(v, Jw), where v, w are vector fields on M , defines a Riemannian metric. The compatibility condition is equivalent to saying that Ω(Jv, Jw) = Ω(v, w)

I.3. Gromov’s nonsqueezing theorem

9

and Ωp (vp , Jvp ) > 0 for 0 6= vp ∈ Tp M . For example, the isomorphism J0 on R2n = Cn introduced in the preceding section is anPalmost complex structure compatible with the standard symplectic form ω = j dxj ∧ dyj , and the associated metric h . , . iJ0 equals the standard metric h . , . i. Note I.3.6. The almost complex structure J is orthogonal with respect to h . , . iJ , that is, hJv, JwiJ = hv, wiJ .

I.3.3

Idea of the proof

Suppose we have a symplectic embedding Φ : Br0 −→ Z. We consider the restriction of Φ to the closed ball Dr := B r for some radius r < r0 , and we write Φ−1 : Φ(Dr ) → Dr for the inverse diffeomorphism. It suffices to show that r ≤ 1, for this entails r0 ≤ 1. As before, we write ω, J0 and h . , . i = h . , . iJ0 for the standard symplectic form, almost complex structure and inner product, respectively, on R2n . Thanks to the fact that Φ : Dr → Φ(Dr ) (and hence Φ−1 ) is a symplectomorphism, the endomorphism5 J := Φ∗ J0 := T Φ ◦ J0 ◦ T Φ−1 of the tangent bundle of Φ(Dr ) is an almost complex structure compatible with ω, since hv, wiJ

:=

ω(v, Jw)

=

ω(T Φ−1 (v), T Φ−1 (Jw))

=

ω(T Φ−1 (v), J0 T Φ−1 (w))

=

hT Φ−1 (v), T Φ−1 (w)i

is an inner product. As we shall see in Section I.7, this J can be extended to an almost complex structure (which we continue to call J) compatible with ω on all of R2n , and such that it coincides with J0 outside a small neighbourhood of Φ(Dr ). The central aim of this course will be to establish a theory of pseudoholomorphic discs. In the setting at hand, this will allow us to establish the existence of a smooth map u ˆ : (D, ∂D) −→ (Z, ∂Z), 5 We use the differential topologist’s notation T Φ for the differential of a differentiable map Φ between manifolds (and Tp Φ for the differential at the point p), except for real-valued functions f , where we tend to write the differential as df .

10

Chapter I. Gromov’s Nonsqueezing Theorem

where  D := z ∈ C : |z| ≤ 1 is the closed unit disc, with the following properties. We write6 0 for the origin of C, and 0 for that of R2n . (D1) (D2)

ˆ(0) = Φ(0), u Z u ˆ∗ ω = π, D

(D3)

ˆy = 0. u ˆx + J(ˆ u) u

Remark I.3.7. By a smooth function on D we mean the restriction to D of a smooth function defined on an open subset of C containing D. In particular, equation (D3) makes sense even in the boundary points of D. Thanks to an extension result of [Seeley 1964], with a surprisingly short and simple proof, this is equivalent to the requirement that the function be smooth on Int(D), and the function and all its derivatives extend continuously to the boundary. Notation I.3.8. We write C ∞ (D, R2n ) for the maps D → R2n that are smooth in this sense. Exercise I.3.9. Show that a differentiable function7 u = g + ih : U → C on an open subset U ⊂ C satisfies the Cauchy–Riemann equations gx = hy , gy = −hx (and hence is holomorphic) if and only if ux + J0 uy = 0. Here (and in (D3) above) the subscripts denote partial derivatives. Observe that this equation entails hux , uy i = 0. Show further that these conditions are equivalent to the differential Tz u : C = Tz U −→ Tu(z) C = C being a complex linear map. If that is the case, show that Tz u is multiplication by uz := 21 (ux − iuy ), i.e. uz is the derivative of u when interpreted as a complexvalued function. Also, verify the formulæ uz = ux = −iuy for u holomorphic. 6 We shall not religiously adhere to this notational convention, but only where it helps to clarify matters. 7 The notation u is a little unfortunate in this context, since most books on elementary complex analysis write a complex-valued function as u + iv, with u denoting the real part. In symplectic topology, however, u has become the standard notation for pseudoholomorphic curves.

11

I.3. Gromov’s nonsqueezing theorem Further consequences are |uz | = |ux | = |uy | for u holomorphic,

and that holomorphic functions (as well as their real and imaginary parts) are harmonic, i.e. ∆u := uxx + uyy = 0. Conclude that the differential of a holomorphic map u : U → C at any given ♦ point z ∈ C, viewed as a real linear map Tz U → Tu(z) C, has rank 0 or 2. In view of this exercise, we recognise condition (D3) as the direct analogue of the holomorphicity condition for a map into an almost complex manifold. Definition I.3.10. Let (M, J) be an almost complex manifold, and U ⊂ C some open subset. A smooth map u : U → M is called a J-holomorphic curve, or a pseudoholomorphic curve with respect to J, if ux + J(u)uy = 0. In particular, J0 -holomorphic curves U → Cn are simply holomorphic maps. As in the holomorphic case, J-holomorphicity is equivalent to saying that the differential Tz u : Tz U → Tu(z) M
is complex linear, where the complex structure on Tu(z) M is defined by J u(z) . Exercise I.3.11. Let ϕ : V → U be a biholomorphism between two open subsets U, V ⊂ C, and u : U → (M, J) a J-holomorphic map. Show that u ◦ ϕ is likewise J-holomorphic. In other words, conformal reparametrisations of J-holomorphic curves are J-holomorphic. ♦ The relevance of condition (D1) is that we can find a J-holomorphic disc passing through the given point Φ(0); the fact that this value is attained in 0 ∈ D is not important and can be arranged by a reparametrisation, i.e. by precomposing Φ with a M¨obius transformation of D. The integral in (D2) is called the symplectic energy of the holomorphic disc. This notion of energy can be defined for any smooth map from a 2-dimensional domain of definition (preferably compact) to a symplectic manifold. In Sections I.4.2 and I.8.4 we shall discuss aspects particular to the symplectic energy of holomorphic and J-holomorphic curves, respectively. Definition I.3.12. Let u : D → M be a smooth map into a symplectic manifold (M, Ω). The symplectic energy of u is Z u∗ Ω. D

ˆ−1 (Φ(Br )), which contains 0 ∈ D On the closure Gr of the subset Gr := u by (D1), we can compose u ˆ with Φ−1 : u := Φ−1 ◦ u ˆ|Gr : Gr −→ Dr .

12

Chapter I. Gromov’s Nonsqueezing Theorem

This gives us the following commutative diagram of maps: Φ

- Z



-

Dr

u

u ˆ | Gr Gr

The map u is J0 -holomorphic (i.e. holomorphic in the usual sense as a map into Cn ), and it passes through the origin: u(0) = 0 ∈ Cn ; see Figure I.2. Beware that on the right of that figure we give a schematic illustration with reduced dimensions, so that u ˆ(Dr ) looks like a real curve, even though it is a disc (possibly with singularities).

u ˆ(D)

Φ(Dr )

Dr u(Gr )

u(0) = 0

Φ

Φ(0) Z

Figure I.2: Proof of the nonsqueezing theorem. We claim that Gr ⊂ Int(D) and u(∂Gr ) ⊂ ∂Dr .

(I.6)

To see this, we make the following observations. First of all, Φ(Dr ) is a compact subset of Cn as the embedded image of a closed ball. Hence, u ˆ−1 (Φ(Dr )) is a −1 closed subset of D. Moreover, u ˆ (Φ(Dr )) is actually contained in Int(D), since Φ(Dr ) ⊂ Z and u ˆ(∂D) ⊂ ∂Z. Points in Gr are mapped by u ˆ to points in the closure

13

I.4. The monotonicity lemma

of Φ(Br ), which equals Φ(Dr ). Hence, Gr ⊂ u ˆ−1 (Φ(Dr )) ⊂ Int(D), which is the first claimed inclusion. The second one follows by observing that ∂Gr = Gr \ Gr is mapped by u ˆ into Φ(Dr ) \ Φ(Br ) = Φ(∂Dr ). It is these properties (I.6) that will allow us to apply the monotonicity lemma, which we are going to prove in Section I.4, to this situation. This lemma says that under these conditions we have Z 2 u∗ ω. (I.7) πr ≤ Gr

But the embedding Φ is symplectic, which means that u ˆ∗ ω = (Φ ◦ u)∗ ω = u∗ Φ∗ ω = u∗ ω on Gr . We also observe that u ˆ∗ ω is a non-negative 2-form on all of D; indeed, using (D3) we find u ˆ∗ ω(∂x , ∂y ) = ω(ˆ ux , u ˆy ) = ω(ˆ ux , J(ˆ u)ˆ ux ) ≥ 0. Thus, the monotonicity lemma says that the area of the holomorphic curve u(Gr ) passing through the centre of Dr and with boundary on ∂Dr — where area is measured with respect to the 2-form ω on u(Gr ) — cannot be smaller than the area of the equatorial disc Dr ∩ {z2 = · · · = zn = 0}. We then compute Z Z Z u∗ ω = u ˆ∗ ω ≤ u ˆ∗ ω = π, Gr

Gr

D

the last equality coming from (D2). With (I.7) we conclude that r ≤ 1. This concludes the sketch proof of Theorem I.3.1. Remark I.3.13. Holomorphicity of u is used in the proof of (I.7) by interpreting the symplectic energy of u as an area with respect to a degenerate metric conformal to the standard metric on R2n . The computation of the energy integral for u ˆ is purely homological. For the compactness property of the moduli space of Jholomorphic curves, one crucial property of the energy integral is its integrand being non-negative.

I.4

The monotonicity lemma

Let G ⊂ C be a domain, i.e. an open, connected subset. We assume that 0 ∈ G. We are going to formulate the monotonicity lemma for a proper holomorphic map u : (G, 0) −→ (Cn , 0). Recall that a map between topological spaces is called proper if the preimage of any compact subset is compact.

14

Chapter I. Gromov’s Nonsqueezing Theorem

As before, we write Br ⊂ Cn for the open ball of radius r about 0 and Dr := B r for its closure. We write Sr = ∂Dr for the (2n − 1)-sphere of radius r. Set Gr := u−1 (Br ). This is an open subset of G, and hence of C. Moreover, Gr is a subset of the — by the properness of u — compact set u−1 (Dr ); hence, Gr ⊂ u−1 (Dr ) ⊂ G. With u being continuous it follows that the boundary ∂Gr = Gr \ Gr is mapped by u into the sphere Sr : u(∂Gr ) ⊂ Sr . This geometric property, as illustrated on the left-hand side of Figure I.2 will be crucial for the monotonicity lemma. If we did not impose this condition, we could easily write down maps that violate inequality (I.7). For example, the inclusion of Br20 , for r0 < r, into the equatorial disc Dr ∩ {z1 -plane} has symplectic energy π(r0 )2 . Theorem I.4.1 (Monotonicity Lemma). If u : (G, 0) → (Cn , 0) is a proper holomorphic map, then for any r > 0 the symplectic energy of u|u−1 (Br ) with respect to the standard symplectic form ω on Cn is bounded from below by πr2 : Z u∗ ω ≥ πr2 . u−1 (Br )

Without the assumption on u being proper, this inequality holds whenever u−1 (Dr ) is compact. This theorem will be proved in Section I.4.3. Exercise I.4.2. Show by an example that the conditions u(0) = 0 is necessary for the monotonicity lemma to hold. ♦ Remark I.4.3. The name ‘monotonicity lemma’ derives from the fact that the inequality can be proved by showing the quotient of the left-hand by the righthand side to be a nondecreasing function of r; see [Jost 2017], for instance. We follow a slightly different route.

I.4.1

The strong maximum principle

Before we attend to the proof of Theorem I.4.1, we want to state and prove a simple version of the maximum principle for subharmonic functions, and use it to gain information about the qualitative topological behaviour of the holomorphic curve u. Our discussion of the maximum principle is inspired by [Jost 2013]. Definition I.4.4. A smooth map f : G → R is called subharmonic if ∆f := fxx + fyy ≥ 0.

15

I.4. The monotonicity lemma

Lemma I.4.5. The function f := |u|2 , where u : G → Cn is holomorphic, is subharmonic. Proof. We compute fxx = 2hu, uxx i + 2hux , ux i, and similarly for fyy . With the harmonicity of u (Exercise I.3.9) we conclude
fxx + fyy = 2 |ux |2 + |uy |2 ≥ 0.



Exercise I.4.6. Show that the function |u| is likewise subharmonic away from the ♦ zeros of u. Consider a smooth function f : G → R, a point z0 ∈ G and a radius r > 0 such that the closed r-disc  Dr (z0 ) := z ∈ C : |z − z0 | ≤ r is contained in G. Write Sr (z0 ) := ∂Dr (z0 ) for the circle of radius r about z0 . We then define the spherical mean Z Z 2π 1 1 S(f, z0 , r) := f ds = f (z0 + reit ) dt. 2πr Sr (z0 ) 2π 0 Observe that the length element in terms of the parametrisation t 7→ z0 + reit of Sr (z0 ) is ds = r dt. The second expression for the spherical mean shows that S(f, z0 , r) extends continuously into r = 0, with S(f, z0 , 0) = f (z0 ). Exercise I.4.7. Deduce from the Cauchy integral formula that u(z0 ) = S(u, z0 , r) for a holomorphic function u : G → C; this is known as the mean value property. (This identity also follows from the harmonicity of u and the next lemma.) ♦ Lemma I.4.8.

Z ∆f dx dy = 2πr Dr (z0 )

∂ S(f, z0 , r). ∂r

Proof. We compute ∂ 2πr S(f, z0 , r) = ∂r = = = =

Z 2π ∂ r f (z0 + reit ) dt ∂r 0 Z 2π r h∇f (z0 + reit ), eit i dt 0 Z h∇f, ni ds Sr (z0 ) Z div(∇f ) dx dy Dr (z0 ) Z ∆f dx dy. Dr (z0 )

16

Chapter I. Gromov’s Nonsqueezing Theorem

Here we have written n for the outer normal along Sr (z0 ), and we have applied  the divergence theorem of Gauß in its classical formulation. Proposition I.4.9. If f is subharmonic, then f (z0 ) ≤ S(f, z0 , r). ∂ Proof. By the preceding lemma we have ∂r S(f, z0 , r) ≥ 0 for r > 0, and the continuous extension of the spherical mean to r = 0 is S(f, z0 , 0) = f (z0 ). 

We can now prove the (strong) maximum principle for subharmonic functions. Theorem I.4.10 (Maximum Principle). Let f : G → R be a subharmonic function. If there exists a point z0 ∈ G with f (z0 ) = sup f (z), z∈G

then f is constant. In particular, if G is bounded and f extends to a continuous function on G, then f (z) ≤ max f (w) for all z ∈ G. w∈∂G

Proof. Suppose there is a point in G where f attains its maximum M . Then,  GM := z ∈ G : f (z) = M 6= ∅. By the preceding proposition, for any z ∈ GM the function f is constant equal to M on any disc Dr (z) contained in G. Thus, GM is open in G. By the continuity of f , the subset GM is also closed in G. It follows that GM = G.  Remark I.4.11. An alternative proof of the maximum principle will be outlined in Section II.1.4. Exercise I.4.12. The purpose of this exercise is to prove the weak maximum principle for subharmonic functions, using a more simplistic argument. Let G ⊂ C be a bounded domain and u : G → R a continuous function on the closure G, which on G is smooth and subharmonic. Show that u attains its maximum on ∂G by considering the function z 7→ u(z) + εex , where z = x + iy and ε > 0, and then passing to the limit ε → 0. ♦ When applied to the function f = |u|2 , the maximum principle says that u : G → Cn cannot touch a boundary sphere Sr2n−1 = ∂Dr2n ⊂ Cn from the inside. Even without the condition u(0) = 0 and u proper, the image of u cannot lie completely in a sphere Sr2n−1 , unless u is constant, as we ask you to show in the next exercise, where we assume n = 2 for notational simplicity. Exercise I.4.13. Let r be the radial coordinate on C2 , and consider the 1-form α := −r dr ◦ J0 .
(a) Show that ker α|T S 3 defines a tangent 2-plane field on S 3 invariant under J0 .

I.4. The monotonicity lemma

17

(b) Verify that the description of α in Cartesian coordinates (x1 , y1 , x2 , y2 ) on R4 = C2 is α = x1 dy1 − y1 dx1 + x2 dy2 − y2 dx2 . (c) Show that the 4-form r dr ∧ α ∧ dα is nowhere zero. This implies that the 3 3-form (α ∧ dα)|T S 3 is nowhere zero (and hence a volume form
on S ). Equivalently, dα does not vanish on the 2-plane field ker α|T S 3 . (d) Let u : G → C2 be a nonconstant holomorphic curve. Show that the image of u cannot be contained in S 3 (or any Sr3 ) by computing u∗ α and u∗ (dα) under the assumption u(G) ⊂ S 3 and deriving a contradiction. The Dark Lady in this exercise is contact geometry [Geiges 2008].

♦

From elementary complex analysis the reader will be aware that holomorphic functions also satisfy a minimum principle. If u : G → C is holomorphic and |u| attains its minimum in z0 ∈ G, then u(z0 ) = 0 or u is constant. This follows by considering the function 1/u. Note I.4.14. Similarly, a harmonic function f : G → R satisfies the mean value property f (z0 ) = S(f, z0 , r) by Proposition I.4.9, applied to f and −f . This immediately implies both the maximum and the minimum principle for harmonic functions. For subharmonic functions, the minimum principle no longer holds, in general, and indeed a holomorphic curve u : G → Cn may well be tangent to Sr2n−1 from the outside. The simplest example would be a complex line tangent to Sr2n−1 . But even with the additional assumption that u pass through 0 ∈ Cn , such tangencies can occur, as the next exercise shows. Exercise I.4.15. Let u : (G, 0) → (C2 , 0) be the holomorphic curve defined by u(z) = (z, sin 4z). Show that the function |u|2 has isolated local minima. ♦ Figure I.3 illustrates our discussion of maximum vs. minimum principle, again with reduced dimensions. possible u(G) Sr

Sr u(G) not possible

Figure I.3: Maximum vs. minimum principle.

18

Chapter I. Gromov’s Nonsqueezing Theorem

I.4.2 Symplectic energy of holomorphic curves Let u : G → Cn be a holomorphic map. The partial derivatives of u with respect to the real coordinates x, y can be written as ux = T u(∂x ),

uy = T u(∂y ).

Then, u∗ ω = u∗ ω(∂x , ∂y ) dx ∧ dy = ω(ux , uy ) dx ∧ dy = ω(ux , J0 ux ) dx ∧ dy, where we have used Exercise I.3.9. Combining this with Exercise I.2.1, we can rewrite ω(ux , J0 ux ) as ω(ux , J0 ux ) = hux , ux i = huy , uy i =


1 1 hux , ux i + huy , uy i =: |∇u|2 . 2 2

This means that the energy integral in the monotonicity lemma (Theorem I.4.1) can be written as Z Z 1 ∗ |∇u|2 dx dy. (I.8) u ω= 2 u−1 (Br ) u−1 (Br ) Remark I.4.16. The identity theorem for holomorphic functions [J¨ anich 1999] implies that the zeros of u, and likewise T u, do not accumulate at any point in G. By Exercise I.3.9, u is an immersion outside the zeros of T u. Notation I.4.17. Instead of h . , . i we use the notation h . , . iR2n for the standard Euclidean inner product on R2n when we want to avoid confusion with other metrics. When we read vectors v, w ∈ R2n as n-tuples v = (v1 , . . . , vn ), w = (w1 , . . . , wn ) of complex numbers, then hv, wiR2n =

n X

Re(vj wj ).

j=1

Set h . , . iu := hT u( . ), T u( . )iR2n . By the preceding remark, this is a Riemannian metric (i.e. a pointwise inner product on tangent spaces) on G\{T u = 0}, and completely degenerate on the discrete set {T u = 0}. Nonetheless, we shall usually speak of the ‘metric’ h . , . iu . The corresponding ‘norm’ on tangent vectors (likewise degenerate at the points where T u = 0) will be denoted by | . |u . Exercise I.4.18. Show that h . , . iu is conformally equivalent to h . , . iR2 , specifically, h . , . iu = |ux |2 h . , . iR2 . Show further that u∗ ω equals the ‘area form’ |ux |2 dx ∧ dy of this metric.

♦

19

I.4. The monotonicity lemma

Remark I.4.19. Let G0 = G \ {T u = 0}, so that u|G0 is an immersion. From a course on elementary differential geometry [Millman & Parker 1977] or analysis [Br¨ocker 1992] you may know that the area element of this immersion, which computes the area on u(G0 ) with respect to the metric induced by h . , . iR2n , is √ gu dx ∧ dy, where gu is the determinant of the Gram matrix of metric coefficients   hux , ux i hux , uy i huy , ux i huy , uy i (which here equals |ux |2 times the unit matrix). Thus, computing area on u(G0 ) w.r.t. h . , . iR2n is the same as computing area on G0 w.r.t. h . , . iu . The same goes for lengths of curves. By Exercise I.4.18, the first way of writing the symplectic energy in (I.8) can be interpreted as an area integral (outside isolated singularities). The second way of writing the symplectic energy is known in the theory of minimal surfaces as the Dirichlet integral [Jost 1994]. So you may also find the symplectic energy referred to as symplectic area or Dirichlet energy. The analogous statement for J-holomorphic curves will be discussed in Section I.8.4. Example I.4.20. We want to verify the monotonicity lemma for the holomorphic function u : C → C, z 7→ az k , with a ∈ C∗ and k ∈ N := {1, 2, 3, . . .} given. We p k −1 0 have u (Br ) = Br0 with r = r/|a|. The complex derivative of u is uz (z) = akz k−1 . By Exercise I.3.9 we have 1 |∇u|2 = |uz |2 = |a|2 k 2 |z|2k−2 . 2 We then compute Z 1 |∇u|2 dx dy 2 u−1 (Br )

=

2 2

2π

Z

r0

Z

ρ2k−1 dρ dϕ

|a| k

0

0

k √ 2 2 1 2k ρ= r/|a| = 2π|a| k ρ ρ=0 2k = kπr2 ,

which is greater than or equal to πr2 . Exercise I.4.21. Express ω = dx ∧ dy in polar coordinates (ρ, ϕ) and show that the pull-back of ω under u equals u∗ ω = |a|2 k 2 ρ2k−1 dρ ∧ dϕ. ♦

I.4.3 Proof of the monotonicity lemma Recall that in the monotonicity lemma (Theorem I.4.1) we are considering a proper holomorphicRmap u : (G, 0) → (Cn , 0), and our aim is to show that for any r > 0 the integral u−1 (Br ) u∗ ω is bounded from below by πr2 .

20

Chapter I. Gromov’s Nonsqueezing Theorem

Let f : G → R0+ be the function defined by f (z) := |u(z)|. This function is continuous on G and smooth on G \ {u = 0}. Define  K := z ∈ G : f (z) = 0 or (f (z) 6= 0 and dz f = 0) . The set {u = 0} ⊂ u−1 (Dr ) is finite by Remark I.4.16 and the properness of u (or the specific assumption that u−1 (Dr ) be compact). Moreover, the set of critical values of f has Lebesgue measure zero by Sard’s theorem [Milnor 1965]. It follows that
f (K) ∩ [0, r] = f u−1 (Dr ) ∩ K is a zero set. Write R := Rr := (0, r) \ f (K) for the set of regular values of f in the open interval (0, r). Note I.4.22. The set K is closed, and u−1 (Dr ) is compact thanks to u being proper. It follows that f (K) ∩ [0, r] is a compact zero set, and hence R is open. This implies that R can be written as the disjoint union of countably many open intervals, R = ∪ν (bν , cν ). Exercise I.4.23. Why does a disjoint union of intervals in the real line always constitute a countable (finite or infinite) collection? ♦ The preimage f −1 (R) does not contain any critical points of u, since the rank of T u is zero in those points (Exercise I.3.9), and dz f = Tu(z) | . | ◦ Tz u. Thus, f −1 (R) ⊂ G \ {T u = 0}, and h . , . iu is a nondegenerate Riemannian metric on f −1 (R). For each t ∈ R, the preimage f −1 (t) is a compact 1-dimensional submanifold
of G, and hence8 a finite collection of embedded circles. Then, u f −1 (t) is a collection of immersed circles in Cn . We write `(t) for the total Euclidean length of this latter collection of circles. On the interval [0, r] we define the strictly increasing function t 7→ a(t) ∈ R0+ as the area of u−1 (Bt ) with respect to the singular area form u∗ ω (Exercise I.4.18): Z u∗ ω. a(t) := u−1 (Bt )

Our aim is to show that a(r) ≥ πr2 . Exercise I.4.24. Convince yourself that the function t 7→ a(t) is indeed strictly ♦ increasing. What role is being played by the properness assumption on a? Lemma I.4.25 and Proposition I.4.27 below will be proved in the next section. 8 See

[Milnor 1965] for the classification of compact 1-manifolds.

21

I.5. Isoperimetric inequalities

Lemma I.4.25. The function a is differentiable at all points of R, and it satisfies a0 (t) ≥ `(t) there. Exercise I.4.26. Show by an example that without the holomorphicity assumption on u, the function t 7→ a(t) need not be continuous in the points t ∈ f (K). What can you say about the continuity of a when u is indeed holomorphic? Exercise I.4.13 is of some relevance here. ♦ Proposition I.4.27 (Isoperimetric Inequality). For all t ∈ R, we have a(t) ≤

`2 (t) . 4π

Combining these two statements, we conclude that p a0 (t) ≥ `(t) ≥ 2 πa(t) for all t ∈ R, and hence

√ d p a0 (t) a(t) = p ≥ π for all t ∈ R. dt 2 a(t)

(Notice that the condition u(0) = 0 guarantees that a(t) > 0 for all t > 0.) Using this inequality on a component (bν , cν ) of R (see Note I.4.22), we find p p √ a(cν ) − a(bν ) ≥ π(cν − bν ), and further  √ X X p p p p p
√ a(r) = a(r) − a(0) ≥ a(cν ) − a(bν ) ≥ π cν − bν = πr, ν

ν

where for the first inequality we have used the monotonicity of a; for the last equality, the fact that R ⊂ (0, r) is of full measure. This proves the monotonicity lemma.

I.5

Isoperimetric inequalities

We keep considering the function f = |u| : G → R+ 0 . In terms of the metric h . , . iu , the functions ` and a on R can be described as `(t)

=

a(t)

=

total length of the circles f −1 (t) w.r.t. h . , . iu ,
area of f −1 [0, t] w.r.t. h . , . iu .

Define ∇u f as the gradient of f with respect to h . , . iu on f −1 (R), that is, df = h∇u f, . iu = hT u(∇u f ), T u( . )iR2n .

(I.9)

Observe that the vector field ∇u f does not have any singularities on f −1 (R).

22

Chapter I. Gromov’s Nonsqueezing Theorem

Note I.5.1. The differential of the norm function | . |R2n : R2n \ {0} −→ R+ does not increase length, that is,
Tz | . |R2n (v) ≤ |v|R2n for z ∈ R2n \ {0} and v ∈ Tz R2n . This follows by considering a smooth curve = v, and computing ˙ γ : (−ε, ε) → R2n \ {0} with γ(0) = z and γ(0) d hγ, γi ˙ R2n ≤ |γ| ˙ R2n . |γ|R2n = dt |γ|R2n Lemma I.5.2. ∇u f u ≤ 1. Proof. Using f = | . |R2n ◦ u and the preceding note, as well as (I.9), we compute u 2

∇ f = df ∇u f ≤ T u ∇u f 2n = ∇u f , u R u which implies the claim.



I.5.1 Proof of Lemma I.4.25 We consider the function a on a component (b, c) of R. Choose a reference point t0 ∈ (b, c). Write φs for the flow of the vector field X := ∇u f /|∇u f |2u on f −1 (R). This flow sends level sets of f to level sets, since df (X) = h∇u f, Xiu ≡ 1. In particular, the number of circles in the collection f −1 (t) is constant as t ranges over a connected component of R. For ease of notation we assume that this number is one for the component (b, c). Choose a regular parametrisation γ0 : S 1 = R/2πZ → f −1 (t0 ) of the embed1 ded circle f −1 (t0 ) in G, that is, γ00 (θ)
6= 0 for all θ ∈ S . We can then parametrise −1 the open subset G(b,c) := f (b, c) of G by Φ:

(b − t0 , c − t0 ) × S 1 (s, θ)

−→ 7−→

G(b,c)
Φ(s, θ) := φs γ0 (θ) .

Notice that θ 7→ γs (θ) := Φ(s, θ) gives a parametrisation of the circle f −1 (t0 + s). Computing length and area in G(b,c) with respect to h . , . iu is the same as computing it in the parameter space (b−t0 , c−t0 )×S 1 with respect to the pull-back metric h . , . iΦ := hT Φ( . ), T Φ( . )iu .

23

I.5. Isoperimetric inequalities We determine the metric coefficients of this new metric: −2 h∂s , ∂s iΦ = hX, Xiu = ∇u f u ; D d E h∂s , ∂θ iΦ = X, γs = 0, dθ u since X is a multiple of ∇u f , and γs parametrises a level set of f ; d 2 h∂θ , ∂θ iΦ = γs . dθ u By Lemma I.5.2, the Gram determinant gΦ of this metric satisfies d 2 gΦ ≥ γs . dθ u The area a(t), for t ∈ (b, c), is then given by 2π

Z

t−t0

Z

a(t) = a(t0 ) + 0

p

gΦ (s, θ) ds dθ.

0

This shows that a is differentiable with derivative Z 2π p Z 2π d a0 (t) = gΦ (t − t0 , θ) dθ ≥ γt (θ) dθ = `(t), dθ u 0 0 which concludes the proof of Lemma I.4.25.



I.5.2 Symplectic action and energy of loops In order to complete the proof of the monotonicity lemma (Theorem I.4.1), it remains to establish the isoperimetric inequality (Proposition I.4.27). The next exercise serves as a preparation. Exercise I.5.3. Let γ : S 1 = R/2πZ → R2n be a smooth loop and λ a primitive 1-form for the standard symplectic form ω on R2n , i.e. dλ = ω. (a) Show that the symplectic action Z A(γ) :=

λ γ

is independent of the choice of λ and remains unchanged under orientationpreserving reparametrisations of γ. If G ⊂ C is a bounded domain with smooth boundary ∂G = S 1 , and u : G → R2n a smooth map with u|∂G = γ, then Z A(γ) = u∗ ω. G

24

Chapter I. Gromov’s Nonsqueezing Theorem

(b) Use the primitive n

λ=

1X xj dyj − yj dxj ) 2 j=1

to show that A(γ) can be computed as Z 1 2π A(γ) = − hγ(t), iγ(t)i ˙ R2n dt. 2 0

♦

Recall that the length 2π

Z

|γ(t)| ˙ R2n dt

L(γ) := 0

of a smooth curve γ : S 1 → R2n does not change under reparametrisations. This L(γ) is the ‘distance travelled’ rather than, in general, the geometric length of the loop γ(S 1 ) ⊂ R2n . The energy of γ is Z 2 1 2π γ(t) ˙ R2n dt. E(γ) := 2 0 Beware that this energy does depend on the parametrisation of the loop. We now prove an action-energy inequality and a more classical formulation of the isoperimetric inequality, whence it is only a small step to Proposition I.4.27. Proposition I.5.4. Let γ : S 1 → R2n be a smooth loop. Then the following inequalities hold: (a) Action-energy inequality: |A(γ)| ≤ E(γ). (b) ‘Classical’ isoperimetric inequality: |A(γ)| ≤

L2 (γ) . 4π

Proof. (a) Identify R2n with Cn , and expand γ as a Fourier series γ(t) =

∞ X

cµ eiµt

µ=−∞

with cµ ∈ Cn . Then, A(γ)

= = =

Z 1 2π hγ(t), iγ(t)i ˙ − R2n dt 2 0 Z 1 X 2π hcµ eiµt , νcν eiνt iR2n dt 2 µ,ν 0 X π ν|cν |2R2n , ν

25

I.5. Isoperimetric inequalities since 2π

Z

he

iµt

,e

iνt

iR2 dt =

0

(

2π

Z

Re e

i(µ−ν)t




dt =

0

2π 0

for µ = ν, otherwise.

Similarly, we can compute the energy of γ, to obtain X E(γ) = π ν 2 |cν |2R2n . ν

The action-energy inequality is an immediate consequence. (b) We first consider the case that γ is actually a regular curve (in other 6= 0 for all t ∈ S 1 . Since A and L do not depend on ˙ words, an immersion), i.e. γ(t) the parametrisation, we may assume that γ is of constant speed |γ| ˙ R2n ≡ L(γ)/2π. Then,  L(γ) 2 1 L2 (γ) E(γ) = · 2π · = , 2 2π 4π and the isoperimetric inequality follows directly from the action-energy inequality. In the general case, we choose a sequence (ak )k∈N in Cn with ak → 0 as k → ∞, and such that each ak does not lie in the compact set  −it ie γ(t) ˙ ∈ Cn : t ∈ S 1 . Then, γk (t) := γ(t) + ak eit defines a sequence of immersed loops that converges to γ in the C ∞ -topology, and the isoperimetric inequality for γ follows from that for the γk by passing to the limit k → ∞.  Note I.5.5. When γ is a simple closed curve in the plane R2 , then A(γ) equals the area of the region bounded by γ. This yields the isoperimetric inequality known from elementary differential geometry. Remark I.5.6. The proof of the isoperimetric inequality via Fourier expansions can be traced back to A. Hurwitz (1902); cf. [Blaschke 1930]. Proof of Proposition I.4.27. Write γ1 , . . . , γN for the collection of loops making up u|f −1 (t) . Notice that the γk are regular curves for t ∈ R. The proof of the inequality a(t) ≤ `2 (t)/4π is now a straightforward computation: Z a(t) = u∗ ω −1 u (Bt ) Z = u∗ λ f −1 (t)

=

N Z X k=1

λ = γk

N X

A(γk )

k=1

≤

N N 2 1 X 2 1 X L (γk ) ≤ L(γk ) 4π 4π

=

`2 (t) , 4π

k=1

k=1

26

Chapter I. Gromov’s Nonsqueezing Theorem

using the theorem of Stokes, the classical isoperimetric inequality, and, in the last line, the fact that the γk are regular (so that L(γk ) is actually the geometric length). 

I.6

Holomorphic curves are minimal surfaces

The observations and results in this section will not be used in the rest of the text. However, they are of independent interest and serve to place the discussion of holomorphic curves in a differential geometric context. For a concise introduction to minimal surfaces, see [Jost 1994] or the class notes [White 2013]. The isoperimetric inequality provides an upper estimate on the area of a holomorphic curve-with-boundary in terms of its boundary length. Thus, it ought not to come as a surprise that holomorphic curves permit an interpretation as minimal surfaces, as we shall now explain. The monotonicity lemma is an expression of this minimal surface property. In this section, h . , . i and | . | denote the standard metric and norm, respectively, on R2n . For v, w vectors in R2n , write area(v, w) for the area of the parallelogram spanned by v and w. Exercise I.6.1. Verify the formula s area(v, w) =

det



hv, vi hv, wi hw, vi hw, wi

 ♦

As before, ω denotes the standard symplectic form on R2n , and J0 the almost complex structure corresponding to the identification of R2n with Cn . Lemma I.6.2. We have the inequality ω(v, w) ≤ area(v, w), where equality holds if and only if (i) v, w are linearly dependent over R, or (ii) v, w span a complex line, with v, w defining the positive orientation. Proof. Both sides of the inequality (and the conditions for equality) remain unchanged when a multiple of v is added to w. We may therefore assume that w is orthogonal to v. Under this assumption, ω(v, w) = hJ0 v, wi ≤ |v| · |w| = area(v, w), with equality if and only if one of v, w is zero, or w is a positive multiple of J0 v.



27

I.7. The space of almost complex structures

Now consider a bounded domain G ⊂ C with smooth boundary, and let v : G → R2n be a smooth map. Write gv for the Gram determinant of v. Notice that
√ gv = area T v(∂x ), T v(∂y ) . With the lemma this shows that v ∗ ω(∂x , ∂y ) ≤

√

gv ,

with equality if (not ‘if and only if’) v is holomorphic when regarded as a map into Cn . The case of equality was shown earlier in Exercise I.4.18. From this observation we easily deduce the following minimality statement for holomorphic curves. Recall that the area of v is, by definition, Z √ area(v) := gv dx ∧ dy; G

if v is an injective immersion, this equals the area of v(G). Theorem I.6.3. Let G ⊂ C be a bounded domain with smooth boundary. Let u : G → Cn be a holomorphic map, and v : G → Cn a smooth one with u|∂G = v|∂G . Then, area(u) ≤ area(v), with equality if v is holomorphic. Proof. We compute √

Z area(u) =

Z gu dx ∧ dy

∗

=

u∗ λ

u ω =

G

G

Z

∂G

v∗ λ =

= ≤

Z

Z∂G √

Z

v∗ ω G

gv dx ∧ dy = area(v),

G

with equality if v is holomorphic.



See Exercise I.8.22 in Section I.8.4 for the generalisation of this result to J-holomorphic curves.

I.7

The space of almost complex structures

One of the tasks left over from Section I.3.3 is to construct an almost complex structure on R2n that coincides with J on Φ(Dr ), and with the standard almost complex structure outside a small neighbourhood, say Φ(Br+ε ), of Φ(Dr ). The required interpolation procedure for almost complex structures is based on the fact, which will be established in this section, that the space of compatible complex structures on a given symplectic vector space is diffeomorphic to a convex space.

28

Chapter I. Gromov’s Nonsqueezing Theorem

I.7.1 Linear algebra of complex structures On the vector space R2n we consider the standard inner product h . , . i and symplectic form ω. As before, we write J0 for the isomorphism of R2n corresponding to multiplication by i under the identification R2n = Cn ; cf. footnote 2 on page 2. For the following discussion we want to think of linear maps of R2n as (2n × 2n)matrices with respect to the canonical basis. Then,  0 1   J0 =   



−1 0 ..

. 0 1

   .  −1 0

Notice that J0−1 = J0t = −J0 . We write E for the unit matrix. Definition I.7.1. A (linear) complex structure on the vector space R2n is an automorphism J : R2n → R2n with J 2 = −E. A complex structure J is said to be compatible with ω if ω(Jv, Jw) = ω(v, w) for all v, w ∈ R2n and ω(v, Jv) > 0 for all v ∈ R2n \ {0}. Notation I.7.2. We write J = J (ω) for the space of ω-compatible linear complex structures on R2n . By Exercise I.2.1, J0 is compatible with ω. Recall from Section I.3.2 that the compatibility condition is equivalent to saying that hv, wiJ := ω(v, Jw) defines an inner product on R2n , for which J is an orthogonal transformation. Also, we have h . , . iJ0 = h . , . i, or ω(v, w) = hv, −J0 wi, which in matrix notation becomes ω(v, w) = −vt J0 w. Notation I.7.3. We write Km×n for the vector space of (m × n)-matrices over the field K. Exercise I.7.4. Given a vector v ∈ R2n , we write vC ∈ Cn for the corresponding complex vector under the natural identification of R2n with Cn . Let A ∈ R2n×2n be a matrix that commutes with J0 . Show the existence of a unique matrix AC ∈ Cn×n such that (Av)C = AC vC for all v ∈ R2n . ♦

I.7. The space of almost complex structures

29

Exercise I.7.5. Show that a real (2n × 2n)-matrix J defines a complex structure if and only if the matrix Z := −J0 J satisfies J0 Z = Z −1 J0 .

(I.10)

Show further that such a J is compatible with ω if and only if Z is symmetric (Z t = Z) and positive definite (which we write as Z > 0). Observe that Z is the matrix representing the inner product h . , . iJ , that is, vt Zw = hv, wiJ .

♦

We denote the space of such matrices by  Z := Z ∈ R2n×2n : Z t = Z, Z > 0, J0 Z = Z −1 J0 . Then the preceding exercise says that we have a bijection ∼ =

J −→ Z,

J 7−→ −J0 J.

Exercise I.7.6. Show that Z can equivalently be described as  Z = Z ∈ R2n×2n : Z t = Z, Z > 0, Z t J0 Z = J0 . The third defining condition is equivalent to ω(Zv, Zw) = ω(v, w). This identifies Z as the space of positive definite symmetric and symplectic matrices. ♦ Remark I.7.7. We shall see presently that the spaces J and Z are manifolds, and the map J 7→ −J0 J a diffeomorphism between them. Our aim will be to transform the space Z in such a way that (I.10) is turned into a linear condition.

I.7.2

The Cayley transformation as a M¨ obius transformation

As you recall from complex analysis, any M¨ obius transformation z 7−→

az + b , cz + d

where a, b, c, d are complex numbers with ad − bc 6= 0, defines an automorphism ˆ := C ∪ {∞}, (i.e. a biholoof the extended complex plane (or Riemann sphere) C ˆ morphism of C onto itself) and these M¨obius transformations constitute the full ˆ The M¨obius transformation group of biholomorphisms of C. fC : z 7−→ is known as the Cayley transformation.

1−z =: w 1+z

30

Chapter I. Gromov’s Nonsqueezing Theorem

ˆ i.e. it squares to the identity Exercise I.7.8. Verify that fC is an involution of C, ♦ map, and fC (±i) = ∓i. Proposition I.7.9. The Cayley transformation restricts to a diffeomorphism ∼ =

{Re(z) > 0} −→ {|z| < 1} = Int(D) from the open right half-plane in C to the open unit disc, and it transforms the involution z 7→ 1/z into w 7→ −w. Proof. In the fraction |1 − z|/|1 + z|, numerator and denominator can be interpreted (for z ∈ C) as the Euclidean distance from the point z to the points ±1, respectively. It follows that |1 − z| = 1 if and only if Re(z) = 0 or z = ∞. |1 + z| Since the point z = 1 in the right half-plane is sent to 0 by the Cayley transformation, the first statement follows. For the second claim, compute that  1 fC ◦ z 7→ ◦ fC−1 (w) = −w.  z Exercise I.7.10.

(a) Find a M¨obius transformation that sends the subset ˆ {Im z ≥ 0} ∪ {∞} ⊂ C

biholomorphically onto the closed unit disc D. (b) Show the following properties of M¨obius transformations: ˆ fixes at most two points, unless it is (i) Any M¨obius transformation of C the identity. ˆ there is a unique M¨ (ii) Given three distinct points z1 , z2 , z3 ⊂ C, obius transformation with z1 7→ 0, z2 7→ 1, and z3 7→ ∞. (iii) Given real numbers z1 < z2 < z3 , where z3 is allowed to equal ∞, the M¨obius transformation as in (ii) is of the form z 7→ (az + b)/(cz + d), with a, b, c, d ∈ R and ad − bc > 0. (c) Use the Schwarz reflection principle9 to show that any biholomorphism of the closed upper half-plane H := {z ∈ C : Im z ≥ 0} extends to a biholomorˆ phism of C (and further to one of C). ♦ Note I.7.11. From this exercise we conclude that the automorphism group Aut(D) of D is of real dimension 3, and it acts simply transitively on ordered triples of boundary points. In other words, given any two such triples, there is a unique automorphism of D that sends one to the other. 9 If

you are not familiar with this result from complex analysis, take a peek at Section II.1.1.

I.7. The space of almost complex structures

31

I.7.3 The Cayley transformation on matrices Inside the vector space of symmetric and complex anti-linear (2n × 2n)-matrices, we define W as the open subset of matrices having operator norm (with respect to the Euclidean norm on R2n ) smaller than 1:  W := W ∈ R2n×2n : W t = W, |W | < 1, J0 W = −W J0 . Thus, W is in a natural way a manifold. Notice that for any W ∈ W we have sW ∈ W for all s ∈ [0, 1]. Notation I.7.12. We tend to write the operator norm on matrices — better: the linear maps on finite-dimensional normed vector spaces defined by those matrices — simply as | . |. For linear operators on infinite-dimensional normed spaces we prefer the notation k . k. Occasionally, when arguments involve several different norms, we clarify which norms are operator norms by using k . kop . Exercise I.7.13. Show that the dimension of the vector space  W ∈ R2n×2n : W t = W, J0 W = −W J0 , and hence that of the manifold W, equals n(n + 1). To this end, think of W as a block matrix made up of (2 × 2)-matrices. Because of the symmetry condition W t = W , we need only consider the blocks on and below the diagonal. Translate the equation J0 W = −W J0 into conditions on the entries of each block. ♦ When the complex anti-linearity condition J0 W = −W J0 is rewritten as −W = −J0 W J0 , one recognises it as the defining equation for the matrices in Z, with Z 7→ Z −1 replaced by W 7→ −W . So the next proposition should not come as too big a surprise. Proposition I.7.14. The map fC : Z 7−→ (E − Z)(E + Z)−1 =: W defines a diffeomorphism from Z onto W. Proof. For Z ∈ Z, the matrix E+Z is symmetric and positive definite, so (E+Z)−1 is defined. Notice that E + Z commutes with E − Z, hence so does (E + Z)−1 , i.e. the order of factors in the definition of fC is not important. Using this observation and Z t = Z, one easily checks that W t = W . Next we want to check that |W | < 1. From the polarisation identity |(E + Z)v|2 − |(E − Z)v|2 = 4hv, Zvi we obtain |w|2 − |W w|2 = 4hv, Zvi for v = (E + Z)−1 w.

(I.11)

From Z > 0 it follows that |W | < 1. With the next exercise, we have shown fC (Z) ⊂ W.

32

Chapter I. Gromov’s Nonsqueezing Theorem

Exercise I.7.15. Observe that Z −1 is likewise symmetric and positive definite. ♦ Show that J0 (E + Z)−1 = (E + Z −1 )−1 J0 , and further that J0 W = −W J0 . In order to show that fC is actually a bijection onto W, we describe an explicit inverse. For |W | < 1, the matrix E + W defines an injective map and hence is invertible. With the observation at the beginning of this proof, one computes that the same map fC , but now defined on W, that is, fC : W 7−→ (E − W )(E + W )−1 =: Z, defines the desired inverse, as was to be expected from Exercise I.7.8. It only remains to check that fC (W) ⊂ Z. From (I.11) we see that Z > 0 when |W | < 1. Symmetry of Z and the equation J0 Z = Z −1 J0 one checks as before. In order to show that Z is a manifold and fC : Z → W a diffeomorphism, we observe that the map Z 7→ (E − Z)(E + Z)−1 is a diffeomorphism between the open neighbourhoods {Z t = Z, Z > 0} and {W t = W, |W | < 1} of Z and W, respectively, inside the vector space of all symmetric (2n × 2n)-matrices. Since W  is a submanifold, so is Z. Remark I.7.16. Similarly, the map J → Z, J → 7 −J0 J is the restriction to J of the diffeomorphism A 7→ −J0 A defined on the vector space R2n×2n . Summarising this discussion, we have a composite diffeomorphism fW :

J J

−→ 7−→

Z −J0 J

−→ 7−→

W fC (−J0 J)

from the manifold J into the convex space W, with fW (J0 ) the zero matrix.

I.7.4

Interpolating almost complex structures

We can now address the interpolation problem. An almost complex structure on R2n is simply a smooth map R2n 3 p 7→ Jp ∈ J , that is, a family of matrices Jp ∈ J with matrix entries varying smoothly in p. In the notation of Section I.3.3, choose ε > 0 small enough such that r + ε < r0 . Then the almost complex structure J = Φ∗ J0 is defined on Φ(Br+ε ). Lemma I.7.17. There is an almost complex structure Jˆ on R2n , compatible with ω, that coincides with J on Φ(Dr ), and with J0 on R2n \ Φ(Br+ε ). Proof. Choose a smooth function χ : R2n → [0, 1] with χ ≡ 1 on Φ(D
r ), and χ ≡ 0 −1 on R2n \ Φ(Br+ε ); see the exercise below. Then, Jˆ := fW χ · fW (J) will do.  We shall continue to write J rather than Jˆ for this extended almost complex structure.

33

I.8. A moduli space of J-holomorphic discs Exercise I.7.18. Starting from the smooth function ( e−1/x for x > 0, R 3 x 7−→ 0 for x ≤ 0,

construct a smooth function R → [0, 1] that is identically 0 on {x ≤ 0} and identically 1 on {x ≥ ε}. Then describe the construction of a function χ with the ♦ desired properties.

I.8

A moduli space of J-holomorphic discs

The final claim from Section I.3.3 that needs to be validated is the existence of a J-holomorphic disc u ˆ with the described properties. In the present section, we set up the relevant moduli space and initiate a discussion of its basic properties. Presumably you know the Schwarz lemma from a course on complex analysis, but since it plays a crucial role in several of our arguments, we begin by recalling its proof and some immediate consequences.

I.8.1

The Schwarz lemma

Write B ⊂ C for the open unit disc, and Dr ⊂ C for the closed disc of radius r. Proposition I.8.1 (Schwarz Lemma). Let f : B → B be a holomorphic map with f (0) = 0. Then, |f (z)| ≤ |z| for all z ∈ B, and |f 0 (0)| ≤ 1. If |f (z0 )| = |z0 | for some 0 6= z0 ∈ B, or if |f 0 (0)| = 1, then f is a rotation, that is, f (z) = eiθ z for some θ ∈ R. Proof. The power series expansion of f about 0 shows that f can be written as f (z) = zg(z) with g : B → C holomorphic. By the maximum principle, the maximum of |g| on Dr , for any given r ∈ (0, 1), is attained on ∂Dr , i.e. for all z ∈ Dr : |g(z)| ≤ max |g(ζ)| = max |ζ|=r

|ζ|=r

|f (ζ)| < 1/r. r

By taking the limit r → 1, we deduce that |g(z)| ≤ 1 for all z ∈ B, which means |f (z)| ≤ |z| and |f 0 (0)| = |g(0)| ≤ 1. If |f (z0 )| = |z0 | for some 0 6= z0 ∈ B, then |g(z0 )| = 1; if |f 0 (0)| = 1, then |g(0)| = 1. In either case, the maximum principle applied to g shows that g ≡ eiθ for some θ ∈ R.  Corollary I.8.2. Every biholomorphic map f : B → B with f (0) = 0 is a rotation. Proof. With the Schwarz lemma applied to f and f −1 we have |f (z)| ≤ |z| = |f −1 (f (z))| ≤ |f (z)|, i.e. |f (z)| = |z| for all z ∈ B. Then apply the second part of the Schwarz lemma. 

34

Chapter I. Gromov’s Nonsqueezing Theorem

I.8.2 Automorphisms of the unit disc Every M¨obius transformation that sends B onto B is obviously a biholomorphism of B. We now show that there are no others. We denote the rotation z 7→ eiθ z by Rθ , and for given z0 ∈ B we define the M¨ obius transformation fz0 by fz0 (z) =

z − z0 . 1 − z0z

The map fz0 indeed restricts to a biholomorphism of B. To see this, it suffices to observe that fz0 (z0 ) = 0, and that |fz0 (z)|2 =

z − z0 z − z0 · 1 − z 0 z 1 − z0 z

equals 1 for |z| = 1. Proposition I.8.3. The automorphism group Aut(B) of B is made up entirely of M¨ obius transformations, specifically,  Aut(B) = Rθ ◦ fz0 : z0 ∈ B, θ ∈ R/2πZ . Proof. Given f ∈ Aut(B), set z0 := f −1 (0). Then, f ◦ fz−1 is an automorphism of 0 B fixing 0, and hence a rotation.  Remark I.8.4. The maps Rθ ◦ fz0 may also be regarded as automorphisms of D, and they constitute the full group Aut(D). In particular, any automorphism of B is the restriction to B of an automorphism of D = B. Exercise I.8.5. Use the description of Aut(B) to show that the automorphism group of the open upper half-plane {Im z > 0} is made up of the real M¨ obius transformations z 7→ (az + b)/(cz + d), a, b, c, d ∈ R, ad − bc > 0. This shows that the extension statement of Exercise I.7.10 (c) also holds for biholomorphisms of the open half-plane. ♦ Exercise I.8.6. Show that Aut(D) acts simply transitively on pairs (z0 , eiθ ) made up of a point z0 ∈ Int(D) and a point eiθ ∈ ∂D; cf. Note I.7.11. ♦

I.8.3 The moduli space M Recall that the target space in Gromov’s nonsqueezing theorem is the cylinder Z = B × Cn−1 in C × Cn−1 . We write the coordinates on this ambient space as (w, x + iy) with x = (x2 , . . . , xn ), y = (y2 , . . . , yn ) in Rn−1 . In the boundary of Z we consider the family of n-dimensional cylinders, t ∈ Rn−1 , Lt := ∂B × Rn−1 × {y = t}. Exercise I.8.7. Compute that the standard symplectic form ω pulls back to the zero form on Lt under the inclusion map Lt → Cn . ♦

I.8. A moduli space of J-holomorphic discs

35

Definition I.8.8. A submanifold L ⊂ (M, Ω) inside a symplectic manifold (M, Ω), with the property that dim L = dim M/2 and Ω|T L = 0 is called Lagrangian. Exercise I.8.9. The condition Ω|T L = 0 by itself forces dim L ≤ dim M/2.

♦

Notation I.8.10. The moduli space M is the set of J-holomorphic discs u : (D, ∂D) −→ (R2n , ∂Z) with the following properties: (M1) (Lagrangian boundary condition) For every u ∈ M there is a t ∈ Rn−1 , called the level of u, such that u(∂D) ⊂ Lt . (M2) (homotopy condition) The boundary map u|∂D : ∂D −→ Lt is (smoothly)10 homotopic in Lt to
eiθ 7−→ eiθ , 0 + it . (M3) (three-point condition) For k = 0, 1, 2 we have u(ik ) ∈ {ik } × Rn−1 × {y = t}.

Remark I.8.11. The Lagrangian boundary condition (M1) is essential for the analytical set-up. Thanks to the homotopy condition (M2), as we shall see in Lemma II.1.18, for every u ∈ M the boundary map u|∂D is an embedding, and in particular injective. Therefore, by Note I.7.11, the three-point condition (M3) fixes the parametrisation of u, that is, for u ∈ M and ϕ ∈ Aut(D), we have u ◦ ϕ ∈ M if and only if ϕ = idD . The boundary of any u ∈ M being embedded is crucial for the transversality theory in Chapter V via the properties of the set of so-called injective points. Remark I.8.12. Occasionally, we identify the target space R2n with Cn , even though it is now equipped with the almost complex structure J rather than J0 . This is often convenient, for instance when using complex coordinates for describing the boundary behaviour of our J-holomorphic discs u; indeed, u(∂D) lies in a region where J = J0 . 10 Under the assumption of a continuous homotopy, we shall see in Lemma II.1.18 that u| ∂D is an embedding. One can then also find a smooth homotopy, so the smoothness assumption is no restriction.

36

Chapter I. Gromov’s Nonsqueezing Theorem

Here is the central theorem of this course. With this theorem (rather, its immediate corollary) and the energy calculation in the next section, we shall have completed the proof of Gromov’s nonsqueezing theorem (Section I.3.3). The meaning of ‘generic almost complex structure’ will be elucidated in Section V.2.4. If you are unfamiliar with the concept of mapping degree, we ask you to bear with us until Section V.7.2. Theorem I.8.13. For a generic choice of the ω-compatible almost complex structure J (Lemma I.7.17), the moduli space M of J-holomorphic discs is a smooth manifold of dimension 2n − 2. The evaluation map ev :

M×D (u, z)

−→ 7−→

R2n u(z)

is smooth, proper, and it takes values in the closed cylinder Z. Regarded as a map into Z, the evaluation map ev has degree (over Z2 ) equal to 1: deg2 ev = 1. Some basic properties of the evaluation map will be worked out in Chapter II; they are summarised in Section II.1.6. The properness of ev is stated again as Theorem II.2.7. Some hard analysis is required for its proof, which will only be complete by the end of Chapter III. Chapters IV and V are concerned with establishing that M is a manifold. In the corollary, Φ denotes the purported embedding Br0 → Z from the proof of Gromov’s theorem. Corollary I.8.14. Possibly after reparametrising with an element of Aut(D), there is a J-holomorphic disc u ˆ : (D, ∂D) → (Z, ∂Z) with u ˆ(0) = Φ(0). Proof. This follows from ev being of nonzero degree (see Note V.7.8), and the reparametrisation being possible by Exercise I.8.6.  Remark I.8.15. As we shall see (Section II.1.5 and Exercise V.3.1), the discs u ∈ M are embeddings near the boundary. We shall not be making any claims about them being globally embedded or pairwise disjoint. In dimension 4, such statements can be proved using positivity of intersection of J-holomorphic curves and an intersection theory adapted to discs; see [Geiges & Zehmisch 2010]. This also implies that M is connected (in dimension 4).

I.8.4 Symplectic energy of J-holomorphic discs Here we show that the J-holomorphic discs in M have the energy property (D2) mentioned in Section I.3.3.

37

I.8. A moduli space of J-holomorphic discs Lemma I.8.16. Every u ∈ M has symplectic energy equal to π: Z u∗ ω = π. D

Proof. Let u ∈ M be a holomorphic disc of level t ∈ Rn−1 defined by the boundary condition (M1). Set u0t (z) := (z, it), z ∈ D (this anticipates Notation I.8.23). Thanks to the homotopy condition (M2), we can find a homotopy H : [0, 1] × ∂D −→ Lt from H(0, . ) = ut0 |∂D to H(1, . ) = u|∂D . Let λ be a primitive 1-form for ω. Then, twice using the theorem of Stokes, Z Z Z Z
∗ H ∗ω + u∗ ω = u∗ λ = ut0 λ. D

∂D

[0,1]×∂D

∂D

t

The first summand vanishes, LR is Lagrangian. The second summand, again
since R ∗  using Stokes, becomes D ut0 ω = D dx ∧ dy = π. Observe that for a J-holomorphic map u with respect to an ω-compatible almost complex structure J, we have

u∗ ω(∂x , ∂y ) = ω T u(∂x ), T u(∂y ) = ω T u(∂x ), JT u(∂x ) ≥ 0. In particular, a J-holomorphic curve has positive symplectic energy, unless it is constant. Notation I.8.17. Let J be an ω-compatible almost complex structure on R2n . Write h . , . iJ := ω( . , J . ) for the induced metric (cf. Section I.3.2), and | . |J for the corresponding norm. Let and u : G → R2n a smooth G ⊂ C be a bounded domain with smooth boundary, R 1 map. The Dirichlet energy of u is defined as 2 G |∇u|2J , where |∇u|2J := |ux |2J + |uy |2J . In the next exercise we generalise the discussion from Section I.4.2. Exercise I.8.18.

(a) Prove the formula |ux + Juy |2J = |∇u|2J − 2ω(ux , uy ),

which implies Z Z Z 1 1 |∇u|2J dx dy − u∗ ω = |ux + Juy |2J dx ∧ dy. 2 G 2 G G Conclude that the Dirichlet energy equals the symplectic energy if and only if u is J-holomorphic. In particular, equation (I.8) holds for any J-holomorphic curve, provided that the Dirichlet energy is measured with respect to | . |J .

38

Chapter I. Gromov’s Nonsqueezing Theorem

(b) Show that the symplectic energy of a not necessarily J-holomorphic map u is invariant under reparametrisations, i.e. the precomposition of a diffeomorphism ϕ : G0 → G. (c) Verify the same for the Dirichlet energy and conformal reparametrisations, i.e. where ϕ is a biholomorphism from G0 to G. ♦ Note I.8.19. For a J-holomorphic curve u we have uy = Jux , and hence |ux |J = |uy |J and hux , uy iJ = 0; cf. Section I.3.3. Exercise I.8.20. Give a proof of Lemma I.8.16 that does not use a primitive of ω. Hint: Interpret
ut0 (D) ∪ H([0, 1] × ∂D) ∪ −u(D) , where the minus sign denotes reversed orientation, as the image of a map S 2 → R2n . Smoothness of this map can be guaranteed by an appropriate choice of H.♦ Here is the promised minimality property of J-holomorphic curves; cf. Section I.6. Exercise I.8.21. Show that a J-holomorphic disc u ∈ M minimises the Dirichlet energy within its (smooth) homotopy class of smooth maps (D, ∂D) → (R2n , ∂Z). Hint: First show that the symplectic energy is an invariant of the homotopy class. ♦ Exercise I.8.22. Prove that Theorem I.6.3 remains valid for J-holomorphic curves, ♦ if the area is measured with respect to the metric h . , . iJ on R2n .

I.8.5

Flat discs

In order to gain a first understanding of the moduli space M, we want to show now that the J-holomorphic discs that stay entirely in the region where J = J0 are ‘flat’ in a sense that we are going to explain. Notation I.8.23. The flat discs in Z are the maps11 uts : D → Z = D × Cn−1 defined by uts (z) := (z, s + it), where s + it is a parameter in Cn−1 . If uts (D) is contained in the region where J = J0 , then obviously uts ∈ M. We want to demonstrate the converse of this observation. 11 We tend to write Int(D) or D for the open and closed unit disc, respectively, as domains of definition, but B and B = D for the same sets in the target space. Occasionally, it becomes impractical to stick to this convention, e.g. right here in the proof that follows.

I.8. A moduli space of J-holomorphic discs

39

Proposition I.8.24. Let u ∈ M be a J-holomorphic disc such that u(D) ⊂ R2n is contained in the region where J = J0 . Then, u = ust for some s, t ∈ Rn−1 . Proof. By the Lagrangian boundary condition (M1), there is a level t = (t2 , . . . , tn ) ∈ Rn−1 such that u(∂D) ⊂ Lt . If we write u = (u1 , u2 , . . . , un ), this implies that for j = 2, . . . , n we have holomorphic functions uj − itj : (D, ∂D) −→ (C, R). This means that ± Im(uj − itj ) are harmonic functions on D that vanish along ∂D. By the maximum principle, Im(uj − itj ) vanishes identically. The Cauchy– Riemann equations then imply that Re(uj − itj ) is constant, so that uj = sj + itj for some sj ∈ R, j = 2, . . . , n. Now consider the holomorphic function u1 : (D, ∂D) −→ (C, ∂D). By the three-point condition (M3), this map is not constant, so the maximum principle implies u1 (B) ⊂ B. The argument principle says that the number of zeros of u1 in B (counted with multiplicities) equals the winding number of the curve u1 |∂D about 0. The latter equals 1 by the homotopy condition (M2). Thus, u1 has a single zero at a point z0 ∈ B, say, and this has to be of order 1. By considering the function u1 − b for any b ∈ B, and observing that the winding number of (u1 −b)|∂D about 0 likewise equals 1, we see that u1 is a bijection of B. Locally, any holomorphic function (up to a coordinate transformation in the domain of definition and a translation in the image) looks like z 7→ z k , so it is locally bijective if and only if its derivative is nonzero, which then means it actually is locally biholomorphic. Thus, we have u1 |B ∈ Aut(B). Since, by Remark I.8.4, the automorphisms of B are given by the restriction of automorphisms of D, we have u1 ∈ Aut(D). The three-point condition (M3) then implies, by Note I.7.11, that u1 is the identity map of D.  The next exercise provides an alternative argument for the penultimate paragraph in the preceding proof. Exercise I.8.25. Let f : (D, ∂D) → (D, ∂D) be a continuous function that is holomorphic on Int(D). Further assume that f has a single zero, which is of order 1. Use the arguments from Sections I.8.1 and I.8.2 to show that f ∈ Aut(D). ♦ Given what we have learned so far, it is still conceivable that the family of J-holomorphic discs that do pass through the region where J 6= J0 does not stay inside a bounded region of R2n . Fortunately, with the geometric a priori bounds that we shall establish in the next chapter, this pathological scenario can be ruled out. In particular, this means that even though the manifold M is noncompact, its ‘ends’ are easy to understand and control.

Chapter II

Compactness The Earth hath bubbles, as the Water ha’s, And these are of them: whither are they vanish’d? William Shakespeare, Macbeth

The moduli space M has infinite ends made up of flat discs, so it is certainly not compact. Nonetheless, the properness of the evaluation map claimed in Theorem I.8.13 is a compactness statement of sorts. The present chapter contains the proof of this statement, except for an analytical ‘bootstrapping’ argument that is relegated to Chapter III. In Section II.1 we look at what we call the nonstandard discs, i.e. those discs u ∈ M whose image u(D) intersects the embedded ball Φ(Br+ε ) where the almost complex structure J (potentially) differs from J0 (Definition II.1.8). We show that the images of the nonstandard discs lie in a bounded region of R2n . This C 0 -bound is the first ingredient in establishing that the noncompactness of the moduli space is caused only by the ends of flat discs. Key ingredients here are the maximum principle and the boundary lemma of E. Hopf, so the reasoning is mostly geometric. The essential results of this discussion are summarised in Section II.1.6. A further source of noncompactness of the moduli space might be the phenomenon of ‘bubbling’ in sequences of holomorphic discs. The analytical arguments in the remainder of this chapter and in Chapter III serve to rule out this limiting behaviour of discs in M. In Section II.2 we first introduce the basic analytical tools (C k -norms, Fr´echet metric, Arzel`a–Ascoli theorem) and show that properness of the evaluation map will be a consequence of establishing uniform C k -bounds on the nonstandard discs. In Section II.2.3 we then describe the bubbling phenomenon for sequences of holomorphic discs (with target spaces other than those in our Theorem I.8.13); in other words, the phenomenon that we should like to rule out in the context of the nonsqueezing theorem. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Geiges, K. Zehmisch, A Course on Holomorphic Discs, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-36064-0_2

41

Chapter II. Compactness

42

A basic observation is that in a manifold with an exact symplectic form ω = dλ there cannot be any nonconstant J-holomorphic spheres for an ω-compatible almost complex structure J. In the remainder of Section II.2 we show that the failure of a uniform C 1 -bound would result in precisely such impossible spheres in R2n . The contradictory properties of purported limit curves that would result from the failure of a C 1 -bound are established by geometric arguments based again on an isoperimetric inequality. These arguments are the content of Sections II.3 and II.4. This concludes the proof of the existence of a uniform C 1 -bound. To get from there to C k -bounds for higher k requires a considerable amount of analysis, as we shall see in Chapter III.

II.1

Geometric a priori bounds

In this section we establish a C 0 -bound on the nonstandard discs, and we show that all discs in M take their image inside the closed cylinder Z. We also discuss the boundary behaviour of these holomorphic discs. Remark II.1.1. An ‘a priori bound’ (or the similar term ‘a priori estimate’ in Chapter III) is an estimate of the ‘size’ of a purported solution to a geometric problem or differential equation, i.e. typically a bound on some appropriate norm of such a solution, prior to any knowledge about the actual existence of any solution. The norm in question may (and in our case, will) involve higher derivatives of the solution. Often, such a priori bounds form part of the subsequent proof that solutions really do exist.

II.1.1 Schwarz reflection in the unit circle We first recall the classical Schwarz reflection principle. Notation II.1.2. Given a subset U ∈ C, in the context of Schwarz reflection we write U := {z ∈ C : z ∈ U } for the conjugate set. There should be little danger of confusing this with the notation for the closure of a set used elsewhere. Proposition II.1.3 (Schwarz Reflection in the Real Line). Let U be an open subset of the closed upper half-plane H := {z ∈ C : Im z ≥ 0}. Let f : U → C be a continuous function that is holomorphic on {z ∈ U : Im z > 0} and takes real values on U ∩ R. Then the function fˆ: U ∪ U → C, well defined by ( f (z) for z ∈ U , fˆ(z) := f (z) for z ∈ U , is holomorphic.



II.1. Geometric a priori bounds

43

For our purposes, we need a variant of this result, where reflection in the real line is replaced by inversion in the unit circle. The proof of this latter result, which we include here, is easily adapted to the previous result. Inversion in the unit circle S 1 ⊂ C is the diffeomorphism of C\{0} that sends a point z to the point w on the same ray from the origin, and such that |zw| = 1. So the map is described by z 7→ z/|z|2 = 1/z. Notice that the fixed point set of this inversion is S 1 . In fact, we may regard the inversion as a diffeomorphism of ˆ sending 0 to ∞ and vice versa. the Riemann sphere C, Proposition II.1.4 (Schwarz Reflection in the Unit Circle). Let U be an open subset of the closed unit disc D. Let f : U → C be a continuous function that is holomorphic on {z ∈ U : |z| < 1} and takes real values on U ∩ ∂D. Then the function fˆ: U ∪ (1/U ) → C, well defined by ( fˆ(z) :=

f (z)

for z ∈ U ,

f (1/z)

for z ∈ (1/U ),

is holomorphic. Recall from Exercise I.3.9 that a differentiable map f : G → C is holomorphic if and only if its differential is complex linear, which means that J0 ◦ T f = T f ◦ J0 . Similarly, we call a linear map L : R2 → R2 complex anti-linear if J0 ◦L = −L◦J0 ; cf. Section I.7.3. Exercise II.1.5. Show that the differentials of the maps z 7→ z and z 7→ 1/z are complex anti-linear. This is true even at the points 0 and ∞; one has to use the holomorphic chart z 7→ 1/z around the point ∞ in the Riemann sphere. ♦ Proof of Proposition II.1.4. The function fˆ is continuous on U ∪ (1/U ), and holomorphic on U ∩ {|z| < 1}. It is also holomorphic on (1/U ) ∩ {|z| > 1}, since the differential of z 7→ f (1/z) is the composition of one complex linear and two complex anti-linear maps. R The theorem of Morera [J¨ anich 1999] says that if the integral ∂∆ fˆ dz of the continuous function fˆ along the boundary curve ∂∆ (oriented counter-clockwise) of any triangle ∆ ⊂ U ∪ (1/U ) vanishes, then fˆ is holomorphic. Thus, consider such a triangle ∆. If ∆ is contained in U ∩ {|z| < 1} or R (1/U ) ∩ {|z| > 1}, where fˆ is holomorphic, then ∂∆ fˆ dz = 0 by the Cauchy integral theorem. Otherwise, consider the regions Qε0 := ∆ ∩ {|z| ≤ 1 − ε} and Qε1 := ∆ ∩ {|z| ≥ 1 + ε}; see Figure II.1. R Then ∂Qε fˆ dz = 0 for j = 0, 1 by the Cauchy integral theorem, but thanks j R to the continuity of fˆ, the sum of these two integrals goes to fˆ dz as ε → 0.  ∂∆

Chapter II. Compactness

44

Qε1 Qε0

Figure II.1: The Morera argument for holomorphicity. ˆ Remark II.1.6. Under the M¨obius transformation z 7→ z−i z+i of C, the closed upper half-plane H is mapped biholomorphically to D \ {1}; cf. Exercise II.2.13 below. This M¨obius transformation conjugates the reflection z 7→ z in the real line to the inversion z 7→ 1/z in the unit circle, and the two versions of Schwarz reflection correspond to each other under this conjugation. Exercise II.1.7. Replace the assumption f (U ∩ ∂D) ⊂ R in Proposition II.1.4 by the condition f (U ∩ ∂D) ⊂ S 1 . Show that the function fˆ: U ∪ (1/U ) → C defined by ( f (z) for z ∈ U , ˆ f (z) := 1/f (1/z) for z ∈ (1/U ), ♦

is holomorphic.

II.1.2 A bound on boundaries As a first step towards proving Theorem I.8.13, we want to establish a bound on the region where the boundaries of the nonstandard discs can lie. For ease of notation, we assume that the almost complex structure J on R2n coincides with J0 outside Φ(Br ), rather than Φ(Br+ε ). In this section, all balls with dimension not specified are understood to be of dimension 2n. As before, we write Dr for the closure of Br . Notice that the closure of Φ(Br ) equals Φ(Dr ), since Φ is an embedding of a slightly larger ball Br0 . Definition II.1.8. The nonstandard discs are the J-holomorphic discs u ∈ M with u(D) ∩ Φ(Br ) 6= ∅. We write Mnst ⊂ M for the subspace of nonstandard discs. 2n−2 Choose K > 0 large enough such that Φ(Dr ) is contained in B 2 × BK ⊂ 2n−2 R ×R . 2

II.1. Geometric a priori bounds

45

Lemma II.1.9. There is a bounded region in R2n that contains the boundaries u(∂D) of all u ∈ Mnst .
Proof. Given u ∈ Mnst , let G be the component of the preimage u−1 R2n \Φ(Dr ) containing ∂D. Notice that G is an open subset of D, the map u is holomorphic on G, and the boundary of G in D (i.e. the set of points in the closure of G but not in G) is mapped under u to Φ(∂Dr ). Let t ∈ Rn−1 be the level of u, that is, u(∂D) ⊂ Lt . Write u on G in complex components as u = (u1 , . . . , un ), and set h := (u2 − it2 , . . . , un − itn ) : (G, ∂D) −→ (Cn−1 , Rn−1 ). ˆ: G ˆ → Cn−1 the holomorphic continuation of h by Schwarz reflection Denote by h ˆ := G ∪ (1/G). to G Choose zm ∈ ∂D such that u(zm ) =: (p1 , p) ⊂ C × Cn−1 maximises the 2n−2 distance — among points in u(∂D) — to the ‘box’ B 2 × BK containing Φ(Dr ). Denote this maximal distance by R. If u(∂D) happens to lie entirely inside the box, we no longer need to be concerned with that u, so we only consider the case R > 0. Our aim is to find a bound on R, independent of u. 2n−2 BR (p)

iRn−1 p 2n−2 BK

prCn−1 (u(D))

Rn−1 prCn−1 (Φ(Dr ))

Figure II.2: The map (u2 , . . . , un )|G . Figure II.2 depicts the situation for u after projection to Cn−1 . Notice that (p1 , p) = u(zm ) ∈ ∂Z, which means that |p1 | = 1. So R equals the distance from p 2n−2 2n−2 to BK . The notation BR (p) is used for the open ball of radius R centred at n−1 ˆ p∈C . The map h obtained by Schwarz reflection is illustrated in Figure II.3. ˆ is an artefact of using The apparent local maximum of the holomorphic curve h reduced dimensions.

Chapter II. Compactness

46 iRn−1

2n−2 BR (p − it)

Rn−1 ˆ G) ˆ h(

ˆ Figure II.3: The map h.
2n−2 Since prCn−1 Φ(Dr ) is contained in BK , the preimage
ˆ −1 D2n−2 (p − it) ⊂ G ˆ h R ˆ and hence compact. We are thus in a position to apply is closed as a subset of C, the monotonicity lemma (Theorem I.4.1) to the map    
ˆ: G ˆ −1 B 2n−2 (p − it) , zm −→ B 2n−2 (p − it), p − it . ˆ R := h h R R
ˆ −1 D2n−2 (p − it) ⊂ G, ˆ R is contained in h ˆ and the Notice that the closure of G R 2n−2 ˆ ˆ boundary ∂ GR is mapped by h to ∂BR (p − it). ˆ ˆ First some notation. Write h| 1/G as h = c ◦ h ◦ τ , where c denotes complex n−1 conjugation on C , and τ the inversion in the unit circle S 1 ⊂ C. We write ωn−1 and ωn for the standard symplectic form on Cn−1 and Cn , respectively. We then compute Z Z Z Z ˆ ∗ ωn−1 = h (τ ∗ ◦ h∗ ◦ c∗ )ωn−1 = − (h∗ ◦ c∗ )ωn−1 = h∗ ωn−1 , 1/G

1/G

G

G

where we have used the transformation formula for the orientation-reversing diffeomorphism τ , and the fact that c∗ ωn−1 = −ωn−1 . Further, with the monotonicity lemma we find Z Z Z Z 2 ∗ ∗ ∗ ˆ ˆ πR ≤ h ωn−1 ≤ h ωn−1 = 2 h ωn−1 ≤ 2 u∗ ωn . (II.1) ˆR G

ˆ G

G

G

For the second and third inequality in this line we have used that u1 and h, as ˆ are holomorphic as maps into C and Cn−1 , respectively. Then with well as h,

47

II.1. Geometric a priori bounds

ˆ ∗ ωn−1 ≥ 0, which gives the second inequality. For the Exercise I.4.18 we have h third we write ωn = ω1 ⊕ ωn−1 , and again with Exercise I.4.18 we have u∗ ωn = u1∗ ω1 + (u2 , . . . , un )∗ ωn−1 = u∗1 ω1 + h∗ ωn−1 ≥ h∗ ωn−1 . In fact, for the inequality u∗ ωn ≥ 0 to hold it suffices that u be J-holomorphic for some ω-compatible J (see page 37), so together with Lemma I.8.16 this tells us that Z Z u∗ ωn ≤ u∗ ωn = π, G

D

which together with (II.1) implies the upper bound R ≤ lemma.

√

2. This proves the 

II.1.3 A bound on nonstandard discs Next we extend Lemma II.1.9 to a bound on the full nonstandard discs. Lemma II.1.10. There is a bounded region in R2n that contains the images u(D) of all u ∈ Mnst . Proof. Choose K > 0 large enough such that the ball BK ⊂ R2n contains Φ(Dr ) and the boundaries u(∂D) of all nonstandard discs. On a neighbourhood of the complement R2n \ BK we have J = J0 . For a given nonstandard disc u, define G := u−1 (R2n \ DK ); this is an open subset of D whose closure is contained in Int(D). The function |u| is subharmonic on G, takes values larger than K on G, and the value K on ∂G. This would contradict the maximum principle (Theorem I.4.10), unless G = ∅. So we must have u(D) ⊂ DK .  Exercise II.1.11. Use the maximum principle and an open-closed argument to show that |u| cannot attain the value K at any interior point of D, and conclude that u(D) ⊂ BK . ♦

II.1.4 The boundary lemma of E. Hopf The Hopf lemma will play a crucial role in some of our arguments. In its proof we use the weak maximum principle (Exercise I.4.12), and then we ask you to derive the (strong) maximum principle (Theorem I.4.10) from it. We say that a domain G ⊂ C has a C 2 -boundary if each point of ∂G has a neighbourhood in C in which ∂G is the graph of a C 2 -function (and G the subgraph). In particular, ∂G is a curve that admits a regular C 2 -parametrisation, that is, with nonvanishing velocity. Proposition II.1.12 (Boundary Lemma of E. Hopf). Let G ⊂ C be a bounded domain with C 2 -boundary, and let f : G → R be a continuous function on the closure G of G that is smooth and subharmonic on G. Suppose there is a boundary

Chapter II. Compactness

48

point z0 ∈ ∂G where f is differentiable, and such that f (z) < f (z0 ) for all z ∈ G close to z0 . Then the directional derivative of f at z0 in the direction of the outward normal is positive. Proof. The C 2 -smoothness of ∂G means that this boundary curve has a welldefined notion of curvature, depending continuously on the point. This allows one to find a closed disc contained in G that touches ∂G in the point z0 only; see Exercise II.1.14 below. By choosing this disc small enough, we may assume that f (z) < f (z0 ) on the whole disc. Moreover, by scaling and translation we may assume this to be the closed unit disc D, and by subtracting f (z0 ) (now with z0 ∈ ∂D) from f we may assume that f (z0 ) = 0, so that f is negative on D \ {z0 }. Exercise II.1.13. Check that the function 2

g(z) := f (z) + ε e−2r − e−2




is subharmonic on the annulus A := {3/4 < |z| < 1} and satisfies g < 0 on ∂A \ {z0 } for ε > 0 small enough. ♦ By the weak maximum principle (Exercise I.4.12) we have max g(z) = max g(z) = 0. z∈A

z∈∂A

With ∂r denoting the radial derivative (which is the derivative in the direction of the outward normal along ∂D), this implies ∂r g(z0 ) ≥ 0, whence
2 ∂r f (z0 ) ≥ −ε∂r |r=1 e−2r − e−2 = 4εe−2 > 0, as we wanted to show.



Exercise II.1.14. In the situation of the Hopf lemma, let γ : R/LZ → ∂G be a C 2 parametrisation by arc length of a component of ∂G, where L is the length of that component. The curvature of γ is defined as κ(s) = |γ 00 (s)|. Set κ0 := maxs κ(s), and let n be the outer normal to ∂G at z0 . Let C be the circle with centre z0 −n/κ0 and radius 1/κ0 . Show that near z0 , the boundary curve ∂G lies outside C (except at the point z0 , where the curves touch). Show that by possibly choosing a smaller radius (depending on the global shape of ∂G), one can find a closed disc contained in G and touching ∂G in z0 only. ♦ In this exercise we have used the boundedness of G, but not in an essential way, and the desired conclusion can be reached under milder conditions. Notice, however, that the definition of C 2 -boundary is a condition not only on ∂G, but also on the behaviour of G near the boundary. Exercise II.1.15. Give an example of a bounded domain G ⊂ C whose boundary is a compact collection of regular C 2 -curves, but where a disc as in the previous exercise cannot be found at every boundary point. ♦

II.1. Geometric a priori bounds

49

Exercise II.1.16. Use the Hopf lemma to prove the strong maximum principle. Hint: Without loss of generality, you may take the domain G to be bounded. Assuming that the subharmonic function f : G → R attains its supremum but is not constant, argue that the subset {z ∈ G : f (z) = max f }, which is closed in G, must have boundary points in G. Then, find an open disc B = Bε2 (z0 ) contained in the open set {z ∈ G : f (z) < max f }, with B ⊂ G and ∂B ∩ {z ∈ G : f (z) = max f } = 6 ∅. Derive a contradiction using the Hopf lemma. ♦

II.1.5 Boundaries are embedded In this section we consider all discs u ∈ M, not just the nonstandard ones. Lemma II.1.17. Every u ∈ M sends Int(D) into the cylinder Z = B 2 × R2n−2 .
Proof. Consider the open subset G := u−1 R2n \Φ(Dr ) of D. Write the boundary of G as ∂G = ∂Dt∂i G, with ∂i G denoting the boundary points of G in the interior of D; this set ∂i G may be empty, in case G = D. Since u(∂i G) ⊂ Φ(Dr ) ⊂ Z, the first component u1 of u satisfies u1 |∂ G < 1 = u1 |∂D . i The map u1 |G : G → C is holomorphic with respect to the standard complex structure on C, so the maximum principle implies u1 |G\∂D < 1, which means u(G \ ∂D) ⊂ Z. By the definition of G, the complement D \ G is sent by u to Φ(Dr ) ⊂ Z.  Lemma II.1.18. Every u ∈ M restricts to an embedding u|∂D : ∂D → ∂Z. Proof. On a neighbourhood of ∂D in D, the map u1 is holomorphic. So the Hopf lemma, applied to the subharmonic function |u1 |, tells us that along u(∂D) the vector T u1 (∂r ) is nonzero and points out of Z. Along ∂D we have ∂θ = i∂r , so the holomorphicity of u1 means that T u1 (∂θ ) = J0 T u1 (∂r ) along u(∂D). Since u1 (∂D) ⊂ ∂B 2 , the vector T u1 (∂θ ) must be tangent to S 1 = ∂B 2 . Then the previous equation means that T u1 (∂r ) must in fact be orthogonal to S 1 (and nonzero), and T u1 (∂θ ) positively tangent to S 1 along u(∂D). Since u1 |∂D is homotopic to the identity map ∂D → S 1 by the homotopy condition (M2), u1 |∂D is a diffeomorphism ∂D → S 1 (rather than a multiple cover). It follows that u|∂D is an embedding ∂D → ∂Z. 

II.1.6 The evaluation map The considerations of the previous sections can be summarised in the following properties of the evaluation map ev : M × D → R2n : (EV1)

ev(M × D) ⊂ Z.

Chapter II. Compactness

50 (EV2)

2n−2 There is a K > 0 such that ev(Mnst × D) ⊂ D2 × BK .

(EV3)

2n−2 Every point p ∈ Z \ (D2 × BK ) has a unique preimage, coming from t the flat disc us with s + it = prCn−1 (p).

II.2

Uniform gradient bounds

We now turn our attention to C 1 -bounds on the holomorphic discs in M.

II.2.1 The Fr´echet metric On the space C ∞ (D, Rm ) of smooth maps u : D → Rm (see Remark I.3.7) we have, for k ∈ N0 := {0, 1, 2, . . .}, the C k -norms kukC 0 := max |u(z)| z∈D

and kukC k :=

X

kDα ukC 0 ,

k ∈ N.

|α|≤k

Here, α = (α1 , α2 ) ∈ N0 × N0 is a double index, |α| := α1 + α2 , and Dα u :=

∂ |α| u . ∂xα1 ∂y α2

We can then define the so-called Fr´echet metric d(u, v) :=

∞ X 1 ku − vkC k · k 2 1 + ku − vkC k

(II.2)

k=0

on C ∞ (D, Rm ). Exercise II.2.1. Convergence with respect to this metric is the same as uniform convergence of all derivatives. ♦ A (possibly degenerate) metric as in (II.2) can be defined starting from any countable sequence of seminorms on a vector space. Definition II.2.2. A Fr´echet space is a vector space E with a family of seminorms | . |k , k ∈ N0 , such that the following conditions hold: (i) If v ∈ E satisfies |v|k = 0 for all k ∈ N0 , then v = 0; this is equivalent to saying that the corresponding Fr´echet metric is nondegenerate. (ii) E is complete with respect to the Fr´echet metric.

51

II.2. Uniform gradient bounds

Exercise II.2.3. Verify the triangle inequality for the Fr´echet metric. Show that the spaces C k (D, Rm ), k ∈ N0 , and C ∞ (D, Rm ) are complete with respect to the C k -norm and the Fr´echet metric, respectively. So the former are Banach spaces; the latter is a Fr´echet space. Hint: Recall (or prove) that the pointwise limit of a C 0 Cauchy sequence is actually the uniform limit. Moreover, it suffices to prove completeness for real-valued C 1 functions on an interval [a, b]. Use the fundamental theorem of calculus to show that uniform convergence of a sequence of functions fν : [a, b] → R and of the sequence (fν0 ) of derivatives implies the differentiability of the pointwise limit f := ♦ lim fν with f 0 = lim fν0 . See also [Wall 2016, Theorem A.4.7]. Remark II.2.4. In the metric space C ∞ (D, Rm ), sequential compactness is the same as compactness in the sense of coverings; see [J¨ anich 1996]. Similarly, in Lemma II.2.6 below, we use that the notions of continuity and sequential continuity are equivalent in metric spaces. In fact, for the equivalence of continuity and sequential continuity, it suffices that the domain of definition be a topological space with a countable local base (a.k.a. neighbourhood base) for its topology, that is, a first-countable space. On C ∞ (D, Rm ), the topology induced by the Fr´echet metric is the C ∞ topology, that is, the union of the compact-open C k -topologies for all k ∈ N0 ; cf. Exercise II.2.1. See [Hirsch 1976] for details on this topology. A countable local base around f0 ∈ C ∞ (D, Rm ) is provided by the sets {f : kf − f0 kC k < 1/k}, k ∈ N. Exercise II.2.5. Let J be an almost complex structure on R2n compatible with the standard symplectic form ω, and u : D → R2n a smooth J-holomorphic disc. Observe from Exercise I.8.18 that u∗ ω = 21 |∇u|2J dx∧dy, and use this to show that the operator norm kTz ukop of Tz u : Tz D → Tu(z) R2n with respect to the metric √ h . , . iJ on R2n equals |∇u(z)|J / 2. Conclude that max kTz ukop = z∈D


1 kukC 1 − kukC 0 . 2

In particular, for a set of J-holomorphic discs to have a uniform C 1 -bound is equivalent to it having a uniform C 0 -bound and a uniform bound on gradients.♦ Lemma II.2.6. The evaluation map ev : C ∞ (D, R2n ) × D → R2n is continuous. Proof. Given sequences (uν ) with uν → u in C ∞ (D, R2n ), and (zν ) with zν → z in D, we have |ev(uν , zν ) − ev(u, z)|

=

|uν (zν ) − u(z)|

≤

|uν (zν ) − u(zν )| + |u(zν ) − u(z)|

≤

kuν − ukC 0 + |u(zν ) − u(z)| → 0

by the C 0 -uniform convergence uν → u and the continuity of u.



Chapter II. Compactness

52

We equip the subset M ⊂ C ∞ (D, R2n ) with the induced topology. Here is the main result of this section. Theorem II.2.7. The evaluation map ev : M × D → R2n is proper. Let K ⊂ M × D be the preimage under ev of a compact subset in R2n . For flat discs uts we have ev(ust , z) = (z, s + it), so that every sequence in K with first component made up of flat discs has a convergent subsequence, since both the sν + itν and the zν lie in a compact set. So it suffices to prove compactness of the closure of the space Mnst of nonstandard discs, since then the closed subset K ∩ (Mnst × D) ⊂ Mnst × D will likewise be compact. Lemma II.1.10 can be read as saying that Mnst is uniformly C 0 -bounded: ∃K0 > 0 ∀u ∈ Mnst : kukC 0 < K0 . Our aim is to establish a uniform C 1 -bound: ∃K1 > 0 ∀u ∈ Mnst : kukC 1 < K1 . In Chapter III we shall see by a ‘bootstrapping’ argument that the C 1 -bound implies boundedness in the C k -norm for all k ∈ N.

II.2.2 The Arzel`a–Ascoli theorem As a preparation, we prove a version of the Arzel`a–Ascoli theorem sufficient for the purpose of reducing the properness of the evaluation map to finding C k -bounds on Mnst . We consider the Banach space C 0 (D, Rm ) of continuous maps D → Rm with the C 0 -norm. Definition II.2.8. A sequence (uν ) in C 0 (D, Rm ) is called uniformly equicontinuous if for every ε > 0 there is a δ > 0 such that |uν (z) − uν (w)| < ε for all z, w ∈ D with |z − w| < δ and all ν ∈ N. Proposition II.2.9 (Arzel`a–Ascoli). Every bounded, uniformly equicontinuous sequence (uν ) in C 0 (D, R2n ) has a uniformly convergent subsequence. Proof. Choose a countable dense subset of points z1 , z2 , . . . in D. The boundedness of (uν ) implies that there is a subsequence (u1ν ) that converges in the point z1 , a further subsequence (u2ν ) that converges in the point z2 , and so on. The diagonal sequence (uνν ) then converges pointwise on a dense subset of points. We write again (uν ) for this diagonal subsequence and claim that it is a Cauchy sequence. Given ε > 0, let δ be as in the definition of equicontinuity. Choose finitely many convergence points a1 , . . . , aN of (uν ) such that the open discs Bδ (aj ) cover D. Then, choose n0 ∈ N such that |uν (aj ) − uµ (aj )| < ε for all ν, µ ≥ n0 and all j = 1, . . . , N . Then, given z ∈ D, choose an aj =: a with |z − a| < δ. It follows that |uν (z) − uµ (z)| ≤ |uν (z) − uν (a)| + |uν (a) − uµ (a)| + |uµ (a) − uµ (z)| < 3ε for ν, µ ≥ n0 , which proves the claim.



53

II.2. Uniform gradient bounds

Corollary II.2.10. (a) Every sequence (uν ) in C ∞ (D, Rm ) that is bounded in the C 0 - and the C 1 -norm has a subsequence uniformly convergent in C 0 (D, Rm ). (b) Every sequence (uν ) in C ∞ (D, Rm ) that is bounded in each of the C k -norms, k ∈ N0 , has a convergent subsequence with respect to the Fr´echet metric. Proof. (a) The boundedness in the C 1 -norm means that the first-order partial derivatives of the uν are bounded uniformly in z ∈ D and ν ∈ N. By the mean value theorem, so is |uν (z) − uν (w)| . |z − w| This says that (uν ) is uniformly equicontinuous, and hence by Arzel` a–Ascoli possesses a subsequence uniformly convergent to some u ∈ C 0 (D, Rm ). (b) The same argument applies to the first partial derivatives of the uν . Hence, passing to a further subsequence we may assume that Dα uν is convergent in C 0 (D, Rm ) for both α = (1, 0) and (0, 1). In particular, (uν ) is then a C 1 Cauchy sequence, which by completeness converges in C 1 (D, Rm ), and the limit must of course still be u. This proves u ∈ C 1 (D, Rm ). Iterating this argument we establish that u ∈ C ∞ (D, Rm ), and the diagonal sequence in the subsequences we have constructed will converge in all C k -norms, k ∈ N.  This corollary, together with the remarks after Theorem II.2.7, says that Theorem II.2.7 will follow from having uniform C k -bounds on Mnst for all k ∈ N0 . Exercise II.2.11. Show that M is a closed subset of C ∞ (D, R2n ).

♦

This exercise implies that the closure of Mnst in M is the same as the closure in C ∞ (D, R2n ).

II.2.3 Examples of bubbling The idea for establishing a uniform C 1 -bound will be, arguing by contradiction, that if such a bound did not exist, a certain bubbling-off phenomenon would have to occur, which in the concrete situation at hand can be ruled out. In order to get a feel for this phenomenon and the reparametrisations of holomorphic discs that are necessary in the process, here we present three simple examples where bubbling does occur. Bubble sphere at an interior point. ˆ given by phic discs in D × C uν :

D z

Consider the sequence (uν )ν∈N of holomor-

−→ 7−→

ˆ D×C (z, ν 2 z).

Chapter II. Compactness

54

Observe that the circle {|z| = 1/ν} in D is mapped by uν to  iθ (e /ν, νeiθ ) : θ ∈ R/2πZ , ˆ as ν → ∞. and this embedded circle ‘converges’ to the point (0, ∞) ∈ D × C Next we consider the restriction of uν to the annulus {1/ν < |z| ≤ 1}. As ν → ∞, the domains of definition form a nested sequence filling out D \ {0}, and the maps converge pointwise to D \ {0} −→ z 7−→

ˆ D×C (z, ∞).

On any compact subset of D \ {0}, the convergence is uniform in all C k -norms; on ˆ the C-factor one should look at this in the chart z 7→ 1/z around the point z = ∞. Exercise II.2.12. Carry out this last step. For the convergence of the derivatives, ♦ consider compact subsets in D \ {0} independent of ν. ∞ We call this, somewhat loosely, Cloc (D \ {0})-convergence. Obviously, the limit map can be completed to the holomorphic disc z 7→ (z, ∞) defined on all of D. Finally, we consider the restriction of uν to the open disc B1/ν = {|z| < 1/ν}. Let vν be the reparametrised map

vν (z) := uν (z/ν 2 ) = (z/ν 2 , z),

z ∈ Bν ⊂ C.

∞ Here we have Cloc (C)-convergence to the map

−→ 7−→

C z

D×C (0, z),

ˆ → D×C ˆ by including the which can be completed to a holomorphic sphere C point z = ∞. This bubble sphere and the disc found as the limit of annuli connect at the limit point (0, ∞) of the ‘waist’ circle. Bubble sphere at a boundary point.

obius transformation The M¨

z 7−→

z−i z+i

 ˆ restricts to a biholomorphism from H ˆ := z ∈ C : Im z ≥ 0 ∪ {∞} to D. of C ♦

Exercise II.2.13. Verify this.

ˆ Consider the By this exercise, we may parametrise a holomorphic disc by H. ˆ ×C ˆ given by sequence (uν ) of holomorphic discs in H uν :

ˆ H z

−→ 7−→



ˆ ×C ˆ H z − (ν 2 + 1)i  z, . z − (ν 2 − 1)i

55

II.2. Uniform gradient bounds

ˆ under uν converges to the point The image of the circle {Im z = ν} ∪ {∞} ⊂ H (∞, 1) as ν → ∞. The strips {0 ≤ Im z < ν} form a nested sequence filling out ∞ (H)H = {z ∈ C : Im z ≥ 0}, and the maps uν on these respective strips Cloc converge to ˆ ˆ ×C H −→ H z 7−→ (z, 1). This completes to a holomorphic disc by ∞ 7→ (∞, 1). As to the restriction of uν to the open disc {z ∈ C : Im z > ν}, the reparametrised map  z − i vν (z) := z + ν 2 , , for z ∈ C with Im(z) > ν − ν 2 , z+i ∞ Cloc (C)-converges to the map

C z

−→ 7−→



ˆ ×C ˆ H z − i , ∞, z+i

which can be completed to a holomorphic sphere by including the point z = ∞. Exercise II.2.14. Study the bubbling behaviour of the sequence (uν ) of holomorphic discs ˆ ×C ˆ uν : D −→ D × C 2 4 z 7−→ (z, ν z, ν z). You should find a ‘double bubble’, one sphere attached to a disc (at an interior point), and a second sphere attached to the first one. ♦ Bubble disc at a boundary point. Consider the sequence (uν ) of holomorphic discs ˆ −→ H ˆ ×H ˆ uν : H 2 z 7−→ (z, ν z). ˆ is mapped by uν to The semicircle {|z| = 1/ν} ∩ H  iθ (e /ν, νeiθ ) : θ ∈ [0, π] , which shrinks to the point (0, ∞) as ν → ∞. ˆ fill out H ˆ \ {0} as ν → ∞, and the maps uν The sets {|z| > 1/ν} ∩ H ∞ ˆ Cloc (H \ {0})-converge to ˆ \ {0} H z

−→ 7−→

ˆ ×H ˆ H (z, ∞).

ˆ i.e. to a This can be completed in the obvious way to a map defined on all of H, holomorphic disc.

Chapter II. Compactness

56

ˆ we reparametrise the map as On the half-discs {|z| < 1/ν} ∩ H vν (z) := uν (z/ν 2 ) = (z/ν 2 , z),

ˆ z ∈ Bν ∩ H.

As ν → ∞, this converges to H z

−→ 7−→

ˆ ˆ ×H H (0, z),

which can be completed to a holomorphic disc by including the point z = ∞. Remark II.2.15. An explicit and elementary example of bubbling for solutions of an inhomogeneous Cauchy–Riemann equation is discussed in [Polterovich 2001, Section 4.4].

II.2.4

The bubbling-off argument

We now want to establish a uniform bound on the gradients of the nonstandard J-holomorphic discs. By Exercise II.2.5, this will yield the desired C 1 -bound. In that exercise we used the metric h . , . iJ corresponding to the almost complex structure J, but for establishing the bound on gradients we may likewise work with the standard metric, thanks to the next exercise. Exercise II.2.16. The metric h . , . iJ on R2n and the standard metric define equivalent norms, i.e. there is a constant C ≥ 1 such that 1 | . | ≤ | . |J ≤ C | . |. C Hint: The largest and smallest eigenvalue of a positive definite symmetric matrix A ∈ Rm×m can be found as the maximum and minimum, respectively, of hAv, vi, as v ranges over the unit sphere S m−1 ⊂ Rm . Use this to show that, for a continuous family At of matrices, these eigenvalues depend continuously on t. Then use the fact that J differs from J0 only on a compact subset of R2n . ♦ Arguing by contradiction, we assume that there is no uniform bound on gradients. This means that we find a sequence (uν ) in Mnst with Rν := max |∇uν (z)| −→ ∞ as ν → ∞. z∈D

Let tν ∈ Rn−1 be the level of uν determined by the boundary condition (M1). By Lemma II.1.9, the sequence (tν ) is bounded, and by passing to a subsequence we may assume that tν −→ t0 as ν → ∞ for some t0 ∈ Rn−1 . For each ν pick a point zν ∈ D with |∇uν (zν )| = max |∇uν (z)| = Rν . z∈D

57

II.2. Uniform gradient bounds

Again without loss of generality, we may assume that the sequence (zν ) converges in D: zν −→ z0 ∈ D as ν → ∞. We now distinguish whether z0 ∈ Int(D) or z0 ∈ ∂D. Case 1: z0 ∈ Int(D). Choose ε > 0 small enough such that Bε (zν ) ⊂ Int(D) for ν sufficiently large. The M¨obius transformation z z 7−→ zν + Rν of C sends the open disc1 BRν ε diffeomorphically onto Bε (zν ), so that for large ν we can define J-holomorphic maps vν :

B Rν ε

−→

z

7−→

(R2n , J)  z  vν (z) := uν zν + . Rν

The following properties of the vν are straightforward; for the estimate in (iii) we use Lemma I.8.16: (i) |∇vν | = |∇uν |/Rν ≤ 1, (ii) |∇vν (0)| = |∇uν (zν )|/Rν = 1, Z Z (iii) vν∗ ω = u∗ν ω ≤ π. B Rν ε

Bε (zν )

The uniform C 0 -bound on the nonstandard discs from Lemma II.1.10 also applies to the vν . A C 1 -bound comes from (i) and Exercise II.2.5. With the bootstrapping argument from Chapter III and Corollary II.2.10, this proves — replacing D by any closed disc Dk , k ∈ N — that (vν ) has a subsequence (vkν )ν∈N convergent in C ∞ (Dk , R2n ). By choosing these subsequences successively as a subsequence of the preceding one, and then taking the diagonal sequence, we arrive at a subsequence, which we now call (vν ) again, that converges on any compact subset of C. Beware that the vν are not defined on all of C, but on any given compact subset they will be defined for sufficiently large ν. The pointwise limit v of the vν is then in C ∞ (C, R2n ). In the language introduced in Section II.2.3, the original sequence has a ∞ Cloc (C)-convergent subsequence. We write (for the convergent subsequence (vν )) ∞ vν −→ v in Cloc (C) as ν → ∞.

Lemma II.2.17. This limit map v ∈ C ∞ (C, R2n ) has the following properties: 1 Discs

with centre not specified are understood to be centred at 0 ∈ C.

58

Chapter II. Compactness

(V1)

v is J-holomorphic.

(V2)

v is not constant. Z v ∗ ω ≤ π. 0
0 small enough such that Bε (zν )∩H ⊂ D+ for ν sufficiently large. Write zν = xν + iyν . Without loss of generality we may assume the existence of the limit r := lim Rν yν ∈ [0, ∞]. ν→∞

Case 2.1: r = ∞. Here we reparametrise as in Case 1, but of course we need to be aware that we work with Bε (zν ) ∩ H, not the full disc Bε (zν ). Thus, we define the J-holomorphic maps  z  , z ∈ BRν ε ∩ {Im z ≥ −Rν yν }. vν (z) := uν zν + Rν By Remark II.2.19, the sequence of norms of gradients |∇vν | is uniformly bounded from above, and by a positive constant from below at the point z = 0. For the symplectic energy, we have Z Z vν∗ ω = u∗ν ω ≤ π. BRν ε ∩{Im z≥−Rν yν }

Bε (zν )∩H

Thus, we have properties completely analogous to (i)–(iii) on page 57. Any given compact subset of C will be contained in the domain of definition of vν for ν suffi∞ ciently large. Thus, as in Case 1, we obtain a Cloc (C)-limit v as in Lemma II.2.17, and the same contradiction as before.

Chapter II. Compactness

60

Case 2.2: r < ∞. Here, the contradiction will not come from an impossible bubble sphere, but from a bubble disc ruled out by the three-point condition (M3) in the definition of the moduli space M. In this case, we reparametrise uν : Bε (zν ) ∩ H → R2n , where we write zν = xν + iyν , as  z  vν (z) := uν xν + , z ∈ BRν ε (iRν yν ) ∩ H. Rν Any compact subset of H will be contained in the domain of definition of vν for ν sufficiently large, since Rν ε goes to infinity, and iRν yν converges to the (finite) point ir. Again by Remark II.2.19, the sequence of norms of gradients |∇vν | is uniformly bounded from above, and the sequence |∇vν (iRν yν )| is bounded from below by a positive constant, i.e. we have properties (i), (ii) as on page 57. The energy estimate (iii) is likewise satisfied: Z Z vν∗ ω = u∗ν ω ≤ π. BRν ε (iRν yν )∩H

Bε (zν )∩H

Thus, analogous to the argument in Case 1, and because of tν → t0 as ν → ∞, ∞ we find a subsequence Cloc (H)-convergent to a J-holomorphic map v : (H, R) −→ (R2n , Lt0 ) with properties (V1)–(V3) as in Lemma II.2.17. As we shall see, such a map v would have contradictory properties. The argument is based on estimates analogous to those in Section II.3 for Case 1, so we defer it to Section II.4.

II.3

An asymptotic isoperimetric inequality

The aim of this section is to show that a J-holomorphic map v : C → R2n with the properties described in Lemma II.2.17 cannot exist, and hence that the gradient of a sequence (uν ) in M cannot blow up at an interior point of D (Case 1). As the pointwise limit of (reparametrised) nonstandard discs, the (purported) map v has its image v(C) contained in a bounded region of R2n by property (EV2) in Section II.1.6.
Case 1.1: For sufficiently large R > 0, the image v C \ DR is contained in the region where J = J0 . In this case we shall appeal to the classical Riemann removable singularities theˆ → R2n orem [J¨anich 1999] in order to extend v to a J-holomorphic sphere vˆ : C and arrive at a contradiction as explained on page 58. First, we lead the reader through a simple proof of Riemann’s theorem.

61

II.3. An asymptotic isoperimetric inequality

Exercise II.3.1. Let G ⊂ C be a domain containing the origin 0 ∈ C, and let f : G \ {0} → C be a bounded holomorphic function. Show that f extends to a holomorphic function G → C by arguing as follows. Set ( z 2 f (z) if z ∈ G \ {0}, g(z) := if z = 0. 0 Show that g is a holomorphic function on G with g 0 (0) = 0. Then use the power ♦ series expansion of g at 0 to define a holomorphic extension of f . In the given case, we can apply this theorem to the components of the holomorphic map z 7−→ v(1/z), z ∈ B1/R \ {0}, to obtain the (contradictory) extension of v into the point ∞. Case 1.2: Without any further assumption on the map v one can still establish a removable singularities theorem in the J-holomorphic setting — a complete proof, relying on the minimality property of J-holomorphic curves, can be found in [McDuff & Salamon 2012, Section 4.5] — and then conclude as in Case 1.1. Instead, we present an ad hoc argument, based on the isoperimetric inequality, showing that a J-holomorphic map C → R2n with finite symplectic energy in fact has energy 0, so a map v as in Lemma II.2.17 cannot exist. Consider the loops γr : S 1 = R/2πZ → R2n , r ∈ R+ , defined by γr (θ) := v

 1  , θ ∈ S1. reiθ

For the following isoperimetric estimates, involving the length of γr and the energy of v, we need an inequality coming from the Cauchy–Schwarz inequality for 2πperiodic functions. Exercise II.3.2. On the vector space of continuous functions S 1 → R we define the scalar product Z 2π 1 f (θ)g(θ) dθ. hf, giS 1 := 2π 0 Use the Cauchy–Schwarz inequality for this inner product to show that for a C 1 function γ : S 1 → Rm and any norm | . | on Rm we have the inequality 1 2 1 L (γ) := 2π 2π

2π

Z

|γ 0 (θ)| dθ 0

2

2π

Z

|γ 0 (θ)|2 dθ.

≤

♦

0

Remark II.3.3. In Proposition I.5.4 we derived the classical isoperimetric inequality from the action-energy inequality. With the preceding exercise we see that, conversely, the action-energy inequality follows from the isoperimetric inequality.

Chapter II. Compactness

62

Set w(x, y) := v(1/z), z = x + iy ∈ C \ {0}, so that γr (θ) = w(r cos θ, r sin θ). Then γr0 (θ) = −wx r sin θ + wy r cos θ. The map w is J-holomorphic; hence, wy = Jwx , hwx , wy iJ = 0, and |wx |J = |wy |J . It follows that |γr0 (θ)|2J = r2 |wx (r cos θ, r sin θ)|J2 =

1 2 r |∇w(r cos θ, r sin θ)|2J . 2

Now, with statement (V3) from Lemma II.2.17, and the fact that the symplectic energy of the J-holomorphic map w equals its Dirichlet energy (Exercise I.8.18), we have Z Z π ≥ v∗ ω = w∗ ω C

= = ≥

1 2 Z

C\{0}

Z

2π

0 2π

Z

Z

∞

|∇w(r cos θ, r sin θ)|2J r dr dθ

0 ∞

dr dθ r 0 0 Z ∞ 1 dr L2J (γr ) , 2π 0 r |γr0 (θ)|2J

where for the last inequality we have used Exercise II.3.2, and LJ denotes the length of curves measured with respect to the metric h . , . iJ . Z 1 dr The integral = − log ε diverges to ∞ as ε → 0, so from the estimate r ε above we conclude that LJ (γr ) cannot be bounded from below for small r. In other words, there must be a sequence rν & 0 with LJ (γrν ) → 0 as ν → ∞. By Exercise II.2.16, the same limit behaviour is shared by the lengths L(γrν ) measured with respect to the standard metric on R2n . By Exercise I.5.3, the action of γrν equals, up to sign, the symplectic energy of v over the domain B1/rν : Z −A(γrν ) = v ∗ ω. B1/rν

The minus sign comes from the fact that γrν is the negatively oriented boundary curve of B1/rν . The isoperimetric inequality (Proposition I.5.4) then yields Z

v∗ ω ≤ B1/rν

L2 (γrν ) −→ 0 as ν → ∞. 4π

It follows that C v ∗ ω = 0. This concludes the discussion of Case 1. The same argument covers Case 2.1. R

63

II.4. Boundary singularities

II.4

Boundary singularities

We now turn to the purported J-holomorphic map v : (H, R) −→ (R2n , Lt0 ) found in Case 2.2 of Section II.2.4 and want to show that such a map cannot actually exist (and hence that boundary bubbling does not occur). ∞ As the Cloc -limit of a sequence of J-holomorphic maps mapping into the closed cylinder Z = D2 × R2n−2 and sending boundary points into ∂Z, the map v shares these properties. Thus, by the maximum principle, the interior of H is mapped by v into the open cylinder Z = B 2 × R2n−2 . This allows us to apply the Hopf lemma, and as in the proof of Lemma II.1.18 we conclude that v wraps ∂H positively around ∂Z. More precisely, write v = (v1 , . . . , vn ) and γ := v1 |∂H :

∂H t

−→ 7−→

∂B v1 (t).

Then γ˙ > 0, where we think of ∂B as S 1 = R/2πZ. Now consider the curves, for r > 0, γr :

[0, π] θ

−→ 7−→

R2n v(reiθ ).

As in Section II.3 we compute Z π ≥ v∗ ω H Z Z 1 π ∞ |∇v(r cos θ, r sin θ)|2J r dr dθ = 2 0 0 Z πZ ∞ dr = |γr0 (θ)|2J dθ r 0 0 Z dr 1 ∞ 2 LJ (γr ) . ≥ π 0 r Exercise II.4.1. Explain the factor 1/π in the last line. ♦ Z R dr The integral = log R diverges to ∞ as R → ∞, so there must be a r 1 sequence rν % ∞ with LJ (γrν ) → 0 as ν → ∞. In particular, this implies that
lim γ(rν ) − γ(−rν ) = 0 ∈ R/2πZ = ∂B.

ν→∞

The map γ being strictly monotone, it must map ∂H surjectively onto ∂B.

Chapter II. Compactness

64

∼ ∂D \ {−z0 } with R sufficiently Now choose an interval I = [−R, R] ⊂ ∂H = large such that γ(I) ⊂ ∂B = R/2πZ has length
15 ` γ(I) ≥ · 2π, 16 ∞ say. Then, by the Cloc (H)-convergence, for ν sufficiently large the domain of definition of vν will include I, and


7 ` pr1 ◦ vν (I) ≥ · 2π. 8 Hence, with h Ri R , xν + Iν := xν − Rν Rν we have


7 ` pr1 ◦ uν (Iν ) ≥ · 2π. 8 But the length of Iν goes to zero as ν → ∞, so for ν sufficiently large it will be disjoint from one of the arcs in ∂H ∪ {∞} corresponding under the M¨ obius transformation gz0 to one of the three arcs in ∂D from 1 to i, from i to −1, or from −1 to 1. By the three-point condition (M3), these arcs are mapped by uν to an arc of length π/2 or π, respectively, in ∂B. This means that
7 π ` pr1 ◦ uν (∂D) ≥ · 2π + > 2π, 8 2 contradicting Lemma II.1.18.

Chapter III

Bounds of Higher Order Gleichwohl sprang ich auch zum zweitenmale noch zu kurz und fiel nicht weit vom andern Ufer bis an den Hals in den Morast. Hier h¨ atte ich unfehlbar umkommen m¨ ussen, wenn nicht die St¨ arke meines eigenen Armes mich an meinem eigenen Haarzopfe, samt dem Pferde, welches ich fest zwischen meine Knie atte. schloß, wieder herausgezogen h¨ Gottfried August B¨ urger, M¨ unchhausens wunderbare Reisen

In this chapter we want to make good on the promise from Chapter II that starting from a C 1 -bound on Mnst one can ‘bootstrap’ (or ‘pigtail’, if one credits M¨ unchhausen with the invention of this method) to a C k -bound for all k ∈ N (page 52), and that one can extract from a C 1 -bounded sequence of holomorphic ∞ discs a subsequence convergent in Cloc (page 57).

III.1

The a priori estimate


First we are going to deal with arbitrary maps u ∈ Cc∞ BR , Rm , that is, smooth Rm -valued maps with compact support inside the open disc of radius R in C. The relevant estimates involve various Sobolev norms. The reason why we cannot directly work with C k -norms or the Fr´echet metric on the space of C ∞ -maps will be discussed in Section III.1.6. In the present chapter we only deal with Sobolev norms on spaces of smooth maps (or maps of class C k ). At a later stage we shall have to work with Sobolev spaces of weakly differentiable maps, but on pedagogical grounds we delay the introduction of these spaces until Chapter IV. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Geiges, K. Zehmisch, A Course on Holomorphic Discs, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-36064-0_3

65

Chapter III. Bounds of Higher Order

66

III.1.1

Sobolev norms

Let G ⊂ C be a domain. The Lp -norm, p ∈ [1, ∞), of a continuous map u : G → Rm is Z 1/p p kukLp (G) := , |u| G

where integration is with respect to the standard Lebesgue measure on C. (This norm may of course take the value ∞ unless we assume compact support, for instance.) Note III.1.1. In this definition, | . | can be any norm on Rm . Since any two such norms are equivalent (Exercise II.2.16), different choices will simply alter the constants in the estimates we shall establish. A convenient choice is the `p -norm |(u1 , . . . , um )|p = |u1 |p + · · · + |um |p


1/p

,

for then the pth power of the Lp -norm is additive, which means one can prove estimates componentwise: p k(u1 , . . . , um )kpLp (G) = ku1 kpLp (G) + · · · + kum kL p (G) .

For the next definition, recall the notation from Section II.2.1. Definition III.1.2. The Sobolev (k, p)-norm, k ∈ N0 , p ∈ [1, ∞) of a smooth map u : G → Rm is 1/p X p kDα ukL kukk,p,G := . p (G) |α|≤k

Notice that kuk0,p,G = kukLp (G) . We usually drop G from the notation when the domain of definition is understood. Note III.1.3. Again, since any two norms on RN are equivalent, the Sobolev (k, p)norm is equivalent to the norm X kDα ukLp (G) . |α|≤k

Thus, whenever we prove estimates, up to a constant, involving various Sobolev norms, we may as well think and argue in terms of the respective equivalent norms. Often this simplifies the reasoning.

III.1.2

The Poincar´e inequality

We consider the linear Cauchy–Riemann operator ∂ :=


1 ∂ x + J0 ∂ y 2

67

III.1. The a priori estimate

on smooth maps u : C → R2n , where, as before, J0 is the endomorphism of R2n corresponding to multiplication by i under the identification R2n = Cn . Observe that u is holomorphic if and only if ∂u = 0. We compute |∂u|2

1 1 1 |ux |2 + hux , J0 uy i + |uy |2 4 2 4 1 1 ∗ 2 |∇u| − u ω(∂x , ∂y ), 4 2

= =

where ω = −h . , J0 . i is the standard symplectic form on R2n ; cf. Exercise I.8.18. Note R ∗ III.1.4. R If u∗ is compactly supported, say in BR , the theorem of Stokes yields u ω = u λ = 0, where λ is any primitive 1-form for ω. Hence, in that case, C ∂BR Z

|∂u|2 = C

1 4

Z

|∇u|2 . C

As we shall see, for the bootstrapping argument we need an estimate of the form kukk,p,BR ≤ c(k, p, R)k∂ukk−1,p,BR for all u ∈ Cc∞ (BR , R2n ),

(III.1)

where p > 2 (this condition on p is required in the Sobolev inequality of Proposition III.1.20). Even so, it is instructive to prove a corresponding estimate for p = 2, which is technically much less involved. Lemma III.1.5 (Poincar´e Inequality). Let u : C → Rm be a C 1 -function with compact support in BR for some R > 0. Then, Z Z |u|2 ≤ 2R2 |∇u|2 . C

C

Proof. First we assume that u is a real-valued function. On the vector space of continuous functions [−R, R] → R we consider, for any given x ∈ (−R, R], the scalar product Z x

hf, gix :=

f (t)g(t) dt. −R

The Cauchy–Schwarz inequality for this inner product, with f ≡ 1 and g(t) = ux (t, y), gives |u(x, y)|2 = hf, gi2x

≤ =

|f |2x |g|2x Z x Z 1 dt · −R

Z ≤

x

|ux (t, y)|2 dt −R

R

|ux (t, y)|2 dt.

2R −R

Chapter III. Bounds of Higher Order

68

By summing over the components of a map u : C → Rm , we see that the same inequality holds in this case. Notice that the right-hand side no longer depends on the x-variable. By integrating this inequality we find Z Z |u|2 = |u(x, y)|2 dx dy [−R,R]2 C Z |ux (t, y)|2 dt dy ≤ 2R · 2R 2 [−R,R] Z 2 = 4R |ux |2 . C

By adding the analogous estimate involving uy , we obtain Z Z 2 |u|2 ≤ 4R2 |∇u|2 , C

C

as claimed.



Here is the promised estimate (III.1) on the (1, 2)-Sobolev norm, which follows easily from Note III.1.4 and Lemma III.1.5. Exercise III.1.6. For u as in the preceding lemma (with m = 2n), show that 2 kuk1,2 ≤ (8R2 + 4)k∂uk2L2 .

♦

This is the base case k = 1 for the (simple) inductive argument that establishes the a priori estimate (III.1) for p = 2 and all k ∈ N; see the proof of Proposition III.1.16.

III.1.3 The inhomogeneous Cauchy–Riemann equation As a further preparation, both for the a priori estimate and for the Fredholm theory in Chapter V, we now want to study the inhomogeneous Cauchy–Riemann equation ∂u = g (iCR) on the space of smooth functions C → Cn . Our presentation here closely follows [Hofer & Zehnder 1994, Appendix 4]. For convenience, we drop the factor 1/2 from the Cauchy–Riemann operator, that is, we take ∂ = ∂x + i∂y : C ∞ (C, Cn ) −→ C ∞ (C, Cn ). The inhomogeneity g is assumed to lie in the vector space Cc∞ (C, Cn ) of compactly supported smooth functions C → Cn .

69

III.1. The a priori estimate Proposition III.1.7. There is a complex linear operator A : Cc∞ (C, Cn ) −→ C ∞ (C, Cn ) such that u := Ag is the unique solution of (iCR) with limz→∞ u(z) = 0. The proof of this proposition will take up the remainder of this section. Definition of the integral operator A Given g ∈ Cc∞ (C, Cn ), we set 1 Ag (z) = − 2π


Z C

g(z + ζ) 2 dλζ , ζ

where dλ2ζ denotes the 2-dimensional Lebesgue measure on C with respect to the variable ζ. We first need to convince ourselves that this integral exists. Exercise III.1.8. Consider the function f : C → C defined by f (z) = 1/z for z 6= 0 (and f (0) arbitrary). (a) Use the monotone convergence R theorem of Beppo Levi to show that |f | is integrable on D with integral D |f | = 2π. (b) Use the Lebesgue dominated convergence theorem to show that f is integrable over compact subsets of C, and that f g is integrable over C for any g ∈ Cc∞ (C, C). ♦ Exercise III.1.9. Show that, in polar coordinates, Ag is given by the formula
1 Ag (z) = − 2π

2π

Z

∞

Z

0

g(z + reiθ ) e−iθ dr dθ,

(III.2)

0

where the integral is to be read as limε&0

R 2π R ∞ 0

ε

.

♦

In fact, this improper integral, also known as the Cauchy principal value at 0, can be defined for functions g of lower regularity, for which the Lebesgue integral over C may no longer exist owing to the singularity of the integrand at 0. For smooth functions this is not an issue, but it is still convenient for computations to think of the integral as a Cauchy principal value. In particular, we can use this to establish the properties of the operator A by ad hoc methods, rather than by appealing to general theory. Thus, for any given ε > 0 we define Aε : Cc∞ (C, Cn ) −→ C ∞ (C, Cn ) by 1 Aε g (z) = − 2π


Z |ζ|≥ε

g(z + ζ) 2 dλζ . ζ

Chapter III. Bounds of Higher Order

70

(i) Aε g is continuous: We show that Aε g is even locally Lipschitz continuous. Since the smooth function g has compact support, it is globally Lipschitz continuous, i.e. there is a constant M such that |g(z2 ) − g(z1 )| ≤ M |z2 − z1 |. Given g, z1 , z2 , choose   R > max |z| : z ∈ supp g + max |z1 |, |z2 | . In polar coordinates ζ = reiθ the area element is r dr dθ; hence, Z g(z2 + ζ) − g(z1 + ζ) 2

Aε g (z2 ) − Aε g (z1 ) ≤ 1 dλζ 2π |ζ|≥ε ζ Z 2π Z R 1 ≤ M |z2 − z1 | dr dθ 2π 0 ε ≤ M R |z2 − z1 |. (ii) Given g, the map R+ → C 0 (C, Cn ) sending ε to Aε g is Lipschitz continuous: For 0 < ε1 ≤ ε2 we compute, as in (i), Z g(z + ζ) 2

dλζ Aε2 g (z) − Aε1 g (z) ≤ 1 2π ε1 ≤|ζ|≤ε2 ζ ≤

kgkC 0 |ε2 − ε1 |.

From (i) and (ii) we conclude that Aε g converges uniformly as ε & 0 and defines a continuous function Ag = lim Aε g ∈ C 0 (C, Cn ). ε&0

Exercise III.1.10. Conclude with (III.2) that the estimate
Ag (z) ≤ diam(supp g) kgkC 0 holds for any g ∈ Cc∞ (C, Cn ).

(III.3) ♦

The operator A commutes with derivatives Exercise III.1.11. Fix a unit vector e ∈ C. For h ∈ R \ {0} we consider the difference quotient
g(z + he) − g(z) ∆h g (z) := . h Verify that A ◦ ∆h = ∆h ◦ A.

♦

71

III.1. The a priori estimate

(iii) For g ∈ Cc∞ (C, Cn ) we have Ag ∈ C ∞ (C, Cn ), and the linear operator A satisfies Dα ◦ A = A ◦ Dα for all derivatives Dα ; cf. Section II.2.1 for this notation. In particular, we have ∂ ◦ A = A ◦ ∂: It suffices to establish the identity De ◦ A = A ◦ De (and the existence of De Ag in the first place) for the directional derivative De in the direction e; the general formula then follows by iteration. By (III.3), for any fixed z we have

∆h Ag (z) − ADe g (z)

=


A ∆h g − De g (z)

≤

ck∆h g − De gkC 0 −→ 0 as h → 0,

where the constant c depends only on supp g. This yields the desired identity. (iv) A ◦ ∂ = id on Cc∞ (C, Cn ):
For a given g ∈ Cc∞ (C, Cn ), we first look at the integrand of Aε ∂g (z) for a fixed z. Writing ζ = ξ + iη, we set k(ξ, η) := g(z + ζ). Then, −

∂k dξ ∧ dη ζ

= = = =

−

kξ + ikη dξ ∧ dη ζ

i (kξ dξ + kη dη) ∧ (dξ + i dη) ζ i dk ∧ dζ ζ   k dζ i·d . ζ

We may choose R > 0 such that the integrand vanishes for |ζ| ≥ R. We then compute, using the theorem of Stokes and observing that the circle {|ζ| = ε} is oriented negatively as the boundary of the annulus {ε ≤ |ζ| ≤ R}, Aε ∂g(z) =

i 2π



Z d ε≤|ζ|≤R

k dζ ζ

 =

1 2πi

Z |ζ|=ε

k dζ . ζ

With the substitution ζ = ε e2πit , whence dζ = ε2πi e2πit dt, this gives 1

Z


g z + εe2πit dt −→ g(z) as ε & 0.

Aε ∂g(z) = 0

From (iii) and (iv) we infer that u := Ag is a solution of (iCR). The asymptotic behaviour of u = Ag Let g ∈ Cc∞ (C, Cn ) be given.

72

Chapter III. Bounds of Higher Order


(v) Ag (z) → 0 as |z| → ∞: Choose R > 0 such that supp g ⊂ BR . For |z| > R we have a situation as in Figure III.1, with R = sin(α/2). |z| Since limθ→0

sin θ θ

= 1, we get R α ≥ for |z| sufficiently large. |z| 4

(III.4)

With (III.2) and |z| > R sufficiently large such that (III.4) holds, we estimate
Ag (z)

≤ ≤ ≤

Z arg(−z)+α/2 Z |z|+R
1 g z + reiθ dr dθ 2π arg(−z)−α/2 |z|−R 1 · α · 2R kgkC 0 2π 4R2 1 kgkC 0 · . π |z|

This implies the claim (v).

 supp ζ 7→ g(z + ζ) −z R |z|

α/2 0


Figure III.1: The asymptotic estimate on Ag (z) as z → ∞.

Uniqueness of the solution Let u0 , u1 be two solutions of (iCR), for a given g, asymptotic to 0 as |z| → ∞. Then the components of u1 − u0 are bounded entire functions, i.e. bounded

73

III.1. The a priori estimate

holomorphic functions
C → C, and hence constant by Liouville’s theorem. Since limz→∞ u1 (z) − u0 (z) = 0, these constants have to be zero. This completes the proof of Proposition III.1.7. 

III.1.4 The Calder´ on–Zygmund inequality In this section we want to prove the following estimate. Proposition III.1.12. For p ∈ (1, ∞) there is a constant c = c(p) such that
k∇ukLp ≤ ck∂ukLp for all u ∈ Cc∞ C, Cn . The main ingredients of the proof are the operator A from the preceding section and the so-called Calder´on–Zygmund inequality. The proof of the latter is beyond the scope of this text, but we discuss the setting of that inequality in some detail. Recall that (changing notation from g to u) Z
1 u(z + ζ) 2 Au (z) = − dλζ for u ∈ Cc∞ (C, Cn ). 2π C ζ With the substitution ζ = w − z this becomes Z
1 u(w) dλ2 . Au (z) = 2π C z − w w

(III.5)

Now consider the function K(z) =

1 log |z|, 2π

z ∈ C \ {0}.

One easily verifies that with ∂ := ∂x − i∂y we have ∂K =

1 1 . 2π z

It follows that (III.5) can be written as a convolution, Z
Au (z) = ∂K(z − w)u(w) dλ2w =: (∂K ∗ u)(z).

(III.6)

C

The Calder´on–Zygmund inequality deals with exactly such convolutions. We denote the partial derivatives of K by K1 := ∂x K and K2 = ∂y K. Theorem III.1.13 (Calder´on–Zygmund Inequality). For p ∈ (1, ∞) there is a constant c = c(p) such that, for j ∈ {1, 2}, k∇(Kj ∗ u)kLp ≤ ckukLp for all u ∈ Cc∞ (C, Cn ).



Chapter III. Bounds of Higher Order

74

For a proof, see [McDuff & Salamon 2012, Section B.2]. Exercise III.1.14. The aim of this exercise is to work through the proof of Theorem III.1.13 for the case p = 2, which is considerably simpler than the general case. Further simplifying the discussion, we assume that u is real-valued, that is, u ∈ Cc∞ (R2 ). Set v = Kj ∗ u for a fixed j ∈ {1, 2}. We want to show that k∇vkL2 ≤ kukL2 for all u ∈ Cc∞ (R2 ).

(III.7)

Beware that v need not have compact support. (i) Convince yourself that vi = Kj ∗ ui for i ∈ {1, 2}. (ii) Verify that Kj (x1 , x2 ) = xj /2π|x|2 . (iii) Use (i) and (ii) to show that there is a constant c = c(u) such that |v(x)| + |∇v(x)| ≤

c for all |x| ≥ c. |x|

(iv) Derive the identity Z

|∇v|2 = − BR

Z

Z v ∆v + BR

v ∂BR

∂v ∂n

(III.8)

from the divergence theorem of Gauß. Here, ∂v/∂n denotes the directional derivative in the direction of the outer normal n along ∂BR . (v) It is known that k := K ∗ h is a solution of Poisson’s equation ∆k = h; see [Evans 2010, Section 2.2.1]. Show that v = K ∗ uj , from which we deduce that ∆v = uj . (vi) Use the estimate in (iii) to show that the second summand on the righthand side of (III.8) converges to zero as R → ∞. Deduce from (v) that the first summand on the right-hand side of (III.8) is independent of R for large R. Conclude that ∇v ∈ L2 (R2 ) and, in terms of the L2 inner product R hf, giL2 := R2 f g (and its obvious analogue for vector-valued functions), we have k∇vk2L2 = −hv, ∆viL2 = −hv, uj iL2 = hvj , uiL2 ≤ k∇vkL2 kukL2 . The claimed estimate (III.7) follows.

♦

Proof of Proposition III.1.12. By point (iv) in Section III.1.3 and (III.6) we have u = A∂u = ∂K ∗ ∂u. The result then follows with the Calder´on–Zygmund inequality.



75

III.1. The a priori estimate

III.1.5

The a priori estimate

In this section we want to derive an a priori estimate, as announced in (III.1), for the (k, p)-norm of a compactly supported map u : BR → R2n in terms of the (k −1, p)-norm of ∂u, where ∂ = 12 (∂x +J0 ∂y ). The existence of such an estimate is quite surprising, since the former norm contains all derivatives of u up to order k, whereas the latter sees only a selection of these derivatives. First, we need an analogue of the Poincar´e inequality for the Lp -norm. Exercise III.1.15. Let p ∈ (1, ∞). Mimic the proof of Lemma III.1.5 — where you replace the Cauchy–Schwarz inequality with the H¨ older inequality kf gk1 ≤ kf kq kgkp , with q the dual coefficient to p, that is, p1 + 1q = 1 — to show that there is a constant c = c(p, R) such that for all C 1 -functions u : C → R2n with compact support in BR we have kukLp ≤ ck∇ukLp . By Note III.1.1 one may argue componentwise as in the proof of Lemma III.1.5.♦ We now formulate the desired estimate. Proposition III.1.16 (The a priori Estimate). For any given natural number k and real numbers p > 1 and R > 0 there is a positive constant c = c(k, p, R) such that kukk,p ≤ c k∂ukk−1,p for all u ∈ Cc∞ (BR , R2n ). Proof. We may replace the Sobolev norms in the claimed inequality by the respective equivalent norms from Note III.1.3. The statement for k = 1 follows by combining the Poincar´e inequality from Exercise III.1.15 with Proposition III.1.12. Now argue by induction on k. Suppose the estimate holds for a given k. Then, X X X kDα ukLp = kDα ukLp + kDα ukLp |α|≤k+1

|α|≤k

≤

|α|=k+1

X

c

kDα ∂ukLp +

X

kDα ukLp .

|α|=k+1

|α|≤k−1

It remains to estimate the second summand in terms of the Lp -norm of the derivatives of ∂u up to order k. The crucial point is that the ∂-operator commutes with derivatives, since J0 is a linear map on R2n . X X
kDα ukLp ≤ kDα ux kLp + kDα uy kLp |α|=k+1

|α|=k

≤

X

c

kDα ∂ux kLp + kDα ∂uy kLp




|α|≤k−1

=

X

c

kDα ∂x ∂ukLp + kDα ∂y ∂ukLp

|α|≤k−1

≤

2c

X |α|≤k

kDα ∂ukLp .




Chapter III. Bounds of Higher Order

76 This completes the inductive step.



Remark III.1.17. We wish to reiterate that for the case p = 2 we have presented a on–Zygmund estimate complete proof of Proposition III.1.16, without the Calder´ as a black box. For the base case k = 1 of the induction, you can use either Exercise III.1.6 or Exercise III.1.14. We quote without proof the following Sobolev inequality; good references are [Adams & Fournier 2003, Theorem 4.12, Part II] or [McDuff & Salamon 2012, Theorem B.1.11]. First, we define the relevant condition on the domain in this inequality. Definition III.1.18. A domain G ⊂ Rm is called a Lipschitz domain if each point of its boundary ∂G has a neighbourhood in Rm in which ∂G is the graph of a Lipschitz continuous function, and G the subgraph. Exercise III.1.19. Show that the open half-disc B1 ∩ {z > 0} ⊂ C is a Lipschitz ♦ domain. In fact, most of the time we apply the following Sobolev inequality (and related results) to domains with smooth boundary such as discs, where the Lipschitz condition is certainly satisfied. The condition p > 2 in the Sobolev inequality is tied to the fact that we are dealing exclusively with 2-dimensional domains. For the requirements on p in dependence on the dimension, see [Adams & Fournier 2003]. Proposition III.1.20 (Sobolev Inequality). For k ∈ N, p > 2 and G ⊂ C a bounded Lipschitz domain, there is a positive constant c = c(k, p, G) such that for all smooth maps u : G → Rm we have kukC k−1 (G) ≤ c kukk,p,G .



III.1.6 Why do we need Sobolev norms? There are two places in the proof of the nonsqueezing theorem where we rely on the theory of Sobolev norms and Sobolev spaces. The first one is the bootstrapping argument in the present chapter, which will give us C k -bounds on the nonstandard discs. The second instance will arise in Chapter IV, where we are going to prove a regularity result for J-holomorphic curves. There, too, we are performing a bootstrapping argument in order to show that solutions of the nonlinear Cauchy– Riemann equation (page 90) are smooth maps, even if a priori we assume smaller degrees of differentiability. The novice is usually perplexed and mystified why one has to go through the rigmarole of Sobolev theory to arrive at these goals. This section is meant to provide some sort of answer to such justifiable queries. As regards the bootstrapping argument in the present chapter, the key to everything that follows is the a priori estimate of Proposition III.1.16. We are

77

III.1. The a priori estimate

going to show by an example how this estimate fails if we replace the Sobolev norms with C k -norms. The bootstrapping argument in Chapter IV is not only concerned with improving the order of the norms involved, but also the degree of differentiability, since there we cannot work with C ∞ -maps from the start, as we shall explain in Section IV.3.6. The example in the present section will also be pertinent to the discussion there. The calculations in the example can best be done in terms of the Wirtinger derivatives of complex-valued functions. As these are not always treated in a first course on complex variables, we introduce them now. Lemma III.1.21. Let U ⊂ C be an open subset. A function f : U → C is real differentiable at z0 ∈ U if and only if there are functions A1 , A2 : U → C, continuous at z0 , such that f (z) = f (z0 ) + A1 (z)(z − z0 ) + A2 (z)(z − z 0 ) for all z ∈ U . In this case we have A1 (z0 )

=

A2 (z0 )

=


1 fx (z0 ) − ify (z0 ) 2
1 fx (z0 ) + ify (z0 ) . 2

Exercise III.1.22. (a) Prove this lemma, starting from the fact that f is real differentiable in z0 = x0 + iy0 if and only if f can be written as f (z) = f (z0 ) + ∆1 (z)(x − x0 ) + ∆2 (z)(y − y0 ), where z = x + iy, with ∆1 , ∆2 continuous at z0 . (b) We define the Wirtinger derivatives ∂f (z0 ) = A1 (z0 ), ∂z

∂f (z0 ) = A2 (z0 ). ∂z

We also write fz and fz , respectively, for these derivatives. The second statement in the lemma then translates into fz =

1 1 (fx − ify ) and fz = (fx + ify ). 2 2

In particular, ∂/∂z equals the linear Cauchy–Riemann operator ∂. (c) Show that f is complex differentiable in z0 if and only if f is real differentiable in z0 and fz (z0 ) = 0; cf. Exercise I.3.9. (d) Verify that ∂f ∂f = ∂z ∂z

and

∂f ∂f = . ∂z ∂z

Chapter III. Bounds of Higher Order

78

(e) Show that ∂/∂z and ∂/∂ z are complex linear operators that satisfy the Leibniz rule. (f) Formulate and prove the chain rule for the Wirtinger derivatives. Also consider compositions g ◦ f where f is real-valued and g a complex-valued function on a real interval, as well as the real derivative of g ◦ f when f is defined on a real interval. ♦ Part (b) of the next exercise shows that the a priori estimate fails for the C k -norms. In fact, as shown in part (a), we may not even be able to write down the estimate in a meaningful way, since ∂u may be of class C k−1 without u being of class C k . The latter will also be the reason why we need Sobolev spaces in the bootstrapping argument of Chapter IV. This example is a refinement of one described by [Sikorav 1994]. Exercise III.1.23. On B1/2 ⊂ C, the open disc of radius 1/2 centred at 0, we consider the following complex-valued function, where log denotes the real natural logarithm: ( z 2 log log |z|−2 for z ∈ B1/2 \ {0}, f (z) = 0 for z = 0. (a) In this part of the exercise we want to show that f is of class C 1 , but not C 2 , whereas fz is of class C 1 . (Strictly speaking, we only ask you to show that the derivatives fzz (z) and fz z (z) at z 6= 0 extend continuously into z = 0. With the notion of a ‘weak’ derivative (Definition IV.2.2) this can be formulated correctly as saying that the weak derivatives of fz exist and are continuous. For further details, see [Geiges, Sa˘glam & Zehmisch 2023].) (i) Prove that fz (0) = 0 by writing down an expression for f as in Lemma III.1.21. Verify that fz (z) = 2z log log |z|−2 −

z log |z|−2

for z 6= 0

by using the differentiation rules for the Wirtinger derivatives. Show that fz is continuous. (ii) Similarly, show that fz (0) = 0 and fz (z) = −

z2 z log |z|−2

for z 6= 0.

This, too, is continuous. (iii) Conclude that f is of class C 1 . (iv) Show that fzz (z) → ∞ as z → 0, so f is not of class C 2 . (v) Show that the derivatives fzz (z) and fz z (z) at z 6= 0 extend continuously into z = 0.

79

III.2. Various Sobolev estimates (b) Now consider the sequence of functions fν : B1/2 → C, ν ∈ N, defined by ( z|z|1/ν log log |z|−2 for z ∈ B1/2 \ {0}, fν (z) = for z = 0. 0 (i) Show that fν is of class C 1 by computing ∂fν and ∂fν .

(ii) Show that k∂fν kC 0 is bounded uniformly in ν, whereas k∂fν kC 0 and hence kfν kC 1 goes to infinity as ν → ∞. (iii) Use a cut-off function to produce a counterexample to the a priori estimate in the C k -norms, i.e. find a sequence of compactly supported C 1 -functions uν : B1/2 → C that violates the inequality kuν kC 1 ≤ ck∂uν kC 0 for any constant c. ♦

III.2

Various Sobolev estimates

In this section we shall assume throughout that G ⊂ C is a bounded Lipschitz domain, p is a real number greater than 2, and k is a natural number, that is, k ≥ 1. These assumptions are essential in Lemma III.2.2, where we use the Sobolev inequality. Exercise III.2.1. Let α be a double index as in Section II.2.1, and h . , . i the standard Euclidean inner product on Rm . Prove the Leibniz rule for the scalar product of two functions u, v ∈ C ∞ (G, Rm ): X
Dα hu, vi = cβγ hDβ u, Dγ vi, β+γ=α

where β, γ are likewise double indices, and the cβγ suitable natural numbers. Give a formula for cβγ . ♦ Lemma III.2.2. There is a constant c = c(k, p, G) such that for any functions u, v ∈ C ∞ (G, Rm ) with finite (k, p)-norm we have khu, vikk,p ≤ c kukk,p kvkk,p . Proof. The Sobolev inequality gives kukC 0 (G) ≤ const. kukk,p , whence Z khu, vikLp (G)

|hu, vi|

=

p

1/p

G

≤

kukC 0 kvkLp

≤

const. kukk,p kvkk,p .

Chapter III. Bounds of Higher Order

80

Now let β, γ be double indices with |β + γ| = k. If |β| < k, we estimate khDβ u, Dγ vikLp

≤

kDβ ukC 0 kDγ vkLp

≤

const. kDβ ukk−|β|,p kvk|γ|,p

≤

const. kukk,p kvkk,p ;

the condition k − |β| ≥ 1 is used when we apply the Sobolev inequality in the second line. For |β| = k the same inequality can be proved by exchanging the roles  of u and v. With the Leibniz rule, the lemma follows. Exercise III.2.3. Consider the function u : B1 → R defined by u(reiθ ) = p

1

for r 6= 0;

r(1 − r)

the value u(0) may be chosen arbitrarily. Show that u ∈ L1 , and that its integral with respect to the area form r dr ∧ dθ (i.e. the Lebesgue measure on R2 ) equals R u = π 2 . Show further that u2 is not in L1 . This illustrates that Lemma III.2.2 B1 fails for k = 0 (and p = 1). How does one have to modify the example for larger p? If you prefer an example where the norms of u, v and the product uv are all finite, you may consider the function uε , for ε > 0, defined by uε (reiθ ) = p

1 r(1 + ε − r)

.

For ε sufficiently small, the inequality in Lemma III.2.2 (with u = v = uε , k = 0 and p = 1) will be violated. ♦ Lemma III.2.4. There is a real polynomial P = Pk,p,G of degree k with nonnegative coefficients such that for any u ∈ C ∞ (G, Rm ) of finite (k, p)-norm and any f ∈ C ∞ (Rm , R) of finite C k -norm we have
kf ◦ ukk,p ≤ kf kC k P kukk,p . Proof. We begin with the estimate
f u(z) − f (0)

≤

Z 1 
 d f tu(z) dt 0 dt Z 1
h∇f tu(z) , u(z)i dt

≤

k∇f kC 0 |u(z)|.

=

0

This estimate yields Z kf ◦ u − f (0)kLp =

|f ◦ u − f (0)| G

p

1/p ≤ kf kC 1 kukLp .

III.3. The C k -norm of nonstandard discs is bounded

81

With the triangle inequality kf ◦ ukLp ≤ kf ◦ u − f (0)kLp + kf (0)kLp and the estimate kf (0)kLp ≤ area(G) kf kC 0 we find
kf ◦ ukLp ≤ const. kf kC 1 kukLp + 1 . For the first derivatives of f ◦ u we observe, with ∂x (f ◦ u) = (df ◦ u) · ∂x u, that k∂x (f ◦ u)kLp ≤ kf kC 1 k∂x ukLp , and similarly for the derivative with respect to the y-coordinate. In total, this gives us the estimate
kf ◦ uk1,p ≤ const. kf kC 1 kuk1,p + 1 , which is the statement of the lemma for k = 1. We now proceed by induction on k, starting from the estimate kf ◦ ukk,p ≤ kf ◦ uk1,p + k∂x (f ◦ u)kk−1,p + k∂y (f ◦ u)kk−1,p . For the second summand on the right-hand side we have, for k ≥ 2, k∂x (f ◦ u)kk−1,p

=

kh∇f ◦ u, ∂x uikk−1,p

≤

const. k∇f ◦ ukk−1,p k∂x ukk−1,p by Lemma III.2.2
const. kf kC k Pk−1 kukk−1,p kukk,p

≤

by the inductive hypothesis. A similar estimate is obtained for ∂y (f ◦u). Collecting terms, we complete the inductive step and establish the lemma.  Exercise III.2.5. Let v ∈ C ∞ (G, Rm ) be a further function satisfying the assumptions made on u in the preceding lemma. By inspecting the proof, show that
kf ◦ u − f ◦ vkk,p ≤ kf kC k P 0 ku − vkk,p with a polynomial P 0 of degree k having non-negative coefficients and no constant ♦ term.

III.3

The C k -norm of nonstandard discs is bounded

Here is the desired C k -bound, for each k ∈ N, on the nonstandard discs. As noted at the end of Section II.2.2, this completes the proof of Theorem II.2.7 on the properness of the evaluation map.

82

Chapter III. Bounds of Higher Order

Proposition III.3.1. For every k ∈ N there is a constant c = ck > 0 such that kukC k (D) ≤ c for all u ∈ Mnst . Thanks to the Sobolev inequality (Proposition III.1.20), it suffices for the proof of this proposition to establish bounds on the Sobolev (k, p)-norms for some p > 2 and all k ∈ N. The estimates that follow hold under the more general assumption p ∈ (1, ∞); the condition p > 1 is required because of our reliance on the a priori estimate. Arguing by contradiction, suppose that Mnst were not bounded in the Sobolev (k, p)-norm for some k. Then we could find a sequence (uν )ν∈N in Mnst such that kuν kk,p,B1 > ν for all ν ∈ N. As shown in Chapter II, the space Mnst is C 1 -bounded, and hence by Corollary II.2.10 (a) to the Arzel` a–Ascoli theorem we may assume (by passing to a subsequence) that uν −→ u ∈ C 0 (D) as ν → ∞. Thus, we arrive at the desired contradiction if we can prove, by bootstrapping, the next lemma. Lemma III.3.2. Let (uν )ν∈N be a C 1 -bounded sequence in M with uniform limit u ∈ C 0 (D, R2n ). Then for every k ∈ N there is a constant c = c(k, p, u, J ) > 0 such that kuν kk,p,B1 ≤ c for sufficiently large ν ∈ N. Wait, you say. Didn’t we appeal to a bootstrapping argument on page 57 to establish the C 1 -bound on Mnst in the first place? It seems we have been caught cheating, or trapped in a circular argument. Lest the reader suspect us of boasting after the manner of M¨ unchhausen, let us clarify the logic of our reasoning. Assuming that Mnst were not C 1 -bounded, we constructed a C 1 -bounded sequence (vν ) of holomorphic discs defined on larger and larger domains BRν ε . Thus, starting from a sufficiently large index ν, we may regard this as a sequence of holomorphic maps defined on some large closed disc. In the first instance, we apply Lemma III.3.2 to this sequence, with a minor modification of the proof, as we shall explain in Remark III.3.11. As argued on page 57 et seq., this leads to a contradiction, which establishes the C 1 -bound on Mnst . Now the lemma also applies to the sequence (uν ) in Mnst . Exercise III.3.3. Use the compactness of D and Note III.1.3 to prove the conclusion of Lemma III.3.2 for k = 1. ♦ Proof of Lemma III.3.2. We prove this by induction on k. The case k = 1 is covered by the preceding exercise, so it remains to perform the inductive step from k to k + 1. Let {Uj : j = 0, 1, . . . , N } be a covering of D by open subsets — these will be specified later — with the property that Uj ∩ ∂D = ∅ for j ≥ 1. Furthermore,

III.3. The C k -norm of nonstandard discs is bounded

83

let {fj : j = 0, 1, . . . , N } be a partition of unity subordinate to this covering, in the sense that the supports supp fj := {z ∈ D : fj (z) 6= 0} of the smooth functions fj : D → [0, 1] satisfy supp fj ⊂ Uj , and

PN

j=0

j = 0, 1, . . . , N,

fj ≡ 1. Then, kuν kk+1,p,B1 ≤

N X

kfj uν kk+1,p,Uj ,

j=0

so it suffices to find a bound on each summand on the right. Localisation near the boundary Recall that the almost complex structure J on R2n coincides with the standard structure J0 outside a compact subset. The uniform limit u of the sequence (uν ) in M is a continuous map u : (D, ∂D) −→ (R2n , Lt ) for a suitable t ∈ Rn−1 . Exercise III.3.4. Use the uniform convergence of (uν ) to show that there is a δ ∈ (0, 12 ), depending only on u, such that every uν with ν sufficiently large maps U0 := D \ D1−δ into the region of R2n where J = J0 ; cf. the proof of Lemma II.1.9.

♦

This means that the component functions of uν |U0 are holomorphic maps U0 → C. By Schwarz reflection in the unit circle (Exercise II.1.7 for the first component, Proposition II.1.4 for the remaining n − 1 components), uν |U0 extends to a holomorphic map u ˆν defined on n 1 o U0 ∪ (1/U 0 ) = 1 − δ < |z| < ⊂ B2 . 1−δ Extend the function f0 , which is supported in U0 and constant equal to 1 near ∂D, to a smooth function (still denoted f0 ) B2 → [0, 1] supported in the annulus U0 ∪ (1/U 0 ). We then have the following estimates: kf0 uν kk+1,p,U0

≤

kf0 u ˆν kk+1,p,B2

≤

const. k∂(f0 u ˆν )kk,p,B2 by Proposition III.1.16

=

const. k∂f0 · u ˆν kk,p,B2 since u ˆν is holomorphic

≤

const. k∂f0 kk,p,B2 kuν kk,p,B1 by Lemma III.2.2;

Chapter III. Bounds of Higher Order

84

uν kk,p,B2 to kuν kk,p,B1 in the last line we appeal to the next for the passage from kˆ exercise. Also notice that we could equally write kf0 kC k+1 instead of k∂f0 kk,p,B2 , since we are only estimating up to constants. Exercise III.3.5. Let ϕ : G1 → G2 be a diffeomorphism of bounded domains G1 , G2 ⊂ C that extends smoothly to a neighbourhood of the closure G1 . Show that for every k ∈ N and p ≥ 1 there is a constant c = c(k, p, ϕ) > 0 such that 1 kukk,p,G2 ≤ ku ◦ ϕkk,p,G1 ≤ ckukk,p,G2 . ♦ c By Exercise III.3.4, the bump function f0 depends on the limit u only, so by the inductive hypothesis we conclude kf0 uν kk+1,p,U0 ≤ const. for ν large, with a constant depending only on u (and k, p, J, of course). Localisation in the interior Here is the key idea of the argument. In the interior, the uν may not be holomorphic (that is, J0 -holomorphic), so when we apply the a priori estimate (Proposition III.1.16) to fj uν , j = 1, . . . , N , we do not lose one derivative of uν . However, by localisation and an affine transformation we are ‘almost’ in a holomorphic situation. This means that we cannot quite get rid of the derivative of uν of order k + 1, but we can make it appear in the estimate with a factor smaller than 1; see (III.12) below. M¨ unchhausen would have been proud of this idea. Consider a point p ∈ R2n in the image u(B1 ). Exercise III.3.6. Use the fact that the tangent vectors ∂x1 , . . . , ∂xn ∈ Tp R2n span a Lagrangian subspace with respect to ω, and the compatibility of J with ω, to prove that ∂x1 , . . . , ∂xn , J(p)∂x1 , . . . , J (p)∂xn is a basis for the vector space Tp R2n .

♦

Let Ap be the endomorphism of R2n defined by Ap ∂xj = ∂xj and Ap J(p)∂xj = ∂yj ,

j = 1, . . . , n.

Then Ap is invertible, and Ap J(p) = J0 Ap . Let ϕp be the affine map z 7−→ Ap (z − p) of R2n . The differential of ϕp is Ap , so the equation Ap J(p) = J0 Ap means that the differential

Tp ϕp : Tp R2n , J(p) −→ T0 R2n , J0 ≡ T0 Cn is complex linear. In other words, the almost complex structure Jp∗ := (ϕp )∗ J satisfies Jp∗ (0) = J0 .

III.3. The C k -norm of nonstandard discs is bounded

85

Exercise III.3.7. Verify this last claim. See Section I.3.3 for the definition of the ♦ push-forward of almost complex structures. Write ck for the constant in the a priori estimate from Proposition III.1.16 (with R = 1 and some fixed p), where we use the (k, p)-norm from Note III.1.3. This choice of norm (and corresponding constant in the estimate) is important for the control of one particular term, as we shall see. We can choose r > 0 so small that, with Br (p) ⊂ R2n denoting the open ball of radius r centred at p, kJ0 − Jp∗ kC 0 (ϕp (Br (p)))
0 chosen such that u(Uj ) ⊂ Brj /2 (pj ), the almost complex structure Jj := (ϕpj )∗ J satisfies kJ0 − Jj kC 0 (ϕpj (Brj (pj )))
0, for ν sufficiently large we may regard vν as being defined on the closed disc DR . The sequence (vν ) is C 1 -bounded (see page 57), and hence has a uniform limit in C 0 (DR , R2n ). The proof of Lemma III.3.2 cannot be used without modification, since the vν are not J0 -holomorphic near ∂DR , so the boundary localisation argument does not apply. Instead, we argue as follows. Choose some δ > 0 and ν0 (δ) sufficiently large such that the vν are defined on BR+δ for ν ≥ ν0 . Cover DR by finitely many open sets U1 , . . . , UN ⊂ BR+δ . Let {fj : j = 1, . . . , N } be a subordinate partition of PN unity, in the sense that supp fj ⊂ Uj , and j=1 fj ≡ 1 on DR . Here the Uj are chosen as in the ‘localisation in the interior’ argument before, and the remainder of the proof is unchanged. The next statement is a corollary to the proof of Lemma III.3.2. Corollary III.3.12. Let (uν )ν∈N be a sequence in M converging to u ∈ M with respect to the (1, p)-norm. Then, uν → u with respect to all (k, p)-norms.

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Chapter III. Bounds of Higher Order

Proof. We apply the inductive argument in the preceding proof to the sequence uν − u, which converges to 0 in W 1,p . The bootstrapping argument for the localisation near the boundary works as before and shows that f0 (uν − u) → 0 in all (k, p)-norms. For the localisation in the interior we observe that the C 0 -convergence to which we appeal after (III.9) is guaranteed by the Sobolev inequality. Further, when we apply the affine maps ϕpj to both uν and u, the constant term drops out in the difference uνj − uj , which means that the summand 1 in (III.10) disappears; cf. Exercise III.2.5. Equation (III.11) then becomes kujν − uj kk,p,Uj ≤ const. kuν − ukk,p,B1 . As you use this information in the remainder of the proof of Lemma III.2.4, you will see that the final estimate in that proof becomes kfj (uνj − uj )kk+1,p,Uj ≤ const. kuν − ukk,p,B1 , which is exactly what we need for the inductive step.



Exercise III.3.13. Verify that the final estimate is as claimed.

♦

Chapter IV

Elliptic Regularity The ancient poetess singeth, that Hesperus all things bringeth, Smoothing the wearied mind. Alfred Lord Tennyson, Leonine Elegiacs

In this chapter we shall describe our moduli space M of holomorphic discs as the set of smooth solutions of a nonlinear differential equation, the so-called nonlinear Cauchy–Riemann equation ∂ J u = 0. In order to show that this space is a manifold, one needs infinite-dimensional versions both of the implicit function theorem and Sard’s theorem. The first theorem enables us to establish the manifold property of subsets of manifolds that arise as preimages of regular values of some differentiable map; the second theorem we use to find such regular values. ‘Regularity’ in this sense will be the topic of Chapter V. The ‘regularity’ in the title of this chapter is of a different nature. The problem we face is that the space of smooth functions we are dealing with is a Fr´echet space, and in such spaces there is, unfortunately, no implicit function theorem. There is, however, such a theorem for Banach spaces, so the idea is to work in a suitable completion of our function space that does carry a Banach space structure. One might simply try to work with spaces of C k -functions for some finite k, which are Banach, and hope that C k -solutions of the equation ∂ J u = 0 are actually smooth. In a course on complex analysis you will have seen this phenomenon of ‘regularity’ for honest holomorphic functions: a C 1 -solution of the (linear) Cauchy– Riemann equation ∂u = 0 is automatically of class C ∞ . As explained in Section III.1.6, and further in Section IV.3.6 below, this kind of bootstrapping is bound to fail, in general, and it does fail for the nonlinear Cauchy–Riemann equation. Instead, one has to work with Sobolev spaces W k,p of functions with merely Lp -integrable weak derivatives up to order k. The miracle of elliptic regularity is that in the realm of W k,p -solutions of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Geiges, K. Zehmisch, A Course on Holomorphic Discs, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-36064-0_4

89

Chapter IV. Elliptic Regularity

90

equation ∂ J u = 0, one can indeed bootstrap with respect to k. Functions that are of class W k,p for all k ∈ N (and some p > 2) are then seen to be smooth by the Sobolev embedding theorem. The term ‘elliptic’ refers to the fact that this kind of bootstrapping is specific to elliptic differential operators, of which the Laplace operator (to which the ∂-operator is closely linked) is the archetypical example. But we shall not pursue this general ‘elliptic’ theory. In Section IV.1 we describe the basic set-up for this approach via the implicit function theorem. We discuss various notions of differentiability in Banach spaces and the concept of Banach manifolds. It will be shown that spaces of maps, even merely continuous ones, between (finite-dimensional) manifolds are Banach manifolds. In Section IV.2 we present some basic material on Sobolev spaces, and we describe the Sobolev completion of the function space on which we plan to study the equation ∂ J u = 0. Regularity of solutions to this equation is then the topic of Section IV.3.

IV.1

The linearisation

Write C for the space of all C ∞ -maps u : (D, ∂D) −→ (R2n , ∂Z) that satisfy conditions (M1), (M2) and (M3) given in Notation I.8.10. As in Exercise II.2.11 one argues that C is a closed subset of the real vector space C ∞ (D, R2n ). The nonlinear Cauchy–Riemann operator ∂ J is the continuous map ∂J :

C u

−→ 7−→

C ∞ (D, R2n ) ux + J(u)uy .

For convenience we have dropped the factor 1/2 present in the classical linear Cauchy–Riemann operator, where J ≡ J0 . The kernel of ∂ J is made up of the J-holomorphic maps in C, so our moduli space M is the solution space of a nonlinear equation, M = {u ∈ C : ∂ J u = 0}. There are two aspects to the nonlinearity of ∂ J : the space C is not a linear subspace (because of the conditions on u|∂D ), and the coefficients of the differential equation ∂ J u = 0 depend on u. Given this description of M as the kernel of ∂ J , our aim will be to establish the differentiability of the operator ∂ J , and to give a description of its linearisation. This will ultimately allow us to equip M with a manifold structure. Along the way, there are a number of quite formidable technical problems we need to overcome. The first one is that on a space of C ∞ maps (even a linear one), there is only a Fr´echet metric, but no norm. So the space C is merely a Fr´echet

91

IV.1. The linearisation

manifold, for which there is no implicit function theorem that one might apply to the equation ∂ J u = 0.

IV.1.1 Notions of differentiability We briefly recall the most important notions of differentiability for maps between Fr´echet or Banach spaces, and we illustrate the failure of the implicit function theorem for Fr´echet spaces. The first notion of differentiability is concerned with directional derivatives. Definition IV.1.1. Let E, F be Fr´echet or Banach spaces and U ⊂ E an open subset. A map f : U → F is called Gateaux1 differentiable at u ∈ U if for each h ∈ E the directional derivative ∂h f (u) := lim

t→0

f (u + th) − f (u) t

exists, and h 7→ ∂h f (u) is a continuous linear2 map. The map f is said to be differentiable of class C 1 on U if f is Gateaux differentiable on U , and the map U u

−→ 7−→

 L(E, F), h 7→ ∂h f (u)

is continuous; here L(E, F) denotes the space of continuous3 linear maps E → F. Remark IV.1.2. When E, F are Banach spaces, the topology on L(E, F) is induced by the operator norm. In particular, this space of linear maps is again a Banach space. In the Fr´echet case, L(E, F) carries the bounded-open topology, i.e. the topology of uniform convergence on bounded sets. In the Banach setting, higher orders of continuous differentiability can be defined analogously, thanks to L(E, F) being a Banach space. In general, L(E, F) is not Fr´echet when E and F are. Therefore, when one wants to define higher orders of continuous differentiability of maps between Fr´echet spaces, one uses the weaker notion we mention below right before Exercise IV.1.6, which can be iterated for higher derivatives. As you know from elementary analysis in finite dimensions, a Gateaux differentiable map (i.e. a map whose partial derivatives exist) that is not of class C 1 need not even be continuous. The following example, illustrating the failure of the implicit function theorem in Fr´echet spaces, goes back to [Eells 1966]. 1 In the birth roll of Vitry-le-Fran¸ cois for the year 1889, and on papers published during his lifetime — cut short by his death in combat in October 1914 —, the name of Ren´ e Gateaux is written without an accent circonflexe on the letter ‘a’. In many posthumous publications his name appears as ‘Gˆ ateaux’. 2 Beware that some authors do not insist on the linearity of this map. 3 Equivalently, this is the space of bounded linear maps. For more on linear operators and the notions bounded/continuous, see Section V.1.1.

Chapter IV. Elliptic Regularity

92

Exercise IV.1.3. On the space C 0 (R) of continuous functions R → R we have a sequence of seminorms | . |k , k ∈ N0 , defined by  |u|k := max |u(x)| : x ∈ [−k, k] . The corresponding Fr´echet metric defines the topology of uniform convergence on compact sets (or compact-open topology), and we write E for the space C 0 (R) with this topology. Show that the map f : E → E, u 7→ exp(u), is of class C 1 , and the map h 7→ ∂h f (u) is invertible at each u ∈ E. Show further that every neighbourhood of the constant function 1 = exp(0) in E contains functions that assume negative values. This implies that no neighbourhood of 1 is the image under f of a neighbourhood ♦ of 0. There is, however, an implicit function theorem for Banach manifolds. In order to have a Banach manifold to work with — where the transition maps are differentiable, by a rather unfortunate nomenclature, in the sense of Fr´echet — we shall pass to a completion of C in a suitable Sobolev space, where we can still make sense of the equation ∂ J u = 0. The remaining steps are then to prove that - the solutions of the equation ∂ J u = 0 are actually of class C ∞ thanks to the miracle of elliptic regularity (this is the content of the present chapter); - for an apt choice of almost complex structure J one has the required transversality to make the solution space a manifold (this point will be addressed in Chapter V). The second differentiability notion for maps between Banach spaces is the usual ‘total’ differentiability as in finite-dimensional analysis. Definition IV.1.4. Let E, F be Banach spaces and U ⊂ E an open subset. A map f : U → F is called Fr´echet differentiable at u ∈ U if there exists a continuous linear map Tu f : E → F such that lim

h→0

f (u + h) − f (u) − Tu f (h) = 0. khkE

Fr´echet differentiability obviously implies continuity. Also, Fr´echet differentiability implies Gateaux differentiability, and then ∂h f (u) = Tu f (h), but not conversely. In finite-dimensional analysis, continuity of the partial derivatives implies total differentiability. This remains true for Banach spaces, if continuity of the partial or directional derivatives is interpreted as in Definition IV.1.1. Proposition IV.1.5. Let E, F be Banach spaces and U ⊂ E an open subset. If f : U → F is differentiable of class C 1 in the sense of Definition IV.1.1, then f is Fr´echet differentiable. 

93

IV.1. The linearisation

The proof of this proposition goes like that in finite dimensions, once one has established an analogue of the mean value theorem, for which one needs the Hahn–Banach theorem. For details, see [Werner 2011, Satz III.5.4], or any other good text on functional analysis. In finite dimensions, continuity of the map (u, h) 7→ ∂h f (u) is equivalent to the continuity of u 7→ {h 7→ ∂h f (u)}. In infinite dimensions this is not true, and the following example illustrates that the first (and weaker) notion of continuity is not sufficient to guarantee Fr´echet differentiability. This example, and some erroneous claims about it that have slipped into the literature, are discussed at length in [Gieraltowska-Kedzierska & Van Vleck 1992]. 1 Exercise IV.1.6. We consider the Banach R π spaces E = L ([0, π]) and F = R, and the map f : E → F defined by f (u) = 0 sin u(x) dx.

(a) Show that f is Gateaux differentiable with derivative π

Z

h(x) cos u(x) dx.

∂h f (u) = 0

(b) Observe that under the usual identification of the dual space (L1 )∗ with L∞ , the linear map h 7→ ∂h f (u) corresponds to cos u ∈ L∞ . Use this, or a direct argument, to show that the map u 7→ {h 7→ ∂h f (u)} is not continuous. (c) Show that the map (u, h) 7→ ∂h f (u) is continuous. Hint: Consider sequences (uν ), (hν ) in L1 with uν → u∞ and hν → h∞ . Write hν cos uν − h∞ cos u∞ = (hν − h∞ ) cos uν + h∞ (cos uν − cos u∞ ), and use Lebesgue’s dominated convergence theorem to control the integral of the second summand. Beware that one cannot apply this theorem directly, since cos uν − cos u∞ may not be pointwise convergent almost everywhere. However, if |h∞ (cos uν − cos u∞ )| did not converge to 0 in L1 , one could pick a subsequence with integrals bounded from below by some ε > 0, and one could pick a subsequence of that subsequence that converges pointwise a.e. to the zero function. Then, Lebesgue’s theorem leads to a contradiction. (d) If R πf were Fr´echet differentiable at u = 0, we would have T0 f (h) = ∂h f (0) = h. Show that f is not Fr´echet differentiable at u = 0 by taking hν — 0 in the defining equation for Fr´echet differentiability — as the characteristic function of the interval [0, 1/ν], ν ∈ N. (e) Show that the restriction of f to the subset L2 ([0, π]) of L1 ([0, π]) is differentiable of class C 1 and hence Fr´echet differentiable. ♦

Chapter IV. Elliptic Regularity

94

IV.1.2

The linearisation of C k

We now consider the spaces C k , k ∈ N0 , which are defined just like the space C at the beginning of this chapter, except that we do not require the maps u to be of class C ∞ , but merely of class C k . We want to show that these spaces are Banach manifolds, and we describe the tangent space at a point of C k . In fact, we proceed the other way round: we shall first introduce a natural concept of ‘tangent space’ and build the manifold structure from there. In order to emphasise that the differentiability of the coordinate changes does not hinge on the differentiability properties of the maps u making up the space, we formulate everything for the space C 0 , but what we say in this and the next section holds for any C k . By the boundary condition (M1), for any given t ∈ Rn−1 we may define the subset Ct0 ⊂ C 0 as the set of maps of level t, that is, those u ∈ C 0 with u(∂D) ⊂ Lt . The tangent space Tu Ct0 at a point u ∈ Ct0 is made up of the vector fields along u such that an infinitesimal shift in the direction of this vector field will preserve the boundary conditions (M1) and (M3). More formally: Definition IV.1.7. The tangent space Tu Ct0 at u = (u1 , . . . , un ) ∈ Ct0 (written in complex components) is the real vector space of all ξ = (ξ1 , . . . , ξn ) ∈ C 0 (D, R2n ) such that (T0)

ξ(z) ∈ Tu(z) R2n for all z ∈ D.

(T1)

ξ(z) ∈ Tu(z) Lt for all z ∈ ∂D, that is, ξ1 (z) is a real multiple of iu1 (z), and ξ2 (z), . . . , ξn (z) are real.

(T2)

ξ1 (ik ) = 0 for k ∈ {0, 1, 2}.

These tangent spaces are Banach spaces with respect to the supremum norm. Our aim will be to equip first Ct0 and then the full space C 0 with the structure of a Banach manifold, that is, a topological space locally modelled on a Banach space, just like a (traditional) manifold is modelled on a finite-dimensional vector space. The local charts will come from an exponential map that we define in analogy with the usual exponential map in Riemannian geometry. Choose a Riemannian metric g1 on R2 = C with g1 = dr2 + dθ2 on an annulus around ∂D in polar coordinates z = reiθ , and Pn g1 Euclidean outside a compact subset of R2 . Let gst be the standard metric i=2 (dx2i + dyi2 ) on R2n−2 , and set g = g1 ⊕ gst on R2 ⊕ R2n−2 . Exercise IV.1.8. Convince yourself that the Riemannian manifold (R2n , g) is geodesically complete (i.e. geodesics exist for all times), and the cylinders Lt are totally geodesic submanifolds (i.e. geodesics in (R2n , g) starting tangentially to Lt stay in this cylinder). ♦ Write expq : Tq R2n → R2n for the exponential map with respect to the metric g at the point q ∈ R2n , that is, expq (v) := γv (1), where γv is the unique geodesic determined by the initial conditions γv (0) = q and γ˙ v (0) = v ∈ Tq R2n .

95

IV.1. The linearisation

Definition IV.1.9. The exponential map on the space Ct0 at the point u ∈ Ct0 is the map Ct0 Tu Ct0 −→ ξ 7−→ expu (ξ) defined by

expu (ξ) (z) := expu(z) ξ(z) ,

z ∈ D.

Exercise IV.1.10. Verify that for ξ ∈ Tu Ct0 , the map expu (ξ) does indeed belong to the space Ct0 . Condition (M1) holds thanks to Exercise IV.1.8, but you also need ♦ to check (M2) and (M3). To define charts on the full space C 0 , containing all levels Ct0 , we use the exponential map on a fixed level t0 , and then compose it with the linear shift on Cn = C ⊕ Cn−1 in the imaginary direction of Cn−1 . Write ϕt , t ∈ Rn−1 , for the shift (w, x + iy) 7→ (w, x + i(y + t)). Then define charts of C 0 by Tu Ct00 ⊕ Rn−1 (ξ, t)

−→ 7−→

C0 ϕ ◦ expu (ξ). t

(IV.1)

Observe that the metric g on R2n = Cn is invariant under the map ϕt . Since g = g1 ⊕ gst is a product metric on R2 ⊕ R2n−2 , and on the R2n−2 -factor the exponential map is simply linear translation, it follows that
ϕt ◦ expu (ξ) = expu ξ + (0, it) . For u of level t0 , we can identify Tu Ct00 ⊕ Rn−1 with the tangent space Tu C 0 via (ξ, t) 7→ ξ + (0, it). So the charts of C 0 are really defined by an exponential map on tangent spaces, just like the charts on Ct0 . The boundary conditions (M1), (M2) and (M3) are irrelevant for the discussion of the manifold structure on C 0 , so we formulate the main result in a slightly more general setting.

IV.1.3

Spaces of continuous maps are smooth Banach manifolds

For concreteness we consider the space C 0 (D, M ) of continuous maps D → M , where M can be any manifold (without boundary). In principle, we could replace D by any compact manifold, with or without boundary. Remark IV.1.11. As mentioned in the preface, our presentation follows unpublished course notes by Kai Cieliebak. Another detailed and rigorous proof of the statement in the section heading can be found in [Wittmann 2019]. On M we pick a Riemannian metric. This gives rise to a metric dM in the usual sense by defining dM (p, q) as the infimum over the lengths of smooth paths between p, q ∈ M . A metric on C 0 (D, M ) is then given by d(u, v) := maxz∈D dM (u(z), v(z)).

Chapter IV. Elliptic Regularity

96

The Riemannian metric on M also gives rise to an exponential map D v

Exp :

−→ 7−→

M ×M
π(v), γv (1) ,

where D is a sufficiently small open neighbourhood of the zero section of the tangent bundle π : T M → M such that the geodesics γv exist until time 1. For the manifold structure on C 0 (D, M ) we shall have to work in any case with a small neighbourhood of the zero section of T M , so there is no advantage in assuming the Riemannian metric to be complete. Observe that Exp maps the zero section of T M diffeomorphically onto the diagonal in M × M ; under the natural identification of the zero section with M , this map is given by p 7→ (p, p). Lemma IV.1.12. For D sufficiently small, Exp |D is a diffeomorphism onto its image. One may choose D such that Exp(D) is invariant under the involution τ : (p, q) 7→ (q, p) of M × M . Proof. Write 0p ∈ Tp M for the zero tangent vector at p ∈ M . The tangent space T0p (T M ) decomposes as a direct sum T0p (T M ) = T hor ⊕ T vert of the horizontal tangent space T hor := Tp (zero section) and the vertical tangent space T vert := T0p (Tp M ), both of which can be canonically identified with Tp M . Consider curves in T M of the form t 7→ 0β(t) , with β(0)
= p, and t 7→ tv, with
v ∈ Tp M
. Under Exp these curves map to t 7→ β(t), β(t) and t 7→ p, γtv (1) = p, γv (t) , respectively, the latter thanks to geodesics being of constant speed. Thus, the differential T0p Exp : T hor ⊕ T vert −→ Tp M ⊕ Tp M, with respect to the mentioned canonical identification, is given by   id 0 T0p Exp = , id id where id denotes the identity on Tp M . So the differential of Exp is invertible along the zero section, and the implicit function theorem yields the first statement of the lemma. If we replace a given D by
Exp−1 Exp(D) ∩ τ (Exp(D)) , the invariance condition will be satisfied.



97

IV.1. The linearisation

For a careful discussion of the manifold charts of C 0 (D, M ) and the differentiability of the coordinate changes it is opportune to define the tangent space a little more formally than in Definition IV.1.7. The tangent space Given u ∈ C 0 (D, M ), we consider the pull-back bundle  u∗ T M = (z, v) ∈ D × T M : u(z) = π(v) . This fits into the following commutative diagram: u∗ T M

u ˜ TM

u∗ π ? D

π u - ? M,

where u∗ π(z, v) = z and u ˜(z, v) = v. Definition IV.1.13. The tangent space Tu := Tu C 0 (D, M )




of C 0 (D, M ) at u is the space Γ(u∗ T M ) of continuous sections of u∗ T M . ˜ Thus, an element ξ˜ ∈ Tu is a continuous map of the form z 7→ ξ(z) = (z, ξ(z)) with ξ(z) ∈ Tu(z) M , which means that Definitions IV.1.7 and IV.1.13 are essentially equivalent. The tangent space Tu becomes a Banach space with the norm ˜ := max |ξ(z)|, kξk z∈D

where |ξ(z)| denotes the norm of ξ(z) ∈ Tu(z) M with respect to the Riemannian metric on M . The manifold charts The set Du := Γ(u∗ D) ⊂ Tu is an open neighbourhood of 0 ∈ Tu , that is, of the zero section z 7→ (z, 0u(z) ) in u∗ T M . As before, except for the formal distinction between ξ˜ and ξ, we define the exponential map expu on Du by

˜ (z) := exp expu (ξ) u(z) ξ(z) , z ∈ D. This map is a homeomorphism onto its image 
Eu := expu (Du ) = v ∈ C 0 (D, M ) : u(z), v(z) ∈ Exp(D) for all z ∈ D .

Chapter IV. Elliptic Regularity

98

Notice that u is the image of the zero section under expu , and Eu is an open neighbourhood of u in C 0 (D, M ). Moreover, v ∈ Eu holds if and only if u ∈ Ev , thanks to the τ -invariance of Exp(D). Remark IV.1.14. These local parametrisations expu

Tu ⊃ Du −−−−−→ Eu ⊂ C 0 (D, M ) are consistent with the interpretation of Tu as the tangent space of C 0 (D, M ) at u, since ˜ T0 expu (ξ)(z) = = =

d ˜ |s=0 expu (sξ)(z) ds d |s=0 expu(z) (sξ(z)) ds ξ(z),

where we have used that T0 expu(z) equals the identity map on Tu(z) M . Coordinate changes are differentiable We are now ready to show that the atlas  0 (Eu , exp−1 u ) : u ∈ C (D, M ) gives C 0 (D, M ) a smooth Banach manifold structure. For v ∈ Eu , the intersection Eu ∩ Ev is an open neighbourhood of both u and v. Proposition IV.1.15. The coordinate change −1 −1 huv := exp−1 v ◦ expu : expu (Eu ∩ Ev ) −→ expv (Eu ∩ Ev )

is infinitely often Fr´echet differentiable. Proof. We first want to interpret the domain of definition exp−1 u (Eu ∩Ev ) as a space of sections. We shall then find that the Fr´echet differential only ‘sees’ the fibrewise direction, so the mere continuity of u and v does not obstruct differentiability. Regarding the first issue, we have  ˜ ∈ Ev exp−1 ξ˜ ∈ Du : expu (ξ) u (Eu ∩ Ev ) = 
= ξ˜ ∈ Γ(u∗ D) : v(z), exp ξ(z) ∈ Exp(D) for all z ∈ D u(z)

=

Γ(O)

with 
O := (z, v) ∈ u∗ D : v(z), expu(z) (v) ∈ Exp(D) . Observe that O is an open neighbourhood of the zero section {(z, 0u(z) ) : z ∈ D} in u∗ T M .

99

IV.1. The linearisation ˜ ∈ Tv is given by Next, for ξ˜ ∈ Γ(O) the image huv (ξ)

˜ (z) = z, exp−1 ◦ exp huv (ξ) u(z) ξ(z) , v(z) ˜ where Φuv is the fibre-preserving map ˜ = Φuv ◦ ξ, so we can write huv (ξ) Φuv :

O (z, v)

−→ 7−→

v∗ T M
◦ expu(z) (v) .

z, exp−1 v(z)


We introduce the notation Φuv (z, v) = z, φuv (z, v) . Then, writing dfib for the derivative with respect to the fibre coordinate v, we define the linear map dξ˜huv : Tu −→ Tv by
dξ˜huv (˜ η )(z) := z, dfib ˜ φuv (η(z)) , ξ(z) with η˜ ∈ Tu written as η˜(z) = (z, η(z)). Exercise IV.1.16. Verify that dξ˜huv is the Fr´echet derivative of huv . It may help ♦ first to consider the Gateaux derivative. The fibre derivative dfib Φuv is the fibre-preserving map dfib Φuv :

O (z, v)

−→ 7−→

Hom(u∗ T M, v ∗ T M )
, z, Tu(z) M 3 w 7→ dfib (z,v) φuv (w) ∈ Tv(z) M 

and we may then write the derivative of huv as ˜ dξ˜huv = dfib Φuv ◦ ξ. Now we can iterate the argument to see that all higher derivatives dk huv exist, and at the point ξ˜ are given by the composition of the k th fibre derivative of Φuv ˜ with ξ. 

IV.1.4

The linearisation of ∂ J

We now want to establish the differentiability of the nonlinear Cauchy–Riemann operator ∂ J : C k → C k−1 (D, R2n ), k ≥ 1. To this end, we first compute the linearisation Du of this operator at u ∈ C k . In Section IV.2 (specifically, Proposition IV.2.29) we then show that Du is a continuous linear operator, and hence constitutes the Gateaux derivative. Furthermore, we show this Gateaux derivative to be continuous, and thus prove Fr´echet differentiability. Remark IV.1.17. For reasons that we shall detail in Section IV.3.6, ultimately we have to replace C k by an appropriate Sobolev space of maps. This Sobolev completion will be introduced in the next section, and the Fr´echet differentiability of ∂ J will be established in that setting.

Chapter IV. Elliptic Regularity

100

Recall the charts for C k defined in (IV.1). Using such a chart at the point u0 ∈ Ctk0 , the differentiable curve s 7−→ us := ϕst ◦ expu0 (sξ) ∈ C k ,

s ∈ (−ε, ε),

is tangent to (ξ, t) ∈ Tu0 C k . The linearisation Du0 of ∂ J is the Gateaux derivative of the composition of ∂ J with this chart, hence Du0 (ξ, t)

= =


d |s=0 ∂x us + J(us )∂y us ds ∂x ξ + J(u0 )∂y ξ + DJ(u0 )(ξ)∂y u0 + DJ(u0 )(t)∂y u0 .

Here, DJ(u0 )(ξ) is a map on D taking values in the linear space R2n×2n of real (2n × 2n)-matrices, sending z ∈ D to the derivative of J : R2n → R2n×2n at u0 (z) in the direction of ξ(z); similarly for DJ(u0 )(t). Define Dut00 (ξ) := ∂x ξ + J(u0 )∂y ξ + DJ(u0 )(ξ)∂y u0 (IV.2) and Ku0 (t) := DJ(u0 )(t)∂y u0 .

(IV.3)

In this notation, we have a splitting Du0 = Dut00 ⊕ Ku0 of the linearisation Du0 — corresponding to the splitting Tu0 C k = Tu0 Ctk0 ⊕ Tt0 Rn−1 of tangent spaces — into a linear operator of first order on a fixed level Ctk0 , and an operator of order zero. Remark IV.1.18. The differential DJ is continuous and hence vanishes on the closure of the set where J = J0 , in particular, outside the open set Φ(Br ); see Definition II.1.8 for notation. This implies that Du = ∂ for all flat discs u. Here we also drop the factor 1/2 from the definition of the linear Cauchy–Riemann operator, that is, we now write ∂ = ∂x + i∂y .

IV.2

The Sobolev completion

The lack of an implicit function theorem for Fr´echet spaces forces us to look at the equation ∂ J u = 0 on a suitable Banach completion of the Fr´echet space of smooth maps. For the reasons given in Section III.1.6 (and Section IV.3.6 below), this requires the introduction of Sobolev spaces. We begin with the definition of Sobolev spaces for arbitrary 2-dimensional domains G ⊂ C. In fact, until the formulation of the Sobolev inequalities, the dimension of G does not really matter. A number of key results hold for bounded Lipschitz domains only, which is what we shall always be dealing with in the applications.

IV.2. The Sobolev completion

101

Remark IV.2.1. We formulate the definition of Sobolev (k, p)-spaces for k ∈ N and p ∈ [1, ∞), and make these assumptions on k and p throughout our discussion. The definition also makes sense for k = 0 (Lp -spaces) or p = ∞, but several of the basic results we rely on fail for k = 0 (e.g. the Banach algebra property) or p = ∞ (e.g. Theorem IV.2.7). Whenever we rely on the Calder´ on–Zygmund inequality (Theorem III.1.13) or the a priori estimate derived from it (Proposition III.1.16 and, in its version for Sobolev spaces, Proposition IV.2.24), we need to assume p > 1. Whenever we appeal to the Sobolev embedding theorem (for 2-dimensional domains), we have to impose the condition p > 2. Most importantly, this is the case for the Banach algebra property of W k,p (Proposition IV.2.17), which is based on Lemma III.2.2 and the Leibniz rule (Proposition IV.2.16), both of which require p > 2. The Sobolev embedding theorem also enters the proof of the Peter Paul inequality (Proposition IV.3.17). The standard monograph on Sobolev spaces is [Adams & Fournier 2003]. Other good references are [Evans 2010, Chapter 5] and [Lieb & Loss 2001]; the latter also contains all the necessary background on measure and integration theory.

IV.2.1

The definition of Sobolev spaces

Write L1loc (G) for the space of measurable functions that are integrable over any compact subset of G. For the time being, we consider real-valued functions, but everything we say extends to complex- or vector-valued functions. Recall the notation Dα for derivatives from Section II.2.1, where α is a double index (or multi-index for functions on higher-dimensional domains). Recall that by Cc∞ (G) we denote the space of smooth functions with compact support in G. This is the so-called space of test functions for the following definition of weak derivatives. 1 Definition IV.2.2. Let u ∈ Lloc (G) and a double index α be given. If there is a 1 function vα ∈ Lloc (G) such that Z Z vα · ϕ for all ϕ ∈ Cc∞ (G), u · Dα ϕ = (−1)|α| G

G

then vα is the weak α-derivative of u. Remark IV.2.3. The terminology ‘the’ weak derivative is justified by the fact that any two such derivatives differ only on a set of measure zero. This follows from the fundamental lemma of the calculus of variations for L1 -functions; see [Alt 2012, 2.22], for instance. If u has continuous partial derivatives of the required order, then — as seen using integration by parts — the actual derivative Dα u coincides with the weak derivative vα . We therefore do not create any confusion by writing, in the general case, the weak derivative as Dα u := vα .

Chapter IV. Elliptic Regularity

102 Exercise IV.2.4.

(a) Compute the weak derivative of the function R 3 t 7→ |t|.

(b) Show that the function u : R → R defined by ( 0 for x < 0, u(x) = 1 for x ≥ 0, ♦

does not have a weak derivative.

We can now introduce the Sobolev spaces. There are two ways of doing so, which turn out to be equivalent. For point (b) of the next definition, observe 1 older inequality; cf. that the inclusion Lp (G) ⊂ Lloc (G) is a consequence of the H¨ [Ambrosio, Da Prato & Mennucci 2011, Remark 3.10]. Definition IV.2.5. Let k ∈ N and p ∈ [1, ∞). (a) The space H k,p (G) is the completion of  u ∈ C k (G) : kukk,p < ∞ with respect to the Sobolev norm k . kk,p ; see Definition III.1.2. (b) The space W k,p (G) is the space of those u ∈ Lp (G) ⊂ L1loc (G) for which the weak derivatives Dα u of order |α| ≤ k exist and satisfy Dα u ∈ Lp (G). Notice that the elements of W k,p (G) are equivalence classes of functions defined and equal up to sets of measure zero. Exercise IV.2.6. For s ∈ R we consider the function z 7→ |z|s on the interval (−1, 1) ⊂ R or on the open unit ball B ⊂ C. For which s is this function an
element of W 1,p (−1, 1) or W 1,p (B), respectively (with p ≥ 1 given)? Observe ♦ the role played by the dimension of the domain. Theorem IV.2.7 (Meyers and Serrin). H k,p (G) = W k,p (G). k,p



This can be read as saying that any function in W (G) can be approximated in the (k, p)-norm by C k -functions. For a proof of this theorem, see Theorem 3.17 in [Adams & Fournier 2003]. From now on, we exclusively use the notation W k,p . One way of interpreting this result is that a function is weakly differentiable if and only if it can be approximated (in the relevant norm) by smooth functions, and the weak derivatives then are the limits of the actual derivatives of the smooth approximation. This will be the key to establishing differentiation rules for weakly differentiable functions in Section IV.2.3. Remark IV.2.8. For Lp -spaces, something stronger is true: the space Cc∞ (G) of compactly supported smooth functions is dense (with respect to the Lp -norm) in Lp (G); see Theorem 2.16 and Lemma 2.19 in [Lieb & Loss 2001]. If G is bounded and Lipschitz, the approximation statement of Theorem IV.2.7 can be strengthened; see Theorem 3.22 in [Adams & Fournier 2003] or Proposition B.1.4 in [McDuff & Salamon 2012]. First we need to introduce some notation.

IV.2. The Sobolev completion

103

Notation IV.2.9. For ` ∈ N0 , let C ` (G) denote the vector space of functions u ∈ C ` (G) for which Dα is bounded and uniformly continuous on G for 0 ≤ |α| ≤ `. This implies that all these derivatives have a unique continuous extension to G. ` We set C ∞ (G) = ∩∞ `=0 C (G). Beware that this notational convention leads to ambiguities for unbounded domains, e.g. C ` (Rm ) 6= C ` (Rm ). By a result of [Seeley 1964] mentioned earlier in Remark I.3.7, for a bounded domain G ⊂ Rm with smooth boundary, the space C ∞ (G) can equivalently be characterised as the space of functions G → R that equal the restriction to G of some smooth function defined in a neighbourhood of G ⊂ Rm . Theorem IV.2.10. If G is a bounded Lipschitz domain, then C ∞ (G) is dense in W k,p (G).  We also record the following result [Adams & Fournier 2003, Theorem 3.6]. Recall that a metric space is called separable if it contains a countable dense subset. Theorem IV.2.11. Let G ⊂ Rm be a bounded domain and p ∈ [1, ∞). Then the Sobolev space W k,p (G) with the Sobolev (k, p)-norm is a separable Banach space. The same is true for Lp (G) with the Lp -norm. 

IV.2.2 The Sobolev embedding theorem When we speak of an embedding W k,p (G) → C ` (G), what we mean is that each element in W k,p (G) has a (unique) representative u ∈ C ` (G), and kukC ` (G) ≤ ckukk,p,G ,

(IV.4)

with a constant c not depending on u, which means that the inclusion map W k,p (G) → C ` (G), [u] 7→ u, is a bounded (or continuous) linear operator between Banach spaces. Notice that (IV.4) is a Sobolev inequality as in Proposition III.1.20, but with weaker differentiability assumptions on u. The following Sobolev embedding theorem is a conflation of Theorem 4.12, Part II in [Adams & Fournier 2003] and, for the compactness statement, the Rellich–Kondrachov theorem [Adams & Fournier 2003, Theorem 6.3, Part III]. Recall that a compact linear operator is one where the image of any bounded sequence has a convergent subsequence; in particular, a compact operator is bounded. Beware that the condition p > 2 is tied up with G being a 2-dimensional domain and the relative degree of differentiability in the Sobolev space vs. the space of continuously differentiable functions under consideration. Theorem IV.2.12. If G ⊂ C is a bounded Lipschitz domain, k ∈ N, and p > 2, there is a compact (and hence continuous) embedding W k,p (G) → C k−1 (G). 

Chapter IV. Elliptic Regularity

104

The inequality corresponding to this embedding theorem is precisely the Sobolev inequality from Proposition III.1.20. In other words, the embedding theorem says that this inequality continues to hold with the smoothness assumption on u weakened to u ∈ W k,p (G). Notation IV.2.13. Whenever we have an embedding W k,p (G) → C 0 (G), that is, a unique continuous representative G → R for each class in W k,p (G), we allow ourselves to write W k,p (G), especially in the case G = D. There are other inequalities in Chapter III that extend to functions of class W k,p . In order to show how the proofs carry over to this more general setting, we need a preparation regarding rules of differentiation in Sobolev spaces.

IV.2.3 Rules of differentiation in Sobolev spaces We begin with some straightforward rules. Proposition IV.2.14. W k,p (G) is a real vector space, and the weak differential Dα is linear. For any u ∈ W k,p (G) and any double indices α, β with |α| + |β| ≤ k, we have β D (Dα u) = Dα+β u. Implicitly, this is saying that Dα u ∈ W k−|α|,p (G). Also, it implies a general ‘Schwarz lemma’: Dβ (Dα u) = Dα (Dβ u). Exercise IV.2.15. Prove this proposition.

♦

Next, we establish the Leibniz rule. We formulate this product rule in the setting most relevant to the application we have in mind, rather than in its most general form. Proposition IV.2.16. Let G ⊂ C be a bounded Lipschitz domain, p > 2, and u, v ∈ W 1,p (G). Then, uv ∈ W 1,p (G), and for α = (1, 0) or (0, 1) the weak derivative Dα satisfies the Leibniz rule Dα (uv) = (Dα u)v + u Dα v. Proof. By the Sobolev embedding theorem, we have u, v ∈ C 0 (G). This implies uv ∈ Lp (G). By Theorem IV.2.10, we can find sequences (uν ), (vν ) in C ∞ (G) with uν → u and vν → v in W 1,p (G). This implies Dα uν → Dα u and Dα vν → Dα v in Lp (G). Using integration by parts and the Leibniz rule for smooth functions we find, for any test function ϕ ∈ Cc∞ (G), Z Z
uν v ν D α ϕ = − (Dα uν )vν + uν Dα vν ϕ. (IV.5) G

G

We claim that in the limit ν → ∞ this formula becomes Z Z
uv Dα ϕ = − (Dα u)v + u Dα v ϕ. G

G

(IV.6)

IV.2. The Sobolev completion

105

From this claim the proposition follows, since the function (Dα u)v + u Dα v is in Lp (G). By the Sobolev inequality kuν − ukC 0 (G) ≤ ckuν − uk1,p,G , we have uniform convergence uν → u ∈ C 0 (G), and likewise for vν → v. This implies that the left-hand side of (IV.5) converges to that of (IV.6) as ν → ∞. Regarding the right-hand side, we start with the observation that Lp (G) ⊂ 1 L (G) (by the H¨older inequality), since G is bounded. We then have the estimate k(Dα uν )vν ϕ − (Dα u)vϕkL1

≤

kDα uν − Dα ukL1 · kvν kC 0 · kϕkC 0 + kDα ukL1 · kvν − vkC 0 · kϕkC 0

→ 0, and a similar one for the second summand. This proves the claim, and hence the proposition.  With the Leibniz rule at our disposal, Lemma III.2.2 now carries over to W k,p -functions. Proposition IV.2.17. For G ⊂ C a bounded Lipschitz domain, k ∈ N, and p > 2, the Sobolev space W k,p (G) is a Banach algebra, that is, the product of two functions u, v ∈ W k,p (G) is again in W k,p (G) and kuvkk,p ≤ c kukk,p kvkk,p with a constant c = c(k, p, G).  Finally, we formulate a chain rule for a composition f ◦ u with f of class C 1 . In the literature, you will usually find some assumption on f 0 being bounded. This is redundant in our case, since our assumptions on u allow us to use the Sobolev embedding theorem. Proposition IV.2.18. Let G ⊂ C be a bounded Lipschitz domain, p > 2, u ∈ W 1,p (G), and f ∈ C 1 (R). Then, f ◦ u ∈ W 1,p (G), and for α = (1, 0) or (0, 1) the weak derivative Dα satisfies the chain rule Dα (f ◦ u) = (f 0 ◦ u)Dα u. Proof. By the Sobolev embedding theorem, we have u ∈ C 0 (G), and hence f ◦ u ∈ C 0 (G) ⊂ Lp (G). Choose a sequence (uν ) in C ∞ (G) with uν → u in W 1,p (G) (and hence in 0 C (G)). Then, for any test function ϕ ∈ Cc∞ (G) we have Z Z (f ◦ uν )Dα ϕ = − (f 0 ◦ uν )(Dα uν )ϕ. G

G

The function f is uniformly continuous on compact sets, which implies f ◦ uν → in C 0 (G), so the left-hand side of the above equation converges to Rf ◦ u uniformly α (f ◦ u)D ϕ. G

Chapter IV. Elliptic Regularity

106

For the right-hand side, we use the estimate k(f 0 ◦ uν )(Dα uν )ϕ − (f 0 ◦ u)(Dα u)ϕkL1 ≤ kf 0 ◦ uν − f 0 ◦ ukC 0 · kDα uν kL1 · kϕkC 0 + kf 0 ◦ ukC 0 · kDα uν − Dα ukL1 · kϕkC 0 → 0, where we use that on compact sets the derivative f 0 is bounded and uniformly continuous. Since the function (f 0 ◦u)Dα u is in Lp (G), this proves the proposition.  Remark IV.2.19. For ease of notation, in the chain rule we have considered a realvalued W k,p -function u, but the result generalises in an obvious fashion to vectorvalued functions. This implies that Lemma III.2.4 holds for u ∈ W k,p (G, Rm ). We shall also need the chain rule for reparametrisations of the domain. Exercise IV.2.20. Let G1 , G2 ⊂ C be bounded Lipschitz domains, τ : G2 → G1 a C 1 -diffeomorphism, and u ∈ W 1,p (G1 ), p > 2. Show that u ◦ τ ∈ W 1,p (G2 ). In fact, you may well find an argument that only uses the boundedness of the Jacobian of τ , so under this assumption the statement remains true for arbitrary domains G1 , G2 ⊂ Rm and p ∈ [1, ∞). ♦ Exercise IV.2.21. Give a proof of Proposition IV.2.17 using the chain rule (Proposition IV.2.18) instead of Lemma III.2.2. ♦

IV.2.4 Sobolev estimates We now reconsider some of the estimates from Chapter III and shall see how they carry over from smooth to weakly differentiable functions. Some of the estimates will then involve the following subspace of W k,p (G). The condition k ≥ 1 is understood; cf. Remark IV.2.1. Definition IV.2.22. The space W0k,p (G) is the closure of Cc∞ (G) in W k,p (G). For a proof of the next result we refer to [Evans 2010, Theorem 5.5.2] (who only deals with C 1 -boundaries) or [Alt 2012, Lemma A6.10]. The proposition holds for G of arbitrary dimension and p ∈ [1, ∞), but then — without the Sobolev embedding to C 0 (G) — one needs to be more careful about what one means by boundary values (the keyword here is ‘trace operator’).4 Proposition IV.2.23. Let G ⊂ C be a bounded Lipschitz domain and p > 2. Then, W0k,p (G) equals the set of those u ∈ W k,p (G) whose continuous representative satisfies u|∂G ≡ 0.  4 Not

only the Sobolev embedding theorem but also the definition of traces requires k ∈ N.

IV.2. The Sobolev completion

107

Here is the new version of the a priori estimate (Proposition III.1.16). This estimate will be instrumental in the Fredholm theory of the ∂-operator in Section V.1.5. Proposition IV.2.24 (The a priori Estimate). For any given natural number k and real numbers p > 1 and R > 0 there is a positive constant c = c(k, p, R) such that
kukk,p ≤ c k∂ukk−1,p for all u ∈ W0k,p BR , R2n .
Proof. Choose a sequence (uν ) in Cc∞ BR , R2n converging to u with respect to the Sobolev (k, p)-norm. The continuity of the norm function allows us to pass to the limit in the inequality of Proposition III.1.16 for uν .  Thanks to the chain rule (Proposition IV.2.18), the proof of the polynomial estimate in Lemma III.2.4 also goes through for Sobolev functions. Lemma IV.2.25. Let G ⊂ C be a bounded Lipschitz domain, k ∈ N, and p > 2. There is a real polynomial P = Pk,p,G of degree k with non-negative coefficients such that for any u ∈ W k,p (G, Rm ) and any f ∈ C ∞ (Rm , R) of finite C k -norm we have f ◦ u ∈ W k,p (G) and
kf ◦ ukk,p ≤ kf kC k P kukk,p . 

IV.2.5 The completion B of C is a Banach manifold We now fix a real number p > 2. Then the space of W 1,p -maps B → R2n embeds into the space of continuous maps D → R2n . Therefore, we can sensibly speak of the subspace B ⊂ C 0 of all W 1,p -maps u : (D, ∂D) −→ (R2n , ∂Z) that satisfy the boundary conditions (M1), (M2) and (M3). The subspace of maps of level t ∈ Rn−1 is denoted by Bt ⊂ B. Likewise, we define the tangent space Tu B = Tu Bt0 ⊕Rn−1 as the vector space of (ξ, t) with ξ a W 1,p -section of u∗ T R2n subject to the boundary conditions (T1) and (T2) in Definition IV.1.7. A metric on B inducing its topology as a Banach manifold is defined by the restriction of the (1, p)-norm. Proposition IV.2.26. The space B is a smooth, separable Banach manifold. Proof. The very same argument as in Sections IV.1.2 and IV.1.3 shows that B is a smooth Banach manifold, where we need to observe the following points: - Lemma IV.2.25 guarantees that the manifold charts do indeed map into the space of W 1,p -maps. - For the homeomorphism property of the manifold charts it is essential that the embedding W 1,p (D) → C 0 (D) is continuous; see Lemma IV.2.31 below for details.

Chapter IV. Elliptic Regularity

108

The separability of B, which will be essential for the application of the Sard– Smale theorem in Chapter V, follows from Theorem IV.2.11 and the next exercise.  Exercise IV.2.27. Let (X, d) be a separable metric space. Show that any subspace A ⊂ X is likewise separable. (This is not true, in general, for topological spaces.) Hint: Let Q ⊂ X be a countable dense subset. For q ∈ Q, n ∈ N, choose aq,n ∈ ♦ A ∩ B1/n (q) whenever this intersection is non-empty. As a consequence of the Sobolev embedding theorem, convergence in W 1,p implies uniform convergence. This means that for a W 1,p -convergent sequence in C, the boundary conditions (M1) to (M3) persist in the limit. In other words, the closure of C in W 1,p (B) is contained in B. Exercise IV.2.28. Show that the closure of C in W 1,p (B) actually coincides with B. Hint: The essential ingredients here are Theorem IV.2.10 and Proposition IV.2.23. Look at the components of u individually, and formulate the boundary condition (M1) as a condition on a function to equal 0 along the boundary. The passage from smooth functions compactly supported in the interior of D to smooth functions equal to zero along the boundary requires a cut-off argument; see Exercise V.5.6. ♦ The boundary condition (M3) requires an ad hoc argument.

IV.2.6

Differentiability of ∂ J

The nonlinear Cauchy–Riemann operator ∂ J on C extends in a natural fashion to an operator ∂ J : B −→ Lp (D, R2n ) by interpreting all derivatives as weak derivatives. Likewise, we can extend the linearisation Du defined in Section IV.1.4 to a linear map Du : Tu B −→ Lp (D, R2n ). Proposition IV.2.29. (a) The linearisation Du is a continuous linear operator and hence constitutes the Gateaux derivative of ∂ J at u. (b) The nonlinear Cauchy–Riemann operator ∂ J on B is differentiable of class C 1 , that is, the map u 7→ Du is continuous. In particular, ∂ J is Fr´echet differentiable. Before we attend to the proof of this proposition, we need to clarify what we mean by u 7→ Du being continuous in statement (b), since, as u varies, so does the space of linear maps in which Du lives. Remark IV.2.30. Most of the complications that we are dealing with in the present section stem from the nonlinear boundary condition on the first component of our maps u; see condition (T1) in Definition IV.1.7. This forces us to define local parametrisations via the exponential map, instead of simply using local linear

IV.2. The Sobolev completion

109

coordinates on the target space (here: R2n , where we would even have global coordinates). We computed the linearisation Du by using the parametrisation expu

Tu ⊃ Du −−−−−→ Eu ⊂ W 1,p (D, R2n ) centred at u; cf. Remark IV.1.14. Here, Tu = Tu Bt ⊕ Rn−1 . We now use the same chart to compute the linearisations Duν for a sequence uν → u. If uν is close to u in the (1, p)-norm, then also in the C 0 -norm by the Sobolev embedding theorem. This means, together with the fact that Exp is a diffeomorphism near the zero section, that we can write uν = expu (ην ) for a uniquely defined ην ∈ Tu . By the chain rule (Proposition IV.2.18) and Lemma IV.2.25, applied to the inverse of Exp near the zero section, ην is indeed of class W 1,p . Conversely, given ην ∈ Tu , the image expu (ην ) under the local parametrisation is of class W 1,p (this we remarked before in Section IV.2.5). The following lemma says that the local parametrisation expu : Du → Eu and its inverse, the chart (Eu , expu−1 ), are continuous at 0 ∈ Du and u ∈ Eu , respectively. Similar arguments can be used to show that the charts Eu → Du of B are homeomorphisms. Lemma IV.2.31. For uν = expu (ην ), the sequence (uν ) converges to u in the (1, p)norm if and only if the sequence (ην ) converges to 0 in the (1, p)-norm. Proof. Write points of T R2n as (q, v) with q ∈ R2n and v ∈ Tq R2n ≡ R2n . On the unit disc bundle DT R2n with respect to the Euclidean metric we consider the function R2n DT R2n −→ F: (q, v) 7−→ expq (v), where the exponential map is the one with respect to the metric g on R2 ⊕ R2n−2 defined in Section IV.1.2. This g is a product metric, invariant under translations in the R2n−2 -direction (on which factor the metric is Euclidean) and, on the R2 factor, equal to the Euclidean metric outside a compact set. This means that (i) F (q + q0 , v) = F (q, v) + q0 for q0 ∈ {0} ⊕ R2n−2 , and (ii) for prR2 (q) outside a suitable compact subset of R2 , we have F (q, v) = q + v for all |v| ≤ 1. Let ∂v F denote the partial Jacobian of F with respect to the v-coordinates, not the directional derivative in the direction of v. Statements (i) and (ii) imply that ∂v F is bounded, in the sense that the following supremum over the operator norm of the partial Jacobian is finite:  k∂v F k := sup |∂v F (q, v)| : q, v ∈ R2n , |v| < 1 < ∞. Now consider a sequence (ην ) in Tu with ην → 0. By the Sobolev embedding theorem, the Euclidean length of ην (z) goes to zero uniformly, and we may assume

Chapter IV. Elliptic Regularity

110

this length to be at most 1 for the whole sequence. We now estimate as in the proof of Lemma III.2.4: |uν (z) − u(z)|

= =

|F (u(z), ην (z)) − F (u(z), 0)| Z 1 d F (u(z), tη (z)) dt ν dt 0

≤

k∂v F k · |ην (z)|,

which implies kuν − ukLp ≤ k∂v F k · kην kLp . Before computing first derivatives, we observe that F (q, 0) = q, and hence the partial Jacobian ∂q F (q, 0) is the identity. Since F is smooth, ∂q F (u, ην ) is C 0 -close to the identity provided that ην is C 0 -close to 0, i.e. the constant map D → 0. We now look at
∂x (uν − u) = ∂q F (u, ην ) − ∂q F (u, 0) ∂x u + ∂v F (u, ην )∂x ην , with a similar expression for the y-derivative. Together with the estimate on the Lp -norm of uν − u, this yields kuν − ukW 1,p ≤ k∂q F (u, ην ) − id kC 0 kukW 1,p + k∂v F k · kην kW 1,p . As ην goes to zero in the (1, p)-norm, it does so, too, in the C 0 -norm by the Sobolev embedding theorem. We conclude that kuν − ukW 1,p → 0. The proof of the converse implication is completely analogous.  Proof of Proposition IV.2.29. We demonstrate part (b) and leave (a) as an exercise. The estimates you need to prove part (a) are actually simpler variants of the ones we are going to establish now. As in Section IV.1.4, equations (IV.2) and (IV.3), we write Du = Dut0 ⊕ Ku , with the linear map Dut0 : Tu Bt0 → Lp (D, R2n ) given by Dut0 (ξ) = ∂x ξ + J(u)∂y ξ + DJ(u)(ξ)∂y u, and the linear map Ku : Rn−1 → Lp (D, R2n ) given by Ku (t) = DJ(u)(t)∂y u. We begin with the second summand Ku in this splitting of Du , for which the relevant estimate is more straightforward. In particular, we need not worry about local charts, since the tangent spaces Tu split off a factor Rn−1 globally.

IV.2. The Sobolev completion

111

Claim 1. The map B 3 u 7→ Ku ∈ L(Rn−1 , Lp (D, R2n )) is continuous. Since we are dealing with metric spaces, it suffices to show that for a sequence uν → u the operator norm kKuν − Ku kop goes to zero as ν → ∞. We begin with the triangle inequality k(Kuν − Ku )(t)kLp

≤

kDJ(uν )(t)(∂y uν − ∂y u)kLp
+ k DJ(uν ) − DJ(u) (t)∂y ukLp ,

and consider the two summands in turn. The first summand we estimate as kDJ(uν )(t)(∂y uν − ∂y u)kLp ≤ kDJk · |t| · kuν − uk1,p , where kDJk := supq∈Cn kDJ(q)kop . Notice that kDJk < ∞, since J coincides with J0 outside a compact set. For the second summand, we write Z 1
d DJ(uν ) − DJ(u) = DJ u + t(uν − u) dt. 0 dt This allows us to estimate
k DJ(uν ) − DJ(u) (t)∂y ukLp ≤ kD2 Jk · kuν − ukC 0 · |t| · kuk1,p , and thanks to the Sobolev embedding theorem we may replace (up to a constant c) the C 0 -norm of uν − u by the Sobolev (1, p)-norm. Putting things together, we obtain the estimate   kKuν − Ku kop ≤ kDJk + c kD2 Jk · kuk1,p · kuν − uk1,p , which proves Claim 1. Claim 2. The map Bt 3 u 7→ Dut ∈ L(Tu Bt , Lp (D, R2n )) is continuous. For the continuity of Dut (with t the level of u), we may ignore the shifts in the t-direction and study a sequence uν → u on a fixed level t. As in Lemma IV.2.31, we write uν = expu (ην ), now with ην ∈ Tu Bt . We then need to estimate the difference Dut ν (ξν ) − Dut (ξ), where ξν := Tην expu (ξ) ∈ Tuν Bt . In this last expression, ξ is an element of Tην (Tu Bt ) ≡ Tu Bt , with a neighbourhood of the origin in Tu Bt regarded as the domain of definition of the local parametrisation expu ; see Figure IV.1, and cf. Remark IV.1.14 for the identification of T0 Tu Bt ≡ Tu Bt in the domain of the chart with the actual tangent space Tu Bt to Bt at u. This means that we may read this difference Dut ν (ξν ) − Dut (ξ) as a linear operator on Tu Bt , and our task is to show that its operator norm goes to zero as uν → u.

Chapter IV. Elliptic Regularity

112

Bt

ξν uν

expu

ξ u

ξ ξ

ην 0

Tu Bt

Figure IV.1: The setting for Claim 2. We consider the term ∂x ξν −∂x ξ in this difference Dut ν (ξν )−Dut (ξ); see (IV.2). The estimates for the other terms are variants of the ones in the proof of Claim 1 and the one we establish now. We begin with k∂x ξν − ∂x ξkLp

=

k∂x (Tην expu − id)(ξ)kLp

≤

k(Tην expu − id)(ξ)k1,p

≤

const. kTην expu − id k1,p · kξk1,p ,

where the last inequality holds thanks to Proposition IV.2.17. Define f : DT R2n → L(R2n , R2n ) by f (q, v) = Tv expq − id. Observe that f (q, 0) = 0. By the same reasoning as at the beginning of the proof of Lemma IV.2.31, we see that k∂v f k < ∞. Our aim now is to show that kf (u, ην )k1,p goes to zero as ην → 0. For w ∈ Tv Tq R2n ≡ R2n and |v| ≤ 1 we have
|Tv expq (w) − w| = f (q, v) − f (q, 0) (w) Z 1 d f (q, tv)(w) dt = dt 0

≤

k∂v f k · |w|.

IV.3. Elliptic regularity

113

This implies kf (u, ην )kLp ≤ k∂v f k · kην kLp . Next we consider first derivatives. Since f (q, 0) is identically equal to the zero map R2n → R2n , we have ∂q f (q, 0) = 0. The map f is smooth, so ∂q f (u, ην ) will be C 0 -close to the zero map provided that ην is C 0 -close to 0. We compute
∂x f (u, ην ) = ∂q f (u, ην )∂x u + ∂v f (u, ην )∂x ην , and similarly for the y-derivative. Together with the estimate on the Lp -norm of f (u, ην ) we have kf (u, ην )k1,p ≤ k∂q f (u, ην )kC 0 kuk1,p + k∂v f k · kην k1,p , which goes to zero as ην → 0. The remaining estimates for the terms in Dut ν (ξν ) − Dut (ξ) we leave as an  exercise. Remark IV.2.32. Using analogous considerations, one can show that ∂ J is actually of class C ∞ . Exercise IV.2.33.

(a) Prove part (a) of the preceding proposition.

(b) Provide the missing estimates in part (b) of the proof. (c) Prove, using similar (but simpler) estimates, that the map u 7→ ∂ J u is continuous. Of course, continuity follows from the Fr´echet differentiability of this map, but a direct proof is a good way to practise various estimates. ♦

IV.3

Elliptic regularity

Recall that we are dealing with the moduli space M = {u ∈ C : ∂ J u = 0}. The aim of the present section is to show that we obtain exactly the same space if we study the equation ∂ J u = 0 on the larger space B of W 1,p -maps for some p > 2. Theorem IV.3.1. Let u ∈ B be a solution of the nonlinear Cauchy–Riemann equation ∂ J u = 0. Then, u is actually of class C ∞ , that is, u ∈ C.

IV.3.1 The topology on M Thus, as a set we have M = {u ∈ B : ∂ J u = 0}. Potentially, the topology on M induced by the C ∞ -topology on C might be different from that induced by the W 1,p -metric on B. The next lemma shows that, in

Chapter IV. Elliptic Regularity

114

fact, the two topologies coincide. The space of C ∞ -maps D → R2n is first-countable and even metrisable (Remark II.2.4), so to establish the homeomorphism property of the identity map on M with respect to one topology on the domain and the other one on the target, it suffices to establish the following lemma about sequential continuity. Lemma IV.3.2. A sequence (uν ) in M converges to u ∈ M with respect to the C ∞ -topology if and only if it does so with respect to the W 1,p -topology. Proof. One direction is straightforward. If uν → u with respect to the C ∞ topology, then a fortiori with respect to the C 1 -norm. Since D is compact, this implies convergence with respect to the (1, p)-norm. Conversely, Corollary III.3.12 tells us that W 1,p -convergence implies that uν → u with respect to all (k, p)-norms. The Sobolev embedding theorem says that W k,p -convergence implies C k−1 -convergence. We conclude to convergence in the C ∞ -topology. 

IV.3.2

A local estimate

The smoothness of u ∈ B with ∂ J u = 0 is a local property. Thus, to check smoothness near a point z ∈ ∂D we can use a conformal identification of D\{eiθ } for some θ ∈ R with the upper half-plane H both in the domain of definition and in the first complex coordinate of the image, such that z is mapped to 0 ∈ ∂H. Thanks to the chain rule (Proposition IV.2.18 and Exercise IV.2.20) and the fact that conformal reparametrisations of J-holomorphic curves are J-holomorphic (Exercise I.3.11), it then suffices to study the following situation: - The almost complex structure J on R2n coincides with J0 along
n Rn = R ⊕ {0} . - For some r0 > 0, we have a map u ∈ W 1,p (Br0 ∩ H, R2n ) weakly solving the equation ∂ J u = 0 (formally speaking: almost everywhere) and satisfying the boundary condition u(Br0 ∩ ∂H) ⊂ Rn . (Recall that Br0 ⊂ C is the open disc of radius r0 centred at 0.) Locally near an interior point of D, we simply study a map u ∈ W 1,p (Br0 , R2n ) on a small open ball, satisfying ∂ J u = 0. Most of what follows deals with the local behaviour near a boundary point, and we shall remark on the simplifications in the interior case at the appropriate places. Two comments are in order. When we write W 1,p (Br0 ∩ H), strictly speaking we mean the space of W 1,p -functions on the interior of Br0 ∩H. Second, the Sobolev embedding theorem guarantees that such functions extend continuously to Br0 ∩H (and its closure in C), so it makes sense to formulate the boundary condition.

IV.3. Elliptic regularity

115

Notation IV.3.3. For G ⊂ C, v ∈ C, and λ ∈ R+ , we write G + v := {z + v : z ∈ G} and λG := {λz : z ∈ G}. Presently we shall have to restrict u to the subset B + := Br ∩ H for some r ∈ (0, r0 ], which we drop from the notation. For λ ∈ R+ , the notation Bλ+ stands for λB + . For r < r0 and h ∈ R with |h| ≤ r0 − r we have B + + (h, 0) ⊂ Br0 ∩ H, so we can define the shift operator τh u : B + → R2n by (τh u)(x, y) = u(x + h, y). For h 6= 0, we then introduce the difference quotient uh =

τh u − u h

on B + . Observe that uh (x, 0) ∈ Rn . If B + is replaced by Bλ+ for some λ < 1, these definitions also make sense for r = r0 , and the condition on h becomes |h| < r0 − λr. As we shall see, Theorem IV.3.1 is a fairly straightforward consequence of the following deceptively simple lemma, whose proof (which will be presented in Section IV.3.5) is surprisingly subtle. Lemma IV.3.4. Let u ∈ W 1,p (Br0 ∩H, R2n ), p > 2, be a map with u(Br0 ∩∂H) ⊂ Rn and ∂ J u = 0. Then there is an r ∈ (0, r0 ] and a positive constant c = c(p, u) such that  r r kuh k1,p,B + ≤ c for all h ∈ − , \ {0}. 1/2 4 4 + In the interior case, Br0 ∩ H is replaced by Br0 , and B1/2 by Br/2 .

IV.3.3 The shift operator Before we derive Theorem IV.3.1 from this lemma, we discuss some basic properties of the shift operator τh . Consider an Lp -function u : G → R on a domain G ⊂ C. For any subdomain G0 ⊂ G and all h 6= 0 with |h| < dist(G0 , ∂G), the difference quotient uh is defined on G0 and of class Lp . The following statement can be thought of as an Lp -version of the mean value theorem. Lemma IV.3.5. If u ∈ W 1,p (G), p ∈ [1, ∞), then kuh kLp (G0 ) ≤ kux kLp (G) , where ux = ∂x u denotes the weak derivative.

Chapter IV. Elliptic Regularity

116

Proof. By Theorem IV.2.7, it suffices to prove this inequality for functions u ∈ C 1 (G) ∩ W 1,p (G). We compute uh (z)

= = =


1 u(x + h, y) − u(x, y) h Z
1 1 d u(x + th, y) dt h 0 dt Z 1 ux (x + th, y) dt. 0

This implies |uh (z)|p ≤

1

Z

|ux (x + th, y)|p dt, 0

where for p > 1 you have to use H¨older’s inequality. By Fubini’s theorem we then have Z Z 1Z Z h p p |u | dx dy ≤ |ux (x + th, y)| dx dy dt ≤ |ux |p dx dy, G0

0

G0

G

where the last inequality simply follows by substitution for each fixed t.



Exercise IV.3.6. For u, v ∈ W 1,p (G), prove that on G0 the following identities hold: (i) ∂x ◦ τh = τh ◦ ∂x or, equivalently, ∂x (uh ) = (∂x u)h in Lp (G0 ); (ii) (uv)h = uh τh v + uv h in W 1,p (G0 ).

♦

Lemma IV.3.7. Let u ∈ Lp (G), p ∈ (1, ∞), be a function with the property that there is a constant K such that for all bounded subdomains G0 ⊂ G and all h 6= 0 with |h| < dist(G0 , ∂G) we have kuh kLp (G0 ) ≤ K. Then the weak derivative ux exists on G and satisfies kux kLp (G) ≤ K. Proof. Let ϕ be a smooth function with compact support in G. For 0 < h < dist(supp ϕ, ∂G), we have Z Z τh u · ϕ = u · τ−h ϕ G

G

by substitution, since τh (u · τ−h ϕ) = τh u · ϕ. It follows that Z Z Z uh ϕ = − uϕ−h −→ − uϕx as h → 0 G

G

G

by the dominated convergence theorem. The required integrable bound comes from the mean value theorem: |uϕ−h | ≤ |u| · maxz∈G |ϕx |.

IV.3. Elliptic regularity

117

Hence, for a given ε > 0 we can find a positive real number h0 smaller than dist(supp ϕ, ∂G) such that Z Z uϕx ≤ ε + uh0 ϕ ≤ ε + kuh0 kLp (supp ϕ) · kϕkLq , G

G

where the second inequality is the H¨ older inequality with q the dual coefficient of p, that is, p1 + 1q = 1. Taking ε & 0, we find Z uϕx ≤ KkϕkLq . G

Exercise IV.3.8. Use Remark IV.2.8 to show that the linear functional Z ∞ Cc (G) 3 ϕ 7−→ − uϕx ∈ R G

extends to a continuous linear functional on Lq (G), with operator norm bounded by K. ♦ Recall that the dual space of Lq (G), that is, the space of continuous linear functionals on Lq (G), for 1 ≤ q < ∞, equals Lp (G); see [Lieb & Loss 2001, Theorem 2.14]. (This result fails for q = ∞, and since q is the dual coefficient to p, we need the requirement p > 1.) This means that R any linear functional Lq (G) 3 w 7→ A(w) ∈ R can be written as A(w) = G vw for some unique v ∈ Lp (G), and the operator norm of A equals kvkLp . Applied to our situation, we find, a fortiori, v ∈ Lp (G) with kvkLp (G) ≤ K and Z Z vϕ = − uϕx for all ϕ ∈ Cc∞ (G). G

G

This is saying that the weak derivative ux exists and equals v.



Exercise IV.3.9. Formulate and prove an analogous statement for domains in the upper half-plane H. ♦

IV.3.4 Proof of Theorem IV.3.1 Let u ∈ B be a map D → R2n satisfying the nonlinear Cauchy–Riemann equation ∂ J u = 0. Our aim is to show that u, which is a priori only of class W 1,p , is actually smooth. By assumption, the weak derivative ux exists and is in Lp . We now work in the local setting of Lemma IV.3.4, describing the map u near a boundary point.
+ This lemma tells us that there is an r ∈ (0, r0 ] such that on B1/2 = 12 Br ∩ H we have the estimate  r r kuhx kLp (B + ) ≤ kuh k1,p,B + ≤ c for all h ∈ − , \ {0}, 1/2 1/2 4 4

118

Chapter IV. Elliptic Regularity

with a similar description in the neighbourhood of an interior point of D. The notation uhx is unambiguous by Exercise IV.3.6. Lemma IV.3.7 (in its version + , tells us that the weak derivative uxx for domains in H), applied to G = B1/2 + p exists on B1/2 and is an L -function. Similarly, by starting with uhy , we find that + uyx ∈ Lp (B1/2 ). Exercise IV.3.10. Let G ⊂ C be a domain and v ∈ W 1,p (G). Suppose that the ♦ weak derivative vyx := ∂x ∂y v exists. Show that vxy exists and equals vyx . + With this exercise we deduce that ux ∈ W 1,p (B1/2 ). From ∂ J u = 0 we + 1,p have uy = J(u)ux , which lies in W (B1/2 ) thanks to the chain rule (Proposition IV.2.18) and W 1,p (G) being a Banach algebra (Proposition IV.2.17). We + conclude that u ∈ W 2,p (B1/2 ). However, the argument up to this point applies locally near any point of D; hence, with the following exercise, in fact we have u ∈ W 2,p (D).

Exercise IV.3.11. Let (λj ) be a partition of unity on D, subordinate to a covering by open subsets Uj ⊂ D, and u : D → R a function with u|Uj of class W 1,p for all j. Show that by summing over the formula for the weak derivatives of λj u one can deduce u ∈ W 1,p (D). ♦ By the Sobolev embedding theorem, u is of class C 1 on D. In particular, the partial derivatives of u (on D or in a local reparametrisation) are derivatives in the classical sense, defined and continuous on D. To continue the bootstrapping, we want to apply this lifting from W 1,p to 2,p W to the partial 1-jet5
w(z) := z, u(z), ux (z) , z ∈ Br0 ∩ H, with the same setting as before, but now the higher-dimensional target space C ⊕ R2n ⊕ R2n . This requires (for the local description near boundary points only) the definition of an appropriate almost complex structure on this target space, which is the content of the next exercise. Exercise IV.3.12. Let J be an structure on R2n that coincides
n almost complex n 2n with J0 along R = R ⊕ {0} . For u, v ∈ R , set
A(u, v) = DJ(u)(v) J(u)v ∈ R2n . Here, DJ(u)(v) is the differential of J at u in the direction of v; this is again a linear operator on R2n , which is then applied to the vector J(u)v. 5 See

[Geiges 2003] for a brief introduction to the language of jets, which are equivalence classes of (local) sections of fibre bundles. A map M → N between two manifolds may be regarded as a section of the trivial bundle M × N → M . An r-jet, r ∈ N0 , at x ∈ M is an equivalence class of smooth local sections around x, determined by the partial derivatives (in some local coordinate system) at x up to order r. In the situation at hand, we only consider one of the two first-order derivatives, hence ‘partial’ 1-jet.

IV.3. Elliptic regularity

119

(a) Show that ˜ u, v)(ζ, h, k) := iζ, J(u)h, J(u)k + A(u, v) Im(ζ) + J(u)A(u, v) Re(ζ) J(z,




defines an almost complex structure on C ⊕ R2n ⊕ R2n that coincides with i ⊕ J0 ⊕ J0 along R ⊕ Rn ⊕ Rn . (b) Let G ⊂ C be a domain and u : G → R2n a J-holomorphic map. Show that
the partial 1-jet w : G → C ⊕ R2n ⊕ R2n defined by w(z) = z, u(z), ux (z) ˜ map. ♦ is a J-holomorphic Returning to our map u : D → R2n , we observe that the boundary condition implies that (in the local setting) w sends Br0 ∩ H to R ⊕ Rn ⊕ Rn , so we are in a completely analogous set-up. Thus, our previous argument implies w ∈ W 2,p (B + ), and in particular ux ∈ W 2,p (B + ). As above, with the J-holomorphicity of u we infer that u ∈ W 3,p (B + ), and then globalise again to u ∈ W 3,p (D). Continuing in this fashion with the partial 1-jet of w, inductively we see that u ∈ W k,p (D) for all k ∈ N, whence u ∈ C ∞ (D) by Sobolev embedding. This concludes the proof of Theorem IV.3.1, up to Lemma IV.3.4.

IV.3.5

Proof of the deceptively simple lemma

For the proof of Lemma IV.3.4 we need two estimates of a more general nature. One is an a priori estimate for maps defined on domains in H with real boundary values; the second is an inequality of ‘Peter Paul’ type (the expression will be explained). + In the a priori estimate we use the notation BR := BR ∩ H. Proposition IV.3.13 (The a priori Estimate With Boundary). For any real numbers p > 1 and R > 0 there is a constant c = c(p, R) such that kuk1,p,B + ≤ c k∂ukLp (B + ) R

R

+ for all u ∈ Cc∞ BR , C with u(BR ∩ ∂H) ⊂ R.




Remark IV.3.14. By passing to the limitas in the proof
of Proposition IV.2.24, + the same equality holds for the closure of u ∈ Cc∞ BR , C : u(BR ∩ ∂H) ⊂ R in
+ W 1,p BR ,C .
+ Proof of Proposition IV.3.13. Given u ∈ Cc∞ BR , C with u(BR ∩ ∂H) ⊂ R, we define u ˆ : BR → C by Schwarz reflection: ( u(z) for Im(z) ≥ 0, u ˆ(z) := u(z) for Im(z) < 0. This function u ˆ is continuous.

Chapter IV. Elliptic Regularity

120

ˆ = f˜ + i˜ g with real-valued functions Exercise IV.3.15. Write u = f + ig and u f, g, f˜, g˜. ˆx and g˜y exist and are continuous, and (a) Show that the partial derivatives u that f˜y exists (and is then continuous) if and only if fy = 0 along the real axis (which happens, for instance, when u is holomorphic). ˆ is weakly differentiable with weak derivative (b) Show that u ( for y ≥ 0, fy (x, y) ˜ fy (x, y) = −fy (x, −y) for y < 0. (c) Show that ∂ u ˆ (in the weak sense), with ∂ = ∂x + i∂y , is given by (
fx − gy + i(gx + fy ) (x, y) for y ≥ 0,
∂u ˆ(x, y) = fx − gy − i(gx + fy ) (x, −y) for y < 0. In particular, |∂ u ˆ(x, −y)| = |∂ u ˆ(x, y)|.

♦

From this exercise it follows that u ˆ is of class W 1,p , and since it is compactly supported in BR , by Proposition IV.2.23 (and the comments before it), u ˆ can be approximated in the Sobolev (1, p)-norm by smooth functions compactly supported in BR . With c = c(1, p, R) the constant from the a priori estimate in Proposition IV.2.24, we then conclude kuk1,p,B + = R

1 1 kˆ uk1,p,BR ≤ ck∂ u ˆkLp (BR ) ≤ ck∂ukLp (B + ) , R 2 2

where we have used the explicit expression of the (weak) derivatives of u ˆ in terms of those of u.  Note IV.3.16. In the proof of Lemma IV.3.4, we need to apply the a priori estimate + to a function of the form f u, with u ∈ W 1,p (BR , C) for some p > 2, u(BR ∩ ∂H) ⊂ + + R, and f : H → [0, 1] a bump function with support in B3R/4 and f ≡ 1 on BR/2 . The estimate does indeed hold in this situation, which we see as follows. Write + u = v + iw with v, w real-valued. Then, f w = 0 on ∂BR , so by Proposition IV.2.23
+ ∞ we find wν ∈ Cc Int(BR ) with wν → f w in the (1, p)-norm. The real part v can
+ be approximated by functions vν ∈ C ∞ BR by Theorem IV.2.10. Then, the a priori estimate holds for uν := f vν + iwν (sic!), and uν → f u. The name of the following inequality refers to the adage that one must ‘rob Peter to pay Paul’. It is a special case of a general class of inequalities where a term is estimated by the sum of two terms, one of which can be made small at the price of making the second summand large.

IV.3. Elliptic regularity

121

Proposition IV.3.17 (Peter Paul Inequality). Let G ⊂ C be a bounded Lipschitz domain and p > 2. Then, for any ε > 0, there is a Cε > 0 such that kukC 0 (G) ≤ εkuk1,p,G + Cε kukLp (G) for all u ∈ W 1,p (G). Proof. We prove this by contradiction. Assume that there is an ε > 0 such that no Cε with the desired property exists. This means that we can find a sequence (uν ) in W 1,p (G) with kuν k1,p,G = 1 for all ν ∈ N and kuν kC 0 (G) > ε + νkuν kLp (G) . As the embedding W 1,p (G) → C 0 (G) is compact by Theorem IV.2.12, the bounded sequence (uν ) in W 1,p (G) has a subsequence (uνj ) that converges uniformly on G to some function u ∈ C 0 (G). The inequality kuνj kLp (G)
2 understood. The conditions on k and p are essential, as we shall frequently rely on the Sobolev embedding theorem and the Banach algebra property of W k,p in the discussion that follows; cf. Remark IV.2.1. Theorem V.1.5. For all u ∈ B, the linearised Cauchy–Riemann operator Du : Tu B −→ Lp (D, R2n ) is a Fredholm operator of index 2n − 2. If u is a flat disc, then Du = ∂ (cf. Remark IV.1.18) is surjective. The proof of this theorem will be given over the course of several subsections, by a successive reduction of the problem. We begin by showing that it suffices to study the operator on a fixed level t ∈ Rn−1 . Recall from Section IV.1.4 that for u ∈ B of level t, we have a splitting Tu B = Tu Bt ⊕ Tt Rn−1 , and a corresponding splitting Du = Dut ⊕ Ku . Lemma V.1.6. The operator Du is Fredholm if and only if Dut is Fredholm, and in this case we have ind Du = ind Dut + (n − 1). 1 Such

a closed complement exists; see Note V.1.15.

Chapter V. Transversality

128

Proof. Both the kernel and the image of Du differ from those of Dut by at most an (n − 1)-dimensional space. This proves the first statement. For the dimension count in the Fredholm situation, we split Rn−1 = Tt Rn−1 as Rn−1 = ker Ku ⊕ A ⊕ B, where A is mapped injectively by Ku into the image of Dut , and the isomorphic image of B has the property Ku (B) ∩ im Dut = {0}. Then, dim ker Du = dim ker Dut + dim ker Ku + dim A and dim coker Du = dim coker Dut − dim B. The lemma follows.



Remark V.1.7. Here we could have argued from more general principles. The operator Ku , being defined on a finite-dimensional space, is compact. The sum of a Fredholm and a compact operator (on the same space) is again Fredholm of the same index (we shall prove this in Section V.1.4, where we cannot do without this fact). From this, Lemma V.1.6 is immediate, where you need to regard Du = Dut ⊕ Ku as a compact perturbation of Dut ⊕ 0 on Tu B; the latter operator has the same image as Dut , but an extra summand Rn−1 in the kernel.

V.1.3

Transformation to a perturbed ∂-operator

We now want to show that, given u ∈ Bt , we can conjugate the operator Dut to a compact perturbation of the linear Cauchy–Riemann operator ∂ on a space V p isomorphic to Tu Bt , where the boundary condition (see Definition IV.1.7) has been ‘straightened’. This, as we shall see in Section V.1.4, will reduce the problem to computing the index of ∂ on V p . In other words, this conjugation simplifies both the Banach space and the operator whose index we wish to compute. Anticipating this transformation, we identify R2n with Cn , even though the operator Dut is the linearised Cauchy–Riemann operator corresponding to some almost complex structure J on R2n . We define V p as the space of those η = (η1 , . . . , ηn ) ∈ W 1,p (D, Cn ) — where the Sobolev embedding theorem allows us to work with continuous representatives — that satisfy η(eiθ ) ∈ eiθ R ⊕ Rn−1 , θ ∈ R/2πZ,

and η1 (ik ) = 0, k ∈ {0, 1, 2}.

Notation V.1.8. Since we are working with a fixed disc u, it is convenient to regard the almost complex structure J along the image u(D) as a matrix-valued function on D, that is, by slight abuse of notation we shall from now on think of J as the function defined by J(z) := J(u(z)).

V.1. Fredholm theory

129

Lemma V.1.9. There is a map B0 : D → GL(2n, R) of class W 1,p (and hence continuous) into the general linear group such that
 ξ 7−→ B0 ξ := z 7→ B0 (z) ξ(z) defines an isomorphism B0 : Tu Bt → V p , and an Lp -map C : D → EndR (Cn ) = R2n×2n to the real linear endomorphisms of Cn = R2n such that with T η := ∂η + Cη the following diagram commutes: Tu B t

Dut

B0 ? Vp

- Lp (D, Cn ) B0

T

? - Lp (D, Cn )

Moreover, the map B0 conjugates the almost complex structure along the holomorphic disc u to the standard structure, that is, with Notation V.1.8 we have B0 JB0−1 = J0 as matrix-valued functions on D. 
Note V.1.10. Since the map f 7→ z 7→ B0 (z) f (z) is an isomorphism of Lp (D, Cn ), with Lemma V.1.6 this lemma reduces the proof of Theorem V.1.5 to showing that T = ∂ + C is a Fredholm operator of index n − 1. Proof of Lemma V.1.9. We begin with the construction of an isomorphism from Tu Bt to V p , defined by a matrix-valued function B on D. The composition B0 of B with a suitable isomorphism of V p will then give us the conjugation formula B0 JB0−1 = J0 . From this we shall derive the formula for the conjugated Cauchy– Riemann operator. We wish to define B : D → GL(2n, R) as a map of the form   b 0 ··· 0  0    B= .  : D −→ GL(C ⊕ R2n−2 ),  ..  E 0 where E denotes the unit matrix of rank 2n − 2. Recall from Definition IV.1.7 that for ξ = (ξ1 , . . . , ξn ) ∈ Tu Bt we have the boundary conditions ξ1 (z) ∈ Riu1 (z) and ξ2 (z), . . . , ξn (z) ∈ R for all z ∈ ∂D, as well as ξ1 (ik ) = 0 for k ∈ {0, 1, 2}. Choose a radius r0 ∈ (0, 1) such that J(z) = J0 and u1 (z) 6= 0 for all z ∈ D e of the annulus A := D \ Br is A e = [r0 , 1] × R. with |z| ≥ r0 . The universal cover A 0

Chapter V. Transversality

130

e → A and R+ × R → C∗ . We write pr : (r, θ) 7→ reiθ for the covering projections A By condition (M3) from Notation I.8.10, the map u1 : A → C∗ has a unique lift e → R+ × R, that is, a map that fits into the commutative diagram u ˜1 : A e A

u ˜1

- R+ × R

pr ? A

pr u1

? - C∗ ,

determined by u ˜1 (1, 0) = (1, 0). The homotopy condition (M2) implies that u ˜1 (r, θ + 2π) = u ˜1 (r, θ) + (0, 2π). Choose a smooth function τ : [r0 , 1] → [0, 1] with τ (r) = 0 near r = r0 , and e: A e → R+ × R by τ (r) = 1 near r = 1. Now define H

e θ) = 1 − τ (r) · u H(r, ˜1 (r, θ) + (0, π/2) + τ (r)(r, θ), e θ). Finally, define and H : A → C∗ by H(reiθ ) = pr ◦ H(r, ( 1 for |z| < r0 , b(z) := H(z) for r0 ≤ |z| ≤ 1. iu1 (z) Exercise V.1.11. Verify that b ≡ 1 on a neighbourhood of {|z| ≤ r0 }, and b(eiθ ) =

eiθ . iu1 (eiθ )

Show that b : D → C is of class W 1,p by the rules of differentiation from Section IV.2.3 and the Banach algebra property (Proposition IV.2.17). ♦ Then multiplication by B (as defined for B0 in the statement of the lemma) defines an isomorphism Tu Bt → V p , thanks to the Banach algebra property of W 1,p . Exercise V.1.12. Show that the matrix-valued function B satisfies BJB −1 = J on D. ♦ Now choose pointwise a conjugation matrix A(z) as on page 84, such that AJA−1 = J0 on D. Notice that the construction of A(p), p ∈ Cn , described there yields a smooth matrix-valued function on Cn . But now we set A(z) := Au(z) , so this is a W 1,p -function on D by the chain rule. Also, A(z) equals the identity matrix for |z| close to 1, so η 7→ Aη defines an isomorphism of V p . Then, B0 := AB has the properties described in the lemma.

V.1. Fredholm theory

131

It remains to compute the transformed operator T . Recall equation (IV.2) for the definition of Dut . We interpret the third summand in that definition in terms of a matrix-valued Lp -map2

(V.1) z 7−→ J 0 (z) := DJ u(z) . ∂y u(z) ∈ R2n×2n
as J 0 (ξ) : D → Cn , z 7→ J 0 (z) ξ(z) . Then, using the identity B0 JB0−1 = J0 , we compute   B0 Dut B0−1 η = B0 ∂x (B0−1 η) + J∂y (B0−1 η) + J 0 B0−1 η =

ηx + iηy + B0 J 0 B0−1 η
+ B0 ∂x B0−1 + J∂y B0−1 η

=

Tη

with T η := ηx + iηy + Cη, where
C := B0 J 0 B0−1 + B0 ∂x B0−1 + J∂y B0−1 .

(V.2)

Each of the three summands is a product of matrix-valued maps, at most one of which is of class Lp , and the others of class W 1,p and hence of class C 0 by Sobolev  embedding. By the compactness of D, such products are of class Lp .

V.1.4

Fredholm plus compact is Fredholm

We now want to show that in fact it suffices to prove that ∂ is a Fredholm operator if we want to demonstrate this property for ∂ + C; cf. Note V.1.10. We begin with a little general Fredholm theory. Recall that an operator K : E → F is called compact if the image under K of the open unit ball in E is relatively compact. Equivalently, the image of every bounded sequence in E has a convergent subsequence. In particular, a compact operator is bounded. Exercise V.1.13. Prove the following statements: (i) The sum of two compact operators is compact. (ii) The pre- or post-composition of a compact operator with a bounded operator is compact. For (i) one can find a direct proof, but you may also use (ii).

♦

Example V.1.14. (1) If idE is compact, then E is finite-dimensional, since the closed unit ball in a Banach space is compact only in finite dimensions; see [Hirzebruch & Scharlau 1971, Lemma 24.2]. 2 Here


we write DJ u(z) rather than DJ(z), since for the differential it is essential that J is 2n a map defined on R .

Chapter V. Transversality

132

(2) If T : E → F is bounded and im T is finite-dimensional, then T is compact. a–Ascoli theorem one can show that the dual K ∗ : F∗ → E∗ of (3) With the Arzel` a compact operator K is likewise compact; see [Hirzebruch & Scharlau 1971, Satz 24.5]. We shall also be using the facts stated in the next note. Note V.1.15. Any finite-dimensional subspace of a Banach space is closed, and it can be shown to possess a complementary closed subspace (use the Hahn–Banach extension theorem to define a projection operator P onto the given subspace; the kernel of P is the complementary subspace). Likewise (and more trivially), any closed subspace of finite codimension has a (closed) complement. See, for instance, Lemmas 24.7 and 24.8 in [Hirzebruch & Scharlau 1971]. ∼ In particular, for a Fredholm operator T : E → F we have splittings E = ∼ ker T ⊕ V and F = im T ⊕ W with dim ker T, dim W < ∞. Exercise V.1.16. Let T : E → F be a bounded linear operator, and T ∗ : F∗ → E∗ the dual operator. Show that if im T is closed (so that coker T is a Banach space), then (coker T )∗ ∼ = ker T ∗ . Hint: Verify that the maps ker T ∗ → (coker T )∗ , ρ 7→ λ with λ(y + im T ) := ρ(y) and (coker T )∗ → ker T ∗ , λ 7→ ρ with ρ defined as the composition λ

F −→ F/ im T −→ R, are well defined and inverses of each other. For a more systematic approach to this statement, see Exercise V.1.21.

♦

Lemma V.1.17. Let K : E → E be a compact operator. Then, T := idE −K is Fredholm. Proof. (i) ker T is finite-dimensional: The identity map on ker T equals idE |ker T = K|ker T , and hence is compact. This implies that ker T is finite-dimensional. (ii) The image im T = T (E) is closed: As a finite-dimensional subspace, ker T ⊂ E is closed and has a complementary closed subspace V. The restriction T |V : V → im T is continuous and bijective. It suffices to show that the inverse (T |V )−1 : im T → V is bounded, because then im T is complete and hence closed. Arguing by contradiction, we assume that (T |V )−1 is not bounded. This means that for every ν ∈ N there is a vν ∈ V with kvν k = 1 and kT vν k ≤ 1/ν. Since K is compact, after passing to a subsequence we may assume that Kvν → x for some x ∈ E. Since T vν → 0, this implies vν → x, hence x ∈ V and kxk = 1. But T x = limν→∞ T vν = 0, so also x ∈ ker T , which is impossible. (iii) coker T is finite-dimensional: By (ii) and Exercise V.1.16 we have (coker T )∗ ∼ = ker T ∗ = ker(idE −K ∗ ). Since K ∗ is compact by Example V.1.14, the claim follows from (i).



V.1. Fredholm theory

133

The next proposition is known as the Atkinson characterisation of Fredholm operators. Proposition V.1.18. A bounded operator T : E → F is Fredholm if and only if T is invertible modulo compact operators, i.e. there exist a bounded operator S : F → E and compact operators KE and KF on E and F, respectively, such that ST = idE −KE and T S = idF −KF . Proof. If S, KE and KF exist as described, then ker T ⊂ ker(idE −KE ) is finitedimensional by Lemma V.1.17. Similarly, from im(idF −KF ) ⊂ im T , we see that dim coker T < ∞. Conversely, if T is Fredholm, we can choose closed complements of ker T and im T , that is, E = ker T ⊕ V and F = im T ⊕ W. The restriction T |V is an isomorphism onto im T , and the operator (
−1 on im T , T |V S := 0 on W is continuous by the open mapping theorem. Exercise V.1.19. Verify that (i) T ST = T and ST S = S; (ii) KE := idE −ST is a projection operator onto ker T , and KF := idF −T S is a ♦ projection operator onto W. As bounded operators with finite-dimensional image, KE and KF are com pact. This concludes the proof of the proposition. Exercise V.1.20. Let T : E → F and S : F → G be bounded linear operators. (a) Show that the sequences T

0 −→ ker T −→ ker ST −→ ker S ∩ im T −→ 0 and

ker S + im T F G G S −→ −→ −→ −→ 0, im T im T im ST im S where the unlabelled maps are the obvious inclusions or projections, are exact. 0 −→

(b) Now suppose that two of T, S and ST are Fredholm operators. Use (a) to show that the third operator is then likewise Fredholm, and ind ST = ind S + ind T.

♦

Chapter V. Transversality

134

Exercise V.1.21. Let T : E → F be a bounded operator with closed image. Consider the exact sequence T

0 −→ ker T −→ E −→ F −→ coker T −→ 0 of Banach spaces. (a) Use the Hahn–Banach theorem to show that the dual sequence T∗

0 −→ (coker T )∗ −→ F∗ −→ E∗ −→ (ker T )∗ −→ 0 is likewise exact, and infer that T ∗ also has closed image. In particular, this means that ker(T ∗ ) ∼ = (coker T )∗ and

coker(T ∗ ) ∼ = (ker T )∗ .

(b) Conclude that if T is Fredholm, then so is T ∗ , with ind T ∗ = − ind T .

♦

In preparation of the next lemma, we recall the following facts. Note V.1.22. Any bijective continuous linear operator has a continuous inverse by the open mapping theorem. The subset of such invertible bounded operators E → F in the space of all bounded operators is open. Indeed, if T : E → F is −1 invertible, then for kSk < kT −1 kP , an explicit inverse of T − S = T (idE −T −1 S) ∞ is given by the geometric series k=0 (T −1 S)k T −1 . Lemma V.1.23. The set of Fredholm operators is an open subset in the space of all bounded linear operators E → F, and the index is constant on the connected components of this subset. Proof. Let T : E → F be a Fredholm operator. Write E = ker T ⊕ V and F = im T ⊕ W as above. Let I : V → E be the inclusion, and P : F → im T the projection defined by the splitting of F. Now, any bounded operator S : E → F gives rise to an operator S 0 := P SI : V → im T . Since T 0 := T |V : V → im T is an isomorphism, by Note V.1.22 the same will hold for S 0 , provided that kS − T k is sufficiently small. This proves the first statement. For the statement about indices, we may interpret such an S 0 = P SI (with S sufficiently close to T ) as a Fredholm operator of index 0. Since P and I are Fredholm operators, so is S (apply Exercise V.1.20 (b) twice), and 0 = ind P SI = ind P + ind S + ind I = dim coker T + ind S − dim ker T, whence ind S = ind T , so the index is a locally constant function on the space of  Fredholm operators. Finally, we arrive at the proposition that will allow us to reduce the claim that ∂ + C is Fredholm to showing that ∂ is Fredholm.

V.1. Fredholm theory

135

Proposition V.1.24. Let T, K : E → F be a Fredholm and a compact operator, respectively. Then, T + K is Fredholm with ind(T + K) = ind T . Proof. Choose S, KE , KF as in the Atkinson characterisation of T being Fredholm. The operator S also does the trick for T + K, since idE −S(T + K) = KE − SK and

idF −(T + K)S = KF − KS

are compact operators by Exercise V.1.13. The statement about indices follows from the preceding lemma, applied to the family T + tK, t ∈ [0, 1], of Fredholm operators.  Exercise V.1.25. Show that a Fredholm operator T has ind T = 0 if and only if it ♦ can be written as T = T0 + K with T0 invertible and K compact. Recall that the map C defined in equation (V.2) is a matrix-valued map p the corresponding operator V p → Lp (D, Cn ) is defined by on D of  class L , and
Cη = z 7→ C(z) η(z) . Lemma V.1.26. The multiplication operator C : V p → Lp (D, Cn ) is compact. Proof. By the Sobolev embedding theorem, we have a compact embedding V p → C 0 (D, Cn ), and the multiplication operator C may be regarded as a linear operator C 0 (D, Cn ) → Lp (D, Cn ) bounded by kCkLp . In other words, we have the factorisation C C : V p −→ C 0 (D, Cn ) −→ Lp (D, Cn ). As the composition of a compact and a bounded linear operator, C is compact.



Note V.1.27. This reduces the proof of Theorem V.1.5 to showing that ∂ is a Fredholm operator of index n − 1.

V.1.5 The components of ∂ We now look at the ∂-operator componentwise, that is, we write V p = V1p ⊕ V2p ⊕ · · · ⊕ Vnp with  V1p := η ∈ W 1,p (D, C) : η(eiθ ) ∈ eiθ R, θ ∈ R/2πZ; η(1) = η(i) = η(−1) = 0 and  V2p = . . . = Vnp := η ∈ W 1,p (D, C) : η(∂D) ⊂ R . We use the notation ∂ (j) := ∂|Vjp : Vjp → Lp (D, C). We need only consider ∂ (1) and ∂ (2) . Theorem V.1.5 is then, by Note V.1.27, a consequence of the next proposition. In the proof it is understood that we always work with the unique continuous representative of any element in Vjp .

Chapter V. Transversality

136

Proposition V.1.28. (a) ∂ (2) is surjective, with ker ∂ (2) = R given by the constant (real-valued) functions. (b) ∂ (1) is invertible. Proof. (a0) Any η ∈ V2p with ∂η = 0 is smooth (and holomorphic on B1 = Int(D)) by elliptic regularity (Theorem IV.3.1). The real boundary values of η allow us to ˆ which must be extend η by Schwarz reflection to a holomorphic function on C, constant by the maximum principle. (Alternatively, apply the maximum principle directly to the harmonic function Im η.) This proves that ker ∂ (2) = R. The surjectivity of ∂ (2) will be proved in three steps. (a1) There is a constant c such that kηk1,p ≤ c k∂ηkLp + kηkLp




for all η ∈ V2p .

(V.3)

(a2) The image of ∂ (2) is closed in Lp (D, C). (a3) The image of ∂ (2) contains the space Cc∞ (B1 , C) of smooth functions on D compactly supported in the interior of D. The estimate in (a1) is known as a semi-Fredholm estimate, for as we shall see it leads directly to statement (a2), which together with the finite-dimensionality of ker ∂ (2) is known as the semi-Fredholm property. Since Cc∞ (B1 , C) is dense in Lp (D, C), the surjectivity of ∂ (2) follows from (a2) and (a3). For a proof that Cc0 (B1 , C) is dense in Lp (D, C), see [Adams & Fournier 2003, Theorem 2.19]; then the density of Cc∞ (B1 , C) follows from the Stone–Weierstraß approximation theorem, or see [Adams & Fournier 2003, Corollary 2.30]. Proof of (a1) The real and imaginary part of η can be approximated separately (in the (1, p)norm) by smooth functions (Theorem IV.2.10) and smooth functions compactly supported in the interior of D (Proposition IV.2.23), respectively. In particular, this means that η can be approximated by smooth complex-valued functions taking real values on ∂D. It therefore suffices to prove (V.3) for smooth functions η in V2p . Choose a smooth bump function f0 : B2 → [0, 1] that is identically equal to 1 near ∂D and supported in a thin annulus around it. Extend a given smooth η by Schwarz reflection in the unit circle (cf. Exercise IV.3.15 and Remark II.1.6) to a ˆ if you wish). In the following estimates, function ηˆ ∈ W 1,p (B2 , C) (or even on C, norms where the domain is not specified refer to functions on D. We estimate f0 η and (1 − f0 )η separately. Compare the estimate for f0 η with the argument on page 83.

V.1. Fredholm theory

kf0 ηk1,p

137

≤ kf0 ηˆk1,p,B2 ≤ ≤ ≤

k(1 − f0 )η)k1,p

const. k∂(f0 ηˆ)kLp (B2 ) by Proposition IV.2.24
const. k(∂f0 )ˆ η kLp (B2 ) + kf0 ∂ ηˆkLp (B2 )
const. kηkLp + k∂ηkLp ;

≤ ≤

const. k∂(1 − f0 )ηkLp by Proposition III.1.16
const. kηkLp + k∂ηkLp .

Proof of (a2) Let υ ∈ Lp (D, C) be a function in the closure of im ∂ (2) , that is, there is a sequence (ην ) in V2p with ∂ην → υ. We need to show that υ ∈ im ∂ (2) . Since ∂ην remains unaltered when we add a real constant to ην , we may assume that ην (1) = 0. We claim that with this assumption the sequence (ην ) is bounded. Arguing by contradiction, if the sequence were unbounded, we would have (after passing to a subsequence) kην k1,p > ν for all ν ∈ N. Consider the sequence of normalised functions ην1 := ην /kην k1,p in V2p . This satisfies k∂ην1 kLp =

k∂ην kLp k∂ην kLp < −→ 0 as ν → ∞, kην k1,p ν


since the sequence k∂ην kLp is convergent and hence bounded. Moreover, the p inclusion V2 → C 0 (D, C) → Lp (D, C) is compact as the composition of a compact and a bounded3 linear map, so (again after passing to a subsequence) there is an η 1 ∈ Lp (D, C) with kην1 − η 1 kLp → 0. Thus, both (∂ην1 ) and (ην1 ) are Cauchy sequences in Lp (D, C). By the semi-Fredholm estimate (V.3), (ην1 ) is a Cauchy sequence in the space 1,p W (D, C), so the limit η 1 is actually of class W 1,p , hence lies in V2p , and satisfies ∂η 1 = 0. By (a0), η 1 must be a constant function, but then the obvious limit properties kη 1 k1,p = 1 and η 1 (1) = 0 are contradictory. This contradiction shows that (ην ) is indeed bounded. By the same argument as for (ην1 ), we now conclude that (a subsequence of) (ην ) converges in the (1, p)norm to a function η ∈ V2p . This function satisfies ∂η = limν→∞ ∂ην = υ (in Lp ), so υ ∈ im ∂ (2) . 3 The inclusion map C 0 (D, C) → Lp (D, C) is bounded by the area π of D, since kηk p ≤ L πkηkC 0 .

Chapter V. Transversality

138 Proof of (a3)

Given f ∈ Cc∞ (B1 , C), we want to find an η ∈ V2p with ∂η = f . As we shall see, this η can be constructed in C ∞ (D, C). 1+z ˆ restricts to a conformal identiThe M¨obius transformation ϕ : z 7→ i 1−z of C ˆ fication D → H; cf. Exercise II.2.13. In order to avoid the points ∞ in domain and image, we regard ϕ as a map C \ {1} → C \ {−i}, and its restriction D \ {1} → H, with limz→1 ϕ(z) = ∞. Define f h := ◦ ϕ−1 |H ∈ Cc∞ (Int(H), C), (V.4) ϕx and also write h for the function in Cc∞ (C, C) obtained by Schwarz reflection. By Proposition III.1.7, there is a unique function κ ∈ C ∞ (C, C) with ∂κ = h and

lim κ(w) = 0.

w→∞

Exercise V.1.29. Set λ = c◦κ◦c, where c denotes complex conjugation on C. Show that limz→∞ λ(z) = 0 and, using the fact that c◦h◦c = h (by construction), verify that ∂λ = h. Then use the uniqueness of κ to conclude that κ(R) ⊂ R. ♦ The chain rule for the Wirtinger derivative, which the avid reader elaborated in Exercise III.1.22, says that for η := κ ◦ ϕ : C \ {1} → C we have ηz = (κ ◦ ϕ)z = (κw ◦ ϕ)ϕz + (κw ◦ ϕ)ϕz = (κw ◦ ϕ)

∂ϕ = (κw ◦ ϕ)ϕx , ∂z

where in the penultimate equality we have used the holomorphicity of ϕ (Exercise III.1.22 (c)) and an identity from Exercise III.1.22 (d); the equality ∂ϕ ∂z = ϕx is a straightforward computation. It follows that ∂η = (∂κ ◦ ϕ)ϕx = (h ◦ ϕ)ϕx on C \ {1}.

(V.5)

By the definition (V.4) of h on H, this gives the desired solution ∂η = f on D \ {1}. But this function h is compactly supported in the interior of H, and so the function h on C obtained by Schwarz reflection vanishes in a neighbourhood of ˆ But then equation (V.5) says that η is holomorphic on an annulus R ∪ {∞} ⊂ C. around ∂D, with the point 1 removed. Since limz→1 η(z) = limw→∞ κ(w) = 0, the Riemann removable singularities theorem allows us to extend η smoothly to a function on C by setting η(1) = 0. The equation ∂η = f holds on D by continuity. (b) To show that ∂ (1) is invertible, we establish the following points: (b1) ∂ (1) is injective. (b2) The image of ∂ (1) is closed in Lp (D, C). (b3) The image of ∂ (1) contains the space Cc∞ (B1 \ {0}, C).

V.1. Fredholm theory

139

Part (b2) is proved by the same semi-Fredholm estimate as in (a2). Together with (b3), the surjectivity of ∂ (1) follows as in (a). Thus, it remains to demonstrate (b1) and (b3). Proof of (b1) As in (a0) we see by elliptic regularity that an η ∈ V1p with ∂η = 0 is holomorphic on B1 . The arguments of Section IV.3 also apply to the function z 7→ η(z)/z — which is defined on D \ {0} and has real boundary values along ∂D — to show that η P extends smoothly to ∂D. Thus, on B1 we may write η as a power series ∞ η(z) = k=0 ak z k with (for any 0 < r < 1) Z Z 2π Z 2π r%1 1 η(z) 1 η(reiθ ) 1 ak = dz = dθ − − − − − → η(eiθ )e−ikθ dθ; 2πi ∂Dr z k+1 2π 0 (reiθ )k 2π 0 the passage to the limit is allowed thanks to η being continuous on D. Thus, the ak are actually the Fourier coefficients in the Fourier series expansion of η|∂D . Notice that for negative integers k, the Fourier coefficients are zero by the Cauchy integral theorem. Since η|∂D is smooth, the Fourier series of η converges uniformly to η, so we have ∞ X η(eiθ ) = ak eikθ , θ ∈ R/2πZ. k=0

The property η(eiθ ) ∈ eiθ R gives us the identity eiθ η(eiθ ) = e−iθ η(eiθ ), which translates into ∞ X

ak ei(1−k)θ =

k=0

∞ X

ak ei(k−1)θ .

k=0

By comparing coefficients, we see that a0 = a2 , a1 = a1 , and ak = 0 for k ≥ 3, so we may write η as η(z) = (α + iβ)z 2 + γz + α − iβ with α, β, γ ∈ R. When we want to prescribe (real!) linear system  2 0 2

values η(ik ) ∈ ik R, k ∈ {0, 1, 2}, we need to solve the 0 2 0

    1 α η(1) −1 β  =  iη(i)  . −1 γ η(−1)

(V.6)

This system has a unique solution for any prescribed values of η at 1, i, −1, and the three-point condition η(1) = η(i) = η(−1) = 0 for elements of V2p yields α = β = γ = 0.

Chapter V. Transversality

140 Proof of (b3)

Given f ∈ Cc∞ (B1 \{0}, C), we wish to find an η1 ∈ V1p with ∂η1 = f . The function g : D → C defined by ( f (z)/z for z ∈ D \ {0}, g(z) := 0 for z = 0, is an element of Cc∞ (B1 , C), and by (a3) there is a solution ζ ∈ V2p of the equation ∂ζ = g. Set η(z) = zζ(z). Then, η satisfies the boundary condition η(eiθ ) ∈ eiθ R, and ∂η(z) = z∂ζ(z) = zg(z) = f (z). Next, set η0 (z) = (α + iβ)z 2 + γz + α − iβ, where (α, β, γ) ∈ R3 is the unique solution of (V.6). Then, η1 := η − η0 ∈ V1p , and ∂η1 = f . 

V.2

Regular values and the Sard–Smale theorem

Ultimately, we want to use the description of the moduli space as M = {u ∈ B : ∂ J u = 0} (cf. Section IV.3.1) to establish its being a manifold. Recall the situation in finite dimensions. Given a smooth function f = (f1 , . . . , fn ) : Rm → Rn , the zero set M0 = {f1 = . . . = fn = 0} ⊂ Rm is a submanifold of codimension n (or empty), provided that the gradients ∇f1 , . . . , ∇fn are linearly independent at all points of M0 . This condition is equivalent to saying that the differential Tp f is surjective at all points p ∈ M0 = f −1 (0). More generally, given a smooth map f : M → N between finite-dimensional manifolds, a point q ∈ N is called a regular value of f if Tp f is surjective for all p ∈ f −1 (q) =: Mq . In this case, Mq ⊂ M is empty or a submanifold of codimension equal to dim N . Even more generally, if Q ⊂ N is a submanifold and f : M → N a map transverse to Q in the sense that Tp f (Tp M ) + Tq Q = Tq N for all q ∈ Q and p ∈ f −1 (q), then f −1 (Q) ⊂ M is a submanifold of codimension equal to the codimension of Q in N . Even though we shall only be considering the preimage of a single point (see Section V.2.4 for an outline of the further strategy), this discussion goes by the name of ‘transversality’ to avoid confusion with the other meaning of ‘regularity’ in Chapter IV.

V.2.1 The implicit function theorem These statements about submanifolds are a direct consequence of the implicit function theorem. This theorem continues to hold for Banach spaces and Banach

V.2. Regular values and the Sard–Smale theorem

141

manifolds, provided that one has an adequate splitting of the domain (or its tangent space, in the case of Banach manifolds) into a direct sum of closed (and hence complete normed) spaces — the reason being that the proof in finite dimensions carries over to infinite dimensions if one can still rely on the Banach fixed point theorem (a.k.a. the contraction mapping theorem). Theorem V.2.1. Let E1 , E2 , F be Banach spaces and k ∈ N. Suppose we have an open set U ⊂ E1 ⊕ E2 containing the point (x01 , x20 ), and a C k -map f : U → F such that (i) f (x01 , x02 ) = 0;
(ii) D2 f (x01 , x20 ) := D f (x01 , . ) (x20 ) : E2 → F is a bounded invertible linear operator. Then there is an open neighbourhood U1 × U2 ⊂ E1 ⊕ E2 of (x10 , x20 ) and a unique continuous map g : U1 → U2 such that   (x1 , x2 ) ∈ U1 × U2 : f (x1 , x2 ) = 0 = (x1 , g(x1 )) : x1 ∈ U1 . Moreover, this map g is of class C k .

V.2.2



Regular values

In infinite dimensions, the definition of regular values takes the assumptions of the infinite-dimensional implicit function theorem into account. Definition V.2.2. Let f : M → N be a C 1 -map between Banach manifolds. A point q ∈ N is a regular value of f if for all p ∈ f −1 (q) the following conditions are satisfied: (R1)

Tp f : Tp M → Tq N is surjective.

(R2)

ker(Tp f ) has a closed complement in the Banach space Tp M .

As in finite dimensions, the preimage f −1 (q) of a regular value q ∈ N of f is empty or a Banach submanifold of M . The tangent space to this submanifold at p ∈ f −1 (q) is ker(Tp f ). Note V.2.3. If ker(Tp f ) is finite-dimensional (so in particular if Tp f is a Fredholm operator), condition (R2) is automatically satisfied by Note V.1.15. The component of f −1 (q) containing p is then a manifold of dimension equal to dim ker(Tp f ). Lemma V.2.4. If condition (R1) in Definition V.2.2 is satisfied, condition (R2) is equivalent to the existence of a right-inverse to Tp f , that is, a bounded linear map R : Tq N → Tp M such that Tp f ◦ R = idTq N , and with R(Tq N ) ⊂ Tp M a closed subspace.

Chapter V. Transversality

142

Proof. Suppose condition (R2) is satisfied, i.e. there is a closed subspace C ⊂ Tp M complementary to ker(Tp f ). Then, Tp f |C : C → Tq N is bijective, and hence
−1 . invertible by the open mapping theorem. Set R := Tp f |C Conversely, if a right-inverse R with the described properties exists, then R(Tq N ) is a complementary subspace to ker(Tp f ). Notice that any v ∈ Tp M can be written as
v = v − (R ◦ Tp f )(v) + (R ◦ Tp f )(v) ∈ ker(Tp f ) ⊕ im R. The sum is direct, for if v ∈ ker(Tp f ) ∩ im R, write v = R(w) and observe that  0 = Tp f (v) = (Tp f ◦ R)(w) = w, hence v = 0.

V.2.3

The Sard–Smale theorem

Sard’s theorem says that the set of regular values of a C ∞ -map f : M → N between finite-dimensional manifolds is dense in N ; in fact, its complement of critical values is of measure zero [Milnor 1965]. An infinite-dimensional version of this theorem for Fredholm maps between separable Banach manifolds was established by [Smale 1965]. Definition V.2.5. A C 1 -map f : M → N between Banach manifolds is called a Fredholm map if the differential Tp f : Tp M → Tf (p) N is a Fredholm operator for all p ∈ M . Remark V.2.6. If the index ind(Tp f ) is independent of the choice of p ∈ M , we call this the index of f and write ind(f ). For instance, by Lemma V.1.23 this will be the case when M is connected. In our application, we shall prove directly that the pointwise index is constant. Theorem V.2.7 (Smale). Let f : M → N be a C ∞ Fredholm map between Banach manifolds, with M separable. Then the regular values of f are dense in N . Smale actually shows that the set of regular values is residual in N , that is, it contains a countable intersection of sets that are open and dense in N . Exercise V.2.8. Show that any countable intersection of residual subsets of a given topological space is again residual; this is often important when one deals with a ♦ succession of constraints. A Baire space is a topological space in which every residual set is dense. Exercise V.2.9. Show that every complete metric space is a Baire space.

♦

Note V.2.10. If q ∈ N is a regular value of a smooth Fredholm map f : M → N , with M connected, then f −1 (q) ⊂ M is a manifold of dimension equal to ind(f ) (or empty). Smale formulates the theorem for Banach manifolds with a countable base of the topology (i.e. second-countable spaces). This is equivalent to separability, as the next exercise shows.

V.2. Regular values and the Sard–Smale theorem

143

Exercise V.2.11. Show that a metric space has a countable base of the topology ♦ if and only if it has a countable dense subset. We do not present a complete proof of Smale’s theorem. Rather, we encourage the reader to take a look at Smale’s beautiful short paper, where this proof covers a mere two pages. The separability of M and the fact (proved in Smale’s paper) that Fredholm maps are locally proper4 are used to reduce the proof to a local statement. The next exercise, whose solution is analogous to the proof of the implicit function theorem (or the equivalent inverse function theorem), then reduces the problem to the finite-dimensional situation. Exercise V.2.12. Let f : M → N be a C 1 Fredholm map. Show that for any p ∈ M there are charts (U, φ) around p and (V, ψ) around f (p) such that ψ ◦ f ◦ φ−1 is of the form E ⊕ ker(Tp f ) −→ E ⊕ coker(T
p f ) (x, y) x, g(x, y) , 7−→ where E is a Banach space, and g is a C 1 -map.

♦


Notice that (x, z) is a regular value of the map (x, y) 7→ x, g(x, y) if and only if z is a regular value of g|{x}×ker(Tp f ) . Thus, Sard’s theorem allows us to find a regular value of f in any given neighbourhood of any f (p), p ∈ M . This shows that the regular values of f lie dense. With the separability of M and the local properness of f one concludes that the set of regular values is actually residual. Here is a simple example, found by [Bonic 1966], illustrating the fact that the Fredholm property is essential in the Sard–Smale theorem. Earlier, a more involved example of this phenomenon was described by [Kupka 1965]. Exercise V.2.13. Let E be the space of real-valued continuous functions x : t 7→ x(t) on the interval [0, 1]. With the C 0 -norm, E is a Banach space. We consider the map f : E → E given by f (x) = x3 . (a) Show that f is of class C ∞ , and compute its derivatives. (b) Why is f not a Fredholm map? (c) Show that x ∈ E is a critical point of f , i.e. a point where Tx f violates (R1) or (R2), if and only if x has a zero, and it is an interior point of the set of critical points if and only if x changes sign. (d) Observe that f sends the set of critical points onto itself, and it sends the interior of that set onto the interior. Conclude that the image of the set of critical points contains an open set. ♦ 4 Smale calls a map locally proper if any point in the domain of definition has a neighbourhood on which the map is proper. (Beware that there are variants of this definition.)

Chapter V. Transversality

144

V.2.4

How to prove that M is a manifold

When we choose any J as in Lemma I.7.17, the nonlinear Cauchy–Riemann operator ∂ J : B → Lp (D, R2n ) is a Fredholm map by Theorem V.1.5, so if 0 were a regular value of this map, we would have succeeded in showing that M is a manifold. Unfortunately, 0 is not, in general, a regular value of ∂ J for an arbitrary choice of J. The idea now is to regard the nonlinear Cauchy–Riemann operator with varying J as a map on the product space B × J , where J is a suitable space of almost complex structures as in Lemma I.7.17. We may not allow all such structures, for then J would not be a separable Banach manifold. On the other hand, the space J needs to be sufficiently large such that the operator ∂• :

B×J (u, J)

−→ 7−→

Lp (D, R2n ) ∂J u

has zero as a regular value. Then, ker ∂ • ⊂ B×J is a (separable) Banach manifold. In Section V.5 we shall use a norm concept introduced by Floer to describe such a separable space J of almost complex structures. The operator ∂ • will be analysed in Section V.6. Finally, in Section V.7 we apply the Sard–Smale theorem to the projection pJ : ker ∂ • → J onto the second factor. Notice that for any J, the fibre p−1 J (J) contains flat discs, so it is non-empty. Any regular value J of pJ then yields a moduli space MJ = ker ∂ J that is a Banach manifold. The set of such J is residual (and in particular dense) in J . This is the meaning of ‘generic J’ in Theorem I.8.13. First, though, we have to deal with some local properties of J-holomorphic discs that are relevant to this line of reasoning.

V.3

The Carleman similarity principle

The aim of this section is to analyse the local behaviour (around interior points of D) of elements of the moduli space M. The boundary behaviour of the holomorphic discs is irrelevant for this local discussion, so we shall be making statements about arbitrary solutions of the nonlinear Cauchy–Riemann equation ∂ J u = 0. In the form stated below (Theorem V.3.3), you will not find the similarity principle in the work of Carleman. However, some of the central ideas can be traced back via the work of [Vekua 1962] and [Bers 1956] to the paper [Carleman 1933].

V.3.1 Behaviour near the boundary First, though, we recall what has been established so far about the local behaviour of holomorphic discs in M. By Lemma II.1.17, any u ∈ M sends the interior of

V.3. The Carleman similarity principle

145

D into the open cylinder Z; in particular, the image of Int(D) under u is disjoint from u(∂D). Furthermore, by Lemma II.1.18, the restriction u|∂D is an embedding ∂D → ∂Z. Exercise V.3.1. Use Lemma II.1.18 (and its proof) and the classical maximum principle to show that for each u ∈ M there is a neighbourhood U ⊂ D of ∂D such ♦ that u|U is an embedding, and u(U ) ∩ u(D \ U ) = ∅.

V.3.2

From a nonlinear to a linear equation

Let J be an almost complex structure on R2n = Cn . We fix a solution u ∈ C ∞ (D, Cn ) of the nonlinear Cauchy–Riemann equation ∂ J u = 0. By the argument on page 84 following Exercise III.3.6, there is a smooth map A : Cn → GL(2n, R) such that (with Ap := A(p) as on page 84) Ap J(p) = J0 Ap for all p ∈ Cn .

(V.7)

Set v := (A ◦ u)u : D → Cn . Using (V.7) and ∂ J u = 0, we compute ∂v = ∂(A ◦ u)u + (A ◦ u)∂ J u = ∂(A ◦ u) · (A ◦ u)−1 v. Thus, with the matrix-valued function B := −∂(A ◦ u) · (A ◦ u)−1 : D −→ R2n×2n we recognise v as a solution of the linear equation ∂v + Bv = 0.

(V.8)

This may look to you like a conjurer’s trick. Bear in mind, though, that we have arrived at this linear equation for v = (A ◦ u)u only after restricting our attention to a single solution u of the nonlinear equation ∂ J u = 0. Remark V.3.2. Likewise, for a fixed u and with J(z) := J(u(z)) (cf. Notation V.1.8), one may regard u as a solution of the linear equation ux (z) + J(z)uy (z) = 0.

V.3.3 The similarity principle and its corollaries The Carleman similarity principle says, in essence, that locally a solution of the linear equation (V.8) looks, up to a pointwise linear transformation of the tangent bundle of R2n = Cn , like a holomorphic map. Even though this transformation is only continuous, the similarity principle allows one to carry over results from complex analysis about the local behaviour of holomorphic functions, such as the identity theorem, to J-holomorphic curves.

Chapter V. Transversality

146

Theorem V.3.3 (Carleman Similarity Principle). Let G ⊂ C be a domain containing the origin 0, and v : G → R2n = Cn a smooth solution of ∂v + Bv = 0, where B is a smooth map G → GL(2n, R). Then there is an ε > 0 such that Bε := Bε (0) is contained in G, a continuous map Φ : Bε −→ GL(n, C), and a holomorphic map h : Bε −→ Cn such that v = Φh on Bε . Applied to v := (A ◦ u)u on G = B1 := Int(D), this theorem tells us that u = (A ◦ u)−1 Φh =: Ψh is of similar form around z0 = 0, but notice that the function Ψ on Bε takes values in GL(2n, R). By precomposing u with a conformal automorphism of B1 before applying the similarity principle, one arrives at the same description around any point z0 ∈ B1 . Before we turn to the proof of theorem V.3.3, we record two corollaries one obtains by applying it to v = (A ◦ u)u. Corollary V.3.4. Given u ∈ M, for any p ∈ R2n the preimage u−1 (p) is a finite set. Proof. Arguing by contradiction, we assume that there is a p0 ∈ R2n for which {u−1 (p0 )} ⊂ D is an infinite set. Then this set has an accumulation point z0 ∈ D. In fact, by Exercise V.3.1, we must have {u−1 (p0 )} ⊂ Int(D) = B1 and z0 ∈ B1 . Now apply Theorem V.3.3 to the translated map z 7→ u(z) − p0 , which is Jp0 -holomorphic for Jp0 (p) := J(p + p0 ). This tells us that u − p0 = Ψh in a neighbourhood of z0 , and z0 is an accumulation point of zeros of the holomorphic map h. From the identity theorem it follows that h ≡ 0, and hence u ≡ p0 in a neighbourhood of z0 ∈ B1 . Now consider the non-empty set {z ∈ B1 : u ≡ p0 in a neighbourhood of z}, which is open by definition. Our argument above shows that this set is closed in B1 , so it must be all of B1 , which by continuity5 means that u ∈ M is constant equal to p0 on all of D, an obvious contradiction.  Corollary V.3.5. Any u ∈ M has a finite set of critical points Crit(u) := {z ∈ D : Tz u = 0}.
Hence, by the preceding corollary, the set u−1 u(Crit(u) of points in D mapping to a critical value is likewise finite. 5 Or

use Exercise V.3.1 once more to arrive at a contradiction.

V.3. The Carleman similarity principle

147

Proof. Again we argue by contradiction. Thus, suppose a given u has an infinite set of critical points. Then, Crit(u) has an accumulation point z0 ∈ D; in fact, by Exercise V.3.1 this point z0 must lie in B1 . By differentiating the equation ∂ J u = 0 with respect to the variable x, we obtain uxx + J(u)uyx + DJ(u)(ux )uy = 0. Set v := ux , C = DJ(u)( . )uy , and J = J ◦ u (by abuse of notation). Then this equation becomes vx + Jvy + Cv = 0. For w := (A ◦ u)v we compute
∂w = ∂(A ◦ u)v + (A ◦ u)∂ J v = ∂(A ◦ u) − (A ◦ u)C (A ◦ u)−1 w, so again we are in the setting of the Carleman similarity principle. We conclude that v is of the form v = Ψh around z0 , with z0 being an accumulation point of zeros of the holomorphic map h. Again by the identity theorem we conclude that ux = v ≡ 0 and uy = −J(u)ux ≡ 0 in a neighbourhood of z0 , which means that u is constant on that neighbourhood, contradicting the previous corollary. 

V.3.4 Proof of the Carleman similarity principle By assumption of Theorem V.3.3, we have a smooth function B on G taking values in the space of real (2n × 2n)-matrices, or the real linear endomorphisms EndR (Cn ), and a smooth solution v : G → Cn of (V.8), that is, ∂v + Bv = 0. Replace B by a map taking values in EndC (Cn ) As a first step, we want to replace B in (V.8) by a map taking values in the complex linear endomorphisms EndC (Cn ) of Cn . With B± :=

1 (B ∓ J0 BJ0 ) 2

we can write B = B+ + B− with B+ (z) a complex linear endomorphism of Cn for all z ∈ G, and B− (z) a complex anti-linear one, since B+ J0 = J0 B+ and B− J0 = −J0 B− . Now define C : G → EndR (Cn ) by  t  v(z) ζ v(z) if v(z) 6= 0, C(z)(ζ) = |v(z)|2  0 if v(z) = 0.

Chapter V. Transversality

148

The map G 3 z 7→ C(z) ∈ EndR (Cn ) is smooth on the open set {z ∈ G : v(z) 6= 0}. By construction, C(z) is complex anti-linear for every z ∈ G, we have Cv = v, and the operator norm of C(z) satisfies ( 1 if v(z) 6= 0, |C(z)| = 0 if v(z) = 0.
Thus, we have C ∈ L∞ G, EndR (Cn ) , and the essential supremum kCkL∞ (G) of this map equals 1, unless v is identically equal to zero. The composition of two complex anti-linear endomorphisms is complex linear, so we obtain a map
B 0 := B+ + B− C ∈ L∞ G, EndC (Cn ) . Since Cv = v, this map satisfies B 0 v = Bv. From now on, we write again B for this new map B 0 . Thus, we have replaced B in (V.8) by a map taking values in EndC (Cn ), at the price of this map being only of class L∞ . An equation on the transformation Φ By reverse engineering we first derive an equation on the sought-after transformation Φ, which we subsequently use to find this transformation. Suppose we have written v as v = Φh on Bε , with Φ and h as described in Theorem V.3.3. Then, on Bε , 0 = ∂v + Bv = ∂(Φh) + BΦh = (∂Φ + BΦ)h + Φ∂h, where we have used the C-linearity of Φ(z). Notice that having arranged B to take values in EndC (Cn ) allows us to regard BΦ as a map of the same kind. If we can find a map Φ (taking values in GL(n, C)) satisfying the equation ∂Φ + BΦ = 0, we shall be able to conclude that ∂h = 0, i.e. that h is holomorphic. Remark V.3.6. As we shall see presently, the map Φ can be chosen to be of class W 1,p . Then, h := Φ−1 v is likewise of class W 1,p . A solution h of class W 1,p of the equation ∂h = 0 is smooth by elliptic regularity (Theorem IV.3.1). For the linear Cauchy–Riemann operator we are dealing with here, this regularity result follows much more directly from the Weyl lemma. We shall discuss this in Section V.6.3. Since ε is not known a priori, we start (without loss of generality, really) with B being defined on the open unit disc B1 , and study the equation ∂Φ + χε BΦ = 0,

(V.9)

where χε denotes the characteristic function of Bε , on a suitable Sobolev space of maps Φ : B1 → EndC (Cn ). Our aim will be to show that, for ε > 0 sufficiently

V.3. The Carleman similarity principle

149

small, a solution Φ does indeed exist, and that it actually takes values in GL(n, C). Restricted to Bε , this defines a solution of the original equation ∂Φ + BΦ = 0. The key, then, is to appeal to the surjectivity of ∂, when regarded as an operator between suitable spaces, as established in Proposition V.1.28 (a). In fact, we extend the image space and the map so as to obtain an invertible operator. Subject to the additional constraints imposed by this extension, we ultimately find a unique solution Φ of (V.9) for ε sufficiently small. Solving equation (V.9) We identify EndC (Cn ) with the space Cn×n of complex (n × n)-matrices. This contains the subspace Rn×n of real (n × n)-matrices. Now define, for some p > 2, the real Banach spaces  V p := Φ ∈ W 1,p (D, Cn×n ) : Φ(∂D) ⊂ Rn×n ⊂ C 0 (D, Cn×n ) and Lp := Lp (D, Cn×n ), and consider the linear operator, for ε ∈ (0, 1), Tε :

Vp Φ

−→ 7−→

Lp ⊕ Rn×n
∂Φ + χε BΦ, Φ(1)

The operator ∂ : V p → Lp is the multidimensional version of ∂ (2) in Proposition V.1.28, so we infer that ∂ : V p → Lp is surjective, with kernel equal to Rn×n given by the constant real- and matrix-valued functions. It follows that the operator T: V p −→ Lp ⊕ Rn×n
Φ 7−→ ∂Φ, Φ(1) is bijective. This operator T is bounded: kT ΦkLp ⊕Rn×n = k∂ΦkLp + |Φ(1)| ≤ kΦk1,p + kΦkC 0 ≤ const. kΦk1,p , where the last inequality comes from the Sobolev embedding theorem (Theorem IV.2.12). By the open mapping theorem, T is invertible. Similarly, for the operator χε B we have the estimate kχε BΦkLp ≤ kχε kLp · kBkL∞ · kΦkC 0 ≤ const. ε2/p kΦk1,p , so the operator norm kχε Bk goes to zero as ε & 0. Then, for kχε Bk < kT −1 k−1 , an explicit inverse of Tε = T + χε B = T (idV p +T −1 χε B) is given by Tε−1 =

∞ X k=0

(−T −1 χε B)k T −1 .

Chapter V. Transversality

150

Hence, for ε > 0 sufficiently small, we find a unique solution Φ = Φε of the equation Tε Φ = (0, E), and in particular a solution of (V.9). Here, E denotes the unit (n × n)-matrix and, below, the constant function D → Rn×n , z 7→ E. This solution Φ is of class W 1,p , and hence continuous by the Sobolev embedding theorem. The map Φ takes values in GL(n, C) It remains to show that for ε > 0 sufficiently small, Φ takes values in the invertible matrices. We begin with the observation that kTε−1 k → kT −1 k as ε & 0, so the operator norm of Tε−1 can be estimated by 2kT −1 k, say, for ε small. Again using the Sobolev embedding theorem, we find kΦ − EkC 0

≤

const. kΦ − Ek1,p

≤

const. kTε−1 k · kTε (Φ − E)kLp ⊕Rn×n

≤

const. kT −1 k · kχε BkLp

≤

const. ε2/p ,

which goes to zero as ε & 0, entailing the invertibility of Φ(z). In the third line we have used
Tε (Φ − E) = (0, E) − ∂E + χε BE, E(1) = (χε B, 0). This concludes the proof of the Carleman similarity principle.



Exercise V.3.7. (a) Show that the Carleman similarity principle also holds under the weaker assumptions that B and v be merely of class W 1,p , p > 2. Observe where in the argument you need the Banach algebra property of W 1,p . (b) Prove the Carleman similarity principle for solutions of equations of the form ∂ J v + Bv = 0, now with Φ taking values in GL(2n, R). Analogous to the discussion concerning B in Section V.3.2, we may assume that J = J(z). Hint: As in Lemma V.1.9, choose a function B0 : G → GL(2n, R) such that B0 JB0−1 = J0 . Show that w := B0 v satisfies an equation of the form ∂w + B 0 w = 0. ♦

V.4

Injective points

The concept of injective points of a holomorphic disc u, and the fact that they constitute an open and dense subset of D, will enter crucially — in conjunction with the Carleman similarity principle — in the proof that the Cauchy–Riemann operator on the universal moduli space, which we are going to define in Section V.6, has a surjective differential. This will allow us to conclude that the universal moduli space is a manifold; see the proof of Proposition V.6.5.

V.4. Injective points

151

Let J be some almost complex structure on R2n , and u : D → R2n a Jholomorphic disc. Definition V.4.1. A point z ∈ D is called an injective point of u if
Tz u 6= 0 and u−1 u(z) = {z}. The set of injective points of u is denoted by Inj(u). Below we shall frequently appeal to the trivial observation that

u−1 u(z) ) {z} for every z ∈ D \ Crit(u) ∪ Inj(u) . Remark V.4.2. From Section V.3.1 it follows that for u ∈ M there is a neighbourhood of ∂D consisting exclusively of injective points.

V.4.1

Inj(u) is open and dense

Here is a basic property of Inj(u). Lemma V.4.3. The set Inj(u) ⊂ D is open. Proof. We want to show that D\Inj(u) is closed. Let (zν ) be a sequence in D\Inj(u) converging to z0 ∈ D as ν → ∞. If Tzν u = 0 for infinitely many zν , then Tz0 u = 0 by continuity, and hence z0 ∈ D \ Inj(u).
Otherwise, we may assume that u−1 u(zν ) ) {zν } for all ν ∈ N, that is, there are points wν 6= zν such that u(wν ) = u(zν ), and by passing to a subsequence we may assume that wν → w0 ∈ D. Then u(w0 ) = lim u(wν ) = lim u(zν ) = u(z0 ). ν→∞

ν→∞

If w0 6= z0 , then z0 ∈ D \ Inj(u). If w0 = z0 , then Tz0 u = 0 (and again z0 ∈ D \ Inj(u)). Indeed, the J-holomorphicity of u means that if Tz0 u 6= 0, then it has rank 2, and u is injective in a neighbourhood of z0 , precluding the existence of the two sequences (zν ), (wν ) with the described properties.  Here is the main result of this section. Proposition V.4.4. For every u ∈ M, the set Inj(u) is dense in D. Proof. Arguing by contradiction, we assume that, for some u ∈ M, the complement D\Inj(u) has an interior point z∗ , which by Remark V.4.2 must lie
in Int(D). Thanks to Corollary V.3.5, we may assume that z∗ 6∈ u−1 u(Crit(u)) . Choose a point z∂ ∈ ∂D such
that the whole straight line segment [z∗ , z∂ ] ⊂ D is disjoint from u−1 u(Crit(u)) ; this, too, is possible by Corollary V.3.5. Let P ⊂ [z∗ , z∂ ] be the set of points that have a neighbourhood in [z∗ , z∂ ] consisting entirely of non-injective points. By construction, this set P is an open subset of [z∗ , z∂ ], and it contains the point z∗ . We claim that it is also closed, and hence equals [z∗ , z∂ ]. This, by Remark V.4.2, is the desired contradiction.

152

Chapter V. Transversality

To prove the claim, we choose a point z0 ∈ [z∗ , z∂ ] in the closure of P , i.e. we have a sequence (zν ) in P converging to z0 . Choose points wν 6= zν in D with u(wν ) = u(zν ). By passing to a subsequence, we may assume wν → w0 ∈ D, with u(w0 ) = u(z0 ). Since z0 is not critical by our choice of the line segment [z∗ , z∂ ], the map u is injective in a neighbourhood of z0 , which implies w0 6= z0 . By Remark V.4.2, w0 lies in Int(D). The lemma in the next section says that in this situation there are disjoint open neighbourhoods U, V ⊂ Int(D) of z0 and w0 , respectively, such that u(U ) = u(V ). In particular, z0 has a neighbourhood of non-injective points in [z∗ , z∂ ]. This proves that z0 ∈ P . 

V.4.2

(Self-)Intersections of holomorphic discs

Observe that in the proof of Proposition V.4.4, the assumption
[z∗ , z∂ ] ∩ u−1 u(Crit(u)) = ∅ implies that Tw0 u 6= 0. This places us in the situation of the following lemma. Lemma V.4.5. Let u : D → R2n be a J-holomorphic curve (for some almost complex structure J on R2n ). Assume that there are two distinct points z0 , w0 ∈ Int(D) with u(z0 ) = u(w0 ) and Tz0 u 6= 0, Tw0 u 6= 0. Moreover, assume that there are sequences (zν ) and (wν ) in D converging to z0 and w0 , respectively, with u(zν ) = u(wν ) and wν 6= w0 for all ν ∈ N, so that w0 is an accumulation point of double points. Then there are disjoint open neighbourhoods U, V ⊂ Int(D) of z0 and w0 , respectively, and a biholomorphism ψ : V → U such that u|V = u ◦ ψ. In particular, z0 has a neighbourhood consisting entirely of non-injective points. The proof of this lemma relies on a local normal form for J-holomorphic curves near a noncritical point. We formulate this separately, because it is of independent interest. Lemma V.4.6. Let G ⊂ C be a domain and u : G → R2n = Cn a J-holomorphic curve. Let z0 ∈ G be a point with Tz0 u 6= 0. (We may then shrink G such that u is an embedding.) Then there is a diffeomorphism ϕ from a neighbourhood W of (z0 , 0) ∈ C ⊕ Cn−1 to a neighbourhood of u(z0 ) such that
W0 := W ∩ C ⊕ {0} ⊂ G, ϕ(z, 0) = u(z) for (z, 0) ∈ W , i.e. z ∈ W0 ,

V.4. Injective points

153

and J(u(z)) ◦ T(z,0) ϕ = T(z,0) ϕ ◦ J0 for all (z, 0) ∈ W ; see Figure V.1. The map v := ϕ−1 ◦ u :

W0 z

−→ 7−→

Cn (z, 0)

is Jϕ -holomorphic for the almost complex structure Jϕ obtained by conjugation, Jϕ (z, w) := Tϕ(z,w) ϕ−1 ◦ J(ϕ(z, w)) ◦ T(z,w) ϕ, (z, w) ∈ W ⊂ C ⊕ Cn−1 . This new almost complex structure Jϕ satisfies Jϕ (z, 0) = J0 . Cn−1 W ϕ ϕ(W0 ) W0 C u(G) ϕ(W ) ⊂ R2n

Figure V.1: A local normal form for J-holomorphic curves. Proof. Assuming without loss of generality, as indicated in the lemma, that u is an embedding, we choose vector fields X2 , . . . , Xn (regarded as functions of z ∈ G) along u(G) such that the vector fields ux , uy = J(u)ux , X2 , J(u)X2 , . . . , Xn , J(u)Xn form a pointwise basis of the tangent spaces of R2n along u(G). Define, with w = (s2 + it2 , . . . , sn + itn ), ϕ(z, w) := u(z) +

n X j=2


sj Xj (z) + tj J(u(z))Xj (z) .

Chapter V. Transversality

154 Then

∂x ϕ(z, 0) = ux (z), ∂y ϕ(z, 0) = uy (z), and ∂sj ϕ(z, 0) = Xj (z),

∂tj ϕ(z, 0) = J(u(z))Xj (z),

j = 2, . . . , n.

This means that the Jacobian of ϕ is of full rank at points (z, 0) ∈ G × Cn−1 . It follows that ϕ is a diffeomorphism in a neighbourhood of (z0 , 0). The equation ϕ(z, 0) = u(z) is clear by construction. Exercise V.4.7. Verify the equation J(u(z)) ◦ T(z,0) ϕ = T(z,0) ϕ ◦ J0 .

♦

From this equation and the definition of Jϕ it is immediate that Jϕ (z, 0) = J0 . Moreover, Jϕ satisfies Jϕ ◦ T ϕ−1 = T ϕ−1 ◦ J, so the J-holomorphicity of u  (T u ◦ i = J ◦ T u) entails that v = ϕ−1 ◦ u is Jϕ -holomorphic. Proof of Lemma V.4.5. By Lemma V.4.6 we may assume (after replacing u with ϕ−1 ◦ u near z0 , and J with Jϕ ) that on a small open disc Bε (z0 ) ⊂ Int(D) the J-holomorphic curve u = (u1 , u) : Bε (z0 ) −→ C ⊕ Cn−1 is of the form u(z) = (z, 0), and that J satisfies J|C⊕{0} = J0 . Since u(z0 ) = u(w0 ), we may likewise replace u near w0 with ϕ−1 ◦ u. Again we write u = (u1 , u) (for the new u near w0 ). Then, by assumption, we have u1 (wν ) = zν and u(wν ) = 0 for all ν ∈ N0 . Furthermore, the condition on J allows us to write, near w0 , J(u) − J0

= = =

J(u1 , u) − J(u1 , 0) Z 1 d J(u1 , tu) dt 0 dt Z 1 D2 J(u1 , tu)(u) dt. 0

Here, D2 denotes the derivative with respect to the 2(n − 1) real variables of the Cn−1 -factor; cf. Section IV.1.4. This means that D2 J(u1 , tu)(u) is a function on D taking values in the linear space EndR (Cn ). With the function B :=

1

Z

 D2 J(u1 , tu)( . ) dt uy

0

on D, which takes values in the homomorphisms from R2n−2 to R2n , the Jholomorphicity of u can be written as 0 = ux + J(u)uy = ux + J0 uy + Bu.

(V.10)

V.5. The Floer space of almost complex structures

155

Write π2 for the projection from C ⊕ Cn−1 onto the Cn−1 -summand. By applying π2 to this last equation, we obtain ∂u + π2 Bu = 0, which places us in the setting of the Carleman similarity principle. The component u has an accumulation point of zeros in w0 , and as in the proof of Corollary V.3.4, one argues that u must be identically zero on a neighbourhood V of w0 . From (V.10) we then see that ∂u1 = 0 on V . The conditions u1 (w0 ) = z0 and Tw0 u 6= 0 guarantee that, possibly after making V a little smaller, ψ := u1 |V is a biholomorphism from V onto U := ψ(V ) ⊂ Bε (z0 ). Since u(z) = (z, 0) for z ∈ Bε (z0 ), we then have u|V = (u1 , 0)|V = u ◦ u1 |V = u ◦ ψ. This proves Lemma V.4.5.



Exercise V.4.8. (a) What can be concluded when the condition Tw0 u 6= 0 in Lemma V.4.5 is dropped? (b) Convince yourself that the lemma can be formulated for two J-holomorphic curves u, v with u(z0 ) = v(w0 ), and the other conditions adapted accordingly. This statement then says that intersection points of two locally distinct Jholomorphic curves (i.e. curves that are not reparametrisations of each other) can only accumulate at a point that is critical for both curves. ♦

V.5

The Floer space of almost complex structures

In this section we find the separable (and yet sufficiently large) space of almost complex structures adapted to the nonsqueezing theorem, as promised in Section V.2.4.

V.5.1 The Floer norm Let f : Rn → R be a smooth function. The differential Df is a map from Rn into the space L(Rn , R) of linear maps Rn → R. Set L0 := R and Lj := L(Rn , Lj−1 ) = L(⊗j Rn , R) for j ∈ N. The higher differentials Dj f := D(Dj−1 f ) are continuous maps from Rn into Lj . Set kDj f kC 0 := sup kDj f (x)kop , x∈Rn

where the operator norm on Lj is induced by the Euclidean norm on Rn .

Chapter V. Transversality

156

Definition V.5.1. Given a sequence κ = (κj )j∈N0 of positive real numbers, the Floer norm on smooth functions f : Rn → R is kf kκ :=

∞ X

κj kDj f kC 0 .

j=0

Notation V.5.2. Let Ω ⊂ Rn be a bounded domain. For any vector space V and any k ∈ N0 ∪ {∞} we write  C0k (Ω, V ) := f ∈ C k (Rn , V ) : f ≡ 0 on Rn \ Ω . We introduce the function space  Cκ∞ (Ω) := f ∈ C0∞ (Ω, R) : kf kκ < ∞ . Exercise V.5.3.

(a) Show that there are continuous embeddings Cκ∞ (Ω) −→ C k (Ω), k ∈ N0 ,

and Cκ∞ (Ω) −→ C ∞ (Ω)

with respect to the Floer norm on Cκ∞ (Ω), the C k -norm on C k (Ω), and the Fr´echet metric on C ∞ (Ω). Find the optimal constant c(κ, k), k ∈ N0 , in the inequality kf kC k ≤ c(κ, k) kf kκ for all f ∈ Cκ∞ (Ω) you need for this purpose. (b) Similarly, show that for the truncated Floer norm kf kκ,m :=

X

κj kDj f kC 0 ,

j≤m

defined for any m ∈ N0 , we have the inequality kf kκ,m ≤ const. kf kC m for all f ∈ Cκ∞ (Ω). Note V.5.4. In the infinite product

Q∞

j=0

♦

C00 (Ω, Lj ) we consider the linear subspace

∞ o n X κj kfj kC 0 < ∞ L := (fj )j≥0 : j=0

with the obvious norm k . kκ suggested by this definition. The prescription f 7→ (Dj f )j≥0 then defines an isometric embedding Cκ∞ (Ω) → L.

V.5. The Floer space of almost complex structures

V.5.2

157

A separable Banach space

Here is the main result of this section, concerning the properties of the Floer norm. Proposition V.5.5. Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω.
(a) The normed vector space Cκ∞ (Ω), k . kκ is a separable Banach space. (b) There is a sequence κ such that Cκ∞ (Ω) lies dense in C0∞ (Ω, R) and in W0k,p (Ω) for all k ∈ N.
Proof. (a1) We first show that Cκ∞ (Ω), k . kκ is complete. Let (fν ) be a Cauchy sequence in Cκ∞ (Ω). Given ε > 0, there is an n0 (ε) ∈ N such that kfν − fµ kκ
0 for all nowhere vanishing vector fields v is satisfied for s in a small open interval containing 0. Thus, we have found a curve (−ε, ε) 3 s 7→ Js ∈ J0 with velocity vector Y at J for s = 0, which demonstrates that Y ∈ TJ J0 .  Exercise V.6.4. (a) Show that condition (Y2) in Proposition V.6.2 is equivalent to saying that Y is self-adjoint with respect to the Riemannian metric h . , . iJ . In particular, with Remark V.5.9 one sees that there is a canonical identification of TJb J0 with T0 W0 ; cf. Notation V.5.11. (b) Show that the diffeomorphism fW : J0 → W0 can equivalently be written as J 7→ (Jb + J)−1 (Jb − J). (c) Let s 7→ Js , s ∈ (−ε, ε), be a smooth path in J0 with Js |s=0 = Jb . Compute the derivative at s = 0 of the identity (J0 + Js )−1 (J0 + Js ) = id. (d) Show that the differential of fW at Jb is given by TJb fW (Y ) = 12 Jb Y , and observe, in agreement with (a), that Y 7→ Jb Y is an automorphism of W0 (and an isomorphism of its tangent space at any W ∈ W0 ). ♦ Implicit in part (d) of the exercise is the observation that the defining equations for the space W0 are linear but for the norm condition |W |Jb < 1, so the tangent space to W0 at any of its points is described precisely by those linear conditions. When we pass to the subspace Wκ ⊂ W0 (cf. Notation V.5.12), the tangent spaces to Wκ also inherit the finiteness condition kW kκ < ∞.

V.6. The universal moduli space

163

The tangent space to Jκ at any J ∈ Jk may then be described as −1 TJ Jκ = TW (fW )(TW Wκ ),

where W = fW (J) ∈ Wκ .

V.6.2 The linearisation of ∂ • Let s 7→ (us , Js ), s ∈ (−ε, ε), be a smooth path in B × Jκ . As in Section IV.1.4, where we computed the linearisation Du of ∂ J , we use a chart of B given by the exponential map and write the velocity vector to the curve s 7→ us in B as d |s=0 us = (ξ, t) ∈ Tu0 Bt0 ⊕ Rn−1 = Tu0 B ⊂ W 1,p (D, R2n ) ⊕ Rn−1 . ds d As in the preceding section we write ds |s=0 Js = Y . Then, the linearisation D(u,J) of ∂ • at (u, J) = (us , Js )|s=0 becomes

D(u,J) (ξ, t; Y ) =


d |s=0 ∂x us + Js (us )∂y us = Du (ξ, t) + Y (u)∂y u. ds

Here Du is the differential of ∂ J at u, but we suppress the dependence on J from the notation. Proposition V.6.5. The linearisation D(u,J) : Tu B ⊕ TJ Jκ −→ Lp (D, R2n ) at any point (u, J) ∈ U is surjective. Here is an exercise in linear algebra we shall use in the proof of this proposition. Exercise V.6.6. Write Y ∈ R2n×2n for an element of TJ J0 evaluated at some point p ∈ R2n (in other words, Y is shorthand for Y (p)). By Remark V.5.9 and Exercise V.6.4, such a matrix Y is characterised by the conditions Y = JY J and Y t = BY B −1 , where B = −J0 J (all at the point p) is the positive definite symmetric matrix that represents the scalar product h . , . iJ ; cf. Exercise I.7.5. Let ξ, η ∈ R2n \ {0} be given. Our aim is to find a Y such that Y ξ = η. (a) We first consider the case that J = J0 , where B is the unit matrix. Set  1  t ηξ + ξη t + J0 (ηξ t + ξη t )J0 Y := 2 |ξ|  1  − 4 hη, ξi(ξξ t + J0 ξξ t J0 ) + hη, J0 ξi(J0 ξξ t − ξξ t J0 ) . |ξ|

Chapter V. Transversality

164 Verify that Y = J0 Y J0 = Y t and Y ξ = η.

Observe how Y is constructed. The first of eight summands is a matrix that sends ξ to η. The second summand makes the matrix symmetric. Together with the third and fourth summands, the resulting matrix Y4 is still symmetric and satisfies Y4 = J0 Y4 J0 , but it no longer sends ξ to η. The remaining four summands also satisfy the two conditions required by Y , and they give the necessary correction so that Y ξ = η. (b) Generalise this strategy to find Y in the case of an arbitrary J. To do this, start with 1 ηξ t B, |ξ|2J and then add correcting terms as in (a). To find these terms, first verify the identity J t = −BJB −1 , and observe that any matrix of the form Z = Z0 + B −1 Z0t B satisfies Z t = BZB −1 . Deduce that JZJ also satisfies this condition. Also use that hξ, JξiJ = 0. ♦ Proof of Proposition V.6.5. Let (u, J) ∈ U be given, i.e. J ∈ Jκ and u ∈ MJ . By Theorem V.1.5, the differential Du : Tu B → Lp (D, R2n ) is a Fredholm operator, so it has a finite-dimensional cokernel. Since Du is a direct summand of D(u,J) , the latter has a finite-dimensional cokernel, too. It follows with Lemma V.1.2 that the image of D(u,J) is closed. We now distinguish the two (not mutually exclusive) cases that u is a flat disc (Notation I.8.23) or a nonstandard disc (Definition II.1.8). In the first case we have Du = ∂ (see Remark IV.1.18), and according to Proposition V.1.28, this linear Cauchy–Riemann operator is surjective. A fortiori, D(u,J) is surjective. In the second case, the image u(D) intersects the domain Ω = Φ(Br+ε \ Dr ) where the almost complex structure is allowed to vary. We shall use in an essential way that the space Jκ of permissible almost complex structures is sufficiently ‘large’, in the sense that the infinitesimal variations of J in Jκ ensure the surjectivity of D(u,J) . By Note V.1.15, the space im D(u,J) , being of finite codimension, has a complementary closed subspace. Our aim is to show that this complement is trivial. If the complement of im D(u,J) ⊂ Lp (D, R2n ) were nontrivial, we could define a nontrivial bounded linear functional ϕ : Lp (D, R2n ) → R with ϕ|im D(u,J) = 0. Therefore, it suffices to show that any element η in the dual space7 Lq (D, R2n ), with q the dual coefficient of p, that is, p1 + 1q = 1, is trivial whenever Z h . , ηi = 0 on im D(u,J) .

(V.12)

D 7 For a proof that Lq is the dual space of Lp , see [Hirzebruch & Scharlau 1971, Satz 19.1] or [Werner 2011, Satz II.2.4]. The dual space of the space of continuous maps, by contrast, is the space of regular Borel measures — yet another reason for working in the realm of Sobolev spaces.

V.6. The universal moduli space

165

With J 0 as defined in (V.1), the differential D(u,J) , applied to a tangent vector (ξ, t; Y ) = (ξ, 0; 0) takes the form D(u,J) (ξ, 0; 0) = ∂x ξ + J(u)∂y ξ + J 0 ξ. Thus, if (V.12) is satisfied we have, in particular, Z

 ∂x ξ + J(u)∂y ξ + J 0 ξ, η = 0 for all ξ ∈ Cc∞ (B1 , R2n ). D

If η were smooth, with the identity J∂y ξ = ∂y (Jξ) − (∂y J)ξ and with integration by parts we would obtain Z


 ξ, ∂x η + J(u)t ∂y η + (∂y J)t − (J 0 )t η = 0. D

This means that η ∈ Lq (D, R2n ) is a weak solution of the differential equation
∂x η + J(u)t ∂y η + (∂y J)t − (J 0 )t η = 0. In Lemma V.6.11 we shall prove that η is then automatically of class W 1,q . Arguing as in Section V.3.2, we can transform η into a solution of a linear equation of the form (V.8), to which one can apply the Carleman similarity principle (in the version of Exercise V.3.7). As in the proof of Corollary V.3.4 one then argues that if η ≡ 0 on the open and non-empty set u−1 (Ω) ⊂ D, then η ≡ 0 on D. Thus, to conclude the proof, we want to show that η|u−1 (Ω) ≡ 0. Arguing by contradiction, we assume that η does not vanish identically on u−1 (Ω). According to Lemma V.4.3 and Proposition V.4.4, the subset Inj(u) ⊂ D of injective points is open and dense in D. This allows us to choose a point z0 ∈ D and an ε > 0 such that Bε (z0 ) ⊂ Inj(u) ∩ u−1 (Ω) and η(z0 ) 6= 0. The fact that z0 ∈ Inj(u) (and hence Tz0 u 6= 0) implies ∂y u(z0 ) 6= 0; cf. Exercise I.3.9. By Exercise V.6.6 we find a tangent vector Y ∈ TJ J0 such that
Y u(z0 ) ∂y u(z0 ) = η(z0 ). Exercise V.6.7. Fill in the minor missing detail in the preceding claim. Conditions (Y1) and (Y2) in Proposition V.6.2 are pointwise linear. This allows one, with J denoting the space of all ω-compatible almost complex structures, to regard TJ J as a vector sub-bundle of the trivial bundle R2n × R2n×2n → R2n . The desired Y ∈ TJ J0 can then be found as a section of this vector bundle with a prescribed ♦ value at u(z0 ), and with support in Ω. Notice that because of Bε (z0 ) ⊂ Inj(u), the support of Y may be chosen so small that it only contains points of u(D) coming from a neighbourhood of z0 , that is,
u−1 supp Y ∩ u(D) ⊂ Bε (z0 ).

Chapter V. Transversality

166

In this way we can ensure that the function hY (u)∂y u, ηi on D is non-negative, and positive near z0 . This yields Z 

Y (u)∂y u, η > 0. D

Since Jκ is dense in J0 , this last positivity condition can also be achieved with a Y picked in the subspace TJ Jκ ⊂ TJ J0 . By (V.12), however, we then have Z Z 



Y (u)∂y u, η = D(u,J) (0, 0; Y ), η = 0, D

D

which contradicts the positivity condition.



Remark V.6.8. Rather than abstractly appealing to the density of Cκ∞ (Ω) in C0∞ (Ω, R), one may observe a little more directly that with the κj defined as in (V.11), one can directly use the bump function ρ to construct a function in Cκ∞ (Ω) with a prescribed value v0 ∈ R at a point x0 ∈ Ω. Indeed, the function ρ constructed in Exercise V.5.7 may be assumed to satisfy ρ(0) = 1, and then the function x 7→ v0 εn0 ρε0 (x − x0 ), with ε0 > 0 sufficiently small such that the support lies in Ω, does the job. Exercise V.6.9. Verify the claims made in this last remark.

V.6.3

♦

The Weyl lemma

The Weyl lemma is a fundamental regularity result in the theory of partial differential equations. Lemma V.6.10 (Weyl Lemma). Let Ω ⊂ Rn be a domain. Let u ∈ L1loc (Ω) be a weak solution of the Laplace equation ∆u = 0, that is, Z u ∆ϕ dλn = 0 for all ϕ ∈ Cc∞ (Ω). Ω

Then, u is harmonic, i.e. an honest real analytic solution of ∆u = 0. Idea of the proof. A nice reference for this result is [Beltrami 1968], where the proof is only marginally longer than the sketch we present here. We have seen in Note I.4.14 that harmonic functions satisfy the mean value 1 (Ω) is harmonic if and property. In fact, it can be shown that a function in Lloc only if it has the mean value property. Given a weak solution u of the Laplace equation, consider uε := ρε ∗ u, where ρε is as in Exercise V.5.7, with ρ chosen to be rotationally symmetric. This convolution uε is harmonic on {x ∈ Ω : Bε (x) ⊂ Ω}, and hence satisfies the mean 1 -convergence uε → u as ε & 0 implies that u likewise value property. The Lloc satisfies the mean value property, and hence is harmonic. 

V.6. The universal moduli space

167

Here is the lemma needed to complete the proof of Proposition V.6.5. Lemma V.6.11. Let η ∈ Lp (D, R2n ) be a weak solution of an equation of the form ∂x η + J∂y η + Aη = 0, where J = J(z), and A = A(z) is a smooth matrix-valued function. Then η is a weak solution of class W 1,p . Proof. We start from the special case of the equation ∂η = 0, to which we can apply the Weyl lemma, and then proceed step by step to the general equation in the lemma. (i) Let η ∈ Lp (D, Cn ) be a weak solution of the equation ∂η = 0, that is, Z hη, ∂ϕi = 0 for all ϕ ∈ Cc∞ (B1 , Cn ). D

Here h . , . i denotes the sesquilinear Hermitian inner product on Cn . Taking ϕ to be of the form ϕ = ∂φ, we see that η is also a weak solution of the Laplace equation, and hence smooth by the Weyl lemma. (ii) Let f ∈ Lp (D, Cn ), and let η be a weak solution of the inhomogeneous equation ∂η = f , that is, Z Z hη, ∂ϕi = − hf, ϕi for all ϕ ∈ Cc∞ (B1 , R2n ). D

D

Proposition V.1.28 allows us to choose a solution ζ ∈ W 1,p (D, R2n ) of this inhomogeneous equation. Then η − ζ is smooth by (i), and hence η of class W 1,p . (iii) If η is a weak solution of the equation ∂η + Aη = 0, we may interpret −Aη ∈ Lp (D, R2n ) as an inhomogeneity and appeal to (ii). (iv) Finally, if η is a weak solution of the general equation in the lemma, we transform the equation to one of type (iii) by conjugating J to J0 as in part (b) of Exercise V.3.7.  For the proof of Proposition V.6.5, instead of appealing to the preceding lemma and the W 1,p -version of the Carleman similarity principle in part (a) of Exercise V.3.7, one may improve on the preceding lemma and show that η is actually an honest smooth solution. This can be achieved by reading the equation on η in the distributional sense, and then reasoning as in the bootstrapping argument on page 118 to show that the partial 1-jet z 7→ (z, η(z), ηx (z)) satisfies an equation of the same kind as η.

V.6.4 The universal moduli space is a Banach manifold Now that the surjectivity of D(u,J) at all points (u, J) ∈ U has been established, the next lemma shows that 0 is a regular value of ∂ • : B × Jκ → Lp (D, R2n ). This concludes the proof that the universal moduli space U is a Banach manifold.

Chapter V. Transversality

168

Lemma V.6.12. For every (u, J) ∈ U the differential D(u,J) has a right-inverse R in the sense of Lemma V.2.4. Proof. The differential Du is a Fredholm operator by Theorem V.1.5. As in Note V.1.15 we write Tu B = ker Du ⊕ V and Lp (D, R2n ) = im Du ⊕ W with dim ker Du , dim W < ∞. Choose a basis f1 , . . . , fm for W, and e1 , . . . , em ∈ T(u,J) (B × Jκ ) with D(u,J) (ei ) = fi , which the surjectivity of D(u,J) at (u, J) ∈ U allows us to do. Define R:

im Du ⊕ W Pm (f , i=1 λi fi )

−→ 7−→

T(u,J) (B × Jκ ) = ker Du ⊕ V ⊕ TJ Jκ
−1 Pm Du |V f + i=1 λi ei .

The image of R is the sum (not, in general, direct) of V and the finite-dimensional space spanned by e1 , . . . , em , and hence closed. One easily verifies that D(u,J) ◦ R is the identity on Lp (D, R2n ). 

V.7 The moduli space M and the evaluation map Continuing with the strategy outlined in Section V.2.4, we now prove that for a suitable choice of J ∈ Jκ , the moduli space M = MJ is a manifold, and we determine the degree of the evaluation map ev : M × D → Z.

V.7.1 The moduli space MJ is a manifold Consider the projection pJ : B × Jκ ⊃ U −→ Jκ . Notice that the projection onto the second factor makes of course sense on all of B × Jκ , but we regard pJ as being defined exclusively on U . Correspondingly, the differential T(u,J) pJ at (u, J) ∈ U has T(u,J) U = ker D(u,J) as its domain of definition. Lemma V.7.1. For all (u, J) ∈ U , the differential T(u,J) pJ : ker D(u,J) = T(u,J) U −→ TJ Jκ satisfies ker(T(u,J) pJ ) ∼ = ker Du and

coker(T(u,J) pJ ) ∼ = coker Du .

By Theorem V.1.5, this lemma shows that the projection pJ : U → Jκ is a Fredholm map of index 2n − 2. Thus, for any regular value J of pJ the (nonempty!) fibre p−1 J (J) = MJ ⊂ U is a manifold of dimension 2n − 2. Such regular values exist in abundance according to the Sard–Smale theorem (Theorem V.2.7).

V.7. The moduli space M and the evaluation map

169

Proof of Lemma V.7.1. We write the differential D(u,J) : Tu B ⊕ TJ Jκ −→ Lp (D, R2n ) as D(u,J) (ξ, t; Y ) = Du (ξ, t) + L(Y ), with L : TJ Jκ −→ Lp (D, R2n ) given by L(Y ) = Y (u)∂y u. We suppress the dependence of L on u from the notation. From T pJ (ξ, t; Y ) = Y we have  ker(T pJ ) = (ξ, t; Y ) ∈ ker D(u,J) : Y = 0 ∼ ker Du . = This is the first statement of the lemma. For the second, we observe that  Y ∈ TJ Jκ : ∃(ξ, t) ∈ Tu B s.t. LY = −Du (ξ, t) T pJ (ker D(u,J) ) = =

L−1 (im Du ).

Hence coker(T pJ ) = TJ Jκ /T pJ (ker D(u,J) ) = TJ Jκ /L−1 (im Du ). It therefore remains to show that the well-defined linear map TJ Jκ /L−1 (im Du ) [Y ]

−→ 7−→

coker Du [L(Y )]

is an isomorphism. Clearly, this map is injective, for [L(Y )] = 0 in coker Du means that L(Y ) ∈ im Du , and hence Y ∈ L−1 (im Du ), i.e. [Y ] = 0. For the surjectivity, consider a class [f ] ∈ coker Du . Write R(f ) = (ξ, t; Y ), where R is the right-inverse of D(u,J) found in Lemma V.6.12. Then the equality D(u,J) ◦ R(f ) = f translates into Du (ξ, t) + L(Y ) = f , which means that [L(Y )] =  [f ] in coker Du .

V.7.2 The mod 2 degree of a smooth map In order to establish the surjectivity of the evaluation map ev :

M×D (u, z)

−→ 7−→

Z u(z),

we need the concept of mapping degree. We want to show that one can define a mod 2 mapping degree of a smooth map f : M → N between equidimensional manifolds without boundary simply by counting the preimages of any regular value, provided that M is compact and N is connected.

170

Chapter V. Transversality

Lemma V.7.2. Let M, N be two equidimensional manifolds with M compact, and let f : M → N be a smooth map. For y ∈ N a regular value of f , the set f −1 (y) is finite, and the function  y 7−→ #f −1 (y) := number of points in f −1 (y) ∈ N0 is locally constant on the (open) set of regular values in N . Proof. The preimage f −1 (y) is a closed subset of the compact space M , and hence compact. In the neighbourhood of any x ∈ f −1 (y), the map f is injective by the inverse function theorem. It follows that the set f −1 (y) is finite. Write f −1 (y) = {x1 , . . . , xk }. Choose pairwise disjoint open neighbourhoods U1 , . . . , Uk of x1 , . . . , xk , respectively, with f |Ui mapping Ui diffeomorphically onto an open subset Vi ⊂ N . Set
V := (V1 ∩ . . . ∩ Vk ) \ f M \ (U1 ∪ . . . ∪ Uk ) . The set M \ (U1 ∪ . . . ∪ Uk ) is closed in M and hence compact. Its image under f is compact, and therefore closed in the Hausdorff space N . It follows that V is an open subset of y in N , consisting entirely of regular values of f , and with #f −1 (y 0 ) = k for all y 0 ∈ V . This proves the lemma.

 −1

We want to show that #f (y) mod 2 does not depend on the choice of regular value. For this we have to make a slight detour via homotopies of maps. Definition V.7.3. Two smooth maps f, g : M → N are (smoothly) homotopic if there is a smooth map F : M × [0, 1] −→ N with F (x, 0) = f (x) and F (x, 1) = g(x) for all x ∈ M . Proposition V.7.4. Let f, g : M → N be homotopic maps, with M compact. Let y ∈ N be a regular value of both f and g. Then, #f −1 (y) ≡ #g −1 (y) mod 2. Proof. Let F be a homotopy between f and g. If y is a regular value of F , then F −1 (y) is a compact 1-dimensional submanifold of M × [0, 1] with boundary

∂ F −1 (y) = F −1 (y) ∩ M × {0} ∪ M × {1} = f −1 (y) × {0} ∪ g −1 (y) × {1}. Then the result follows from the well-known fact8 that the only compact 1dimensional manifolds are compact intervals (with two endpoints) or circles (with none); see Figure V.2. 8 As

noted earlier, a proof of this fact can be found in [Milnor 1965].

V.7. The moduli space M and the evaluation map

171

F −1 (y)

M × {0}

M × {1}

Figure V.2: The submanifold F −1 (y) ⊂ M × [0, 1]. If y is not a regular value of F , choose an open neighbourhood V of y in N consisting entirely of regular values for both f and g, and with #f −1 (y 0 ) = #f −1 (y), #g −1 (y 0 ) = #g −1 (y) for all y 0 ∈ V ; this is possible by the proof of Lemma V.7.2. Then choose a regular value y 0 ∈ V of F (using Sard’s theorem) and proceed as in the first case.  Exercise V.7.5. Given two points y0 , y1 in the same connected component of a manifold N , show that one can find a diffeomorphism h of N that is isotopic (i.e. homotopic via diffeomorphisms of N ) to the identity and satisfies h(y0 ) = y1 . Hint: Choose a smooth path γ : [0, 1] → N with γ(0) = y0 , γ(1) = y1 , and without self-intersections. Extend the vector field t 7→ γ(t) ˙ along γ to a compactly supported smooth vector field on N , using local charts and bump functions. Then, the time-1 map of the flow of this extended vector field is the desired diffeomorphism. ♦ Proposition V.7.6. Let f : M → N be a smooth map between equidimensional manifolds, with M compact and N connected. Let y0 , y1 be regular values of f . Then, #f −1 (y0 ) ≡ #f −1 (y1 ) mod 2. Definition V.7.7. The well-defined class of #f −1 (y) in Z/2Z (for any regular value y of f ) is called the mapping degree mod 2 of f , and denoted by deg2 (f ). Proof of Proposition V.7.6. Let h be a diffeomorphism of N , isotopic to the identity, sending y0 to y1 . Then, y1 is a regular value of h ◦ f , and this composition is
homotopic to f via (x, t) 7→ H f (x), t , where H is a homotopy from h to idM . With Proposition V.7.4 we conclude that #f −1 (y0 ) = #(h ◦ f )−1 (y1 ) ≡ #f −1 (y1 ) mod 2.



Note V.7.8. If deg2 (f ) = 1, then f is surjective. This follows immediately from the fact that any nonvalue is a regular value.

Chapter V. Transversality

172

Exercise V.7.9. Show that the mapping degree deg2 (f ) depends only on the ♦ smooth homotopy class of f . Exercise V.7.10. Adapt the arguments of this section to the situation where M is a compact manifold with boundary, N a connected manifold with boundary, and f : (M, ∂M ) → (N, ∂N ) a smooth map. Observe that a point y ∈ ∂N with a preimage in M \ ∂M cannot be a regular value. The compactness assumption on M may be dropped if instead one requires ♦ f to be a proper map. Exercise V.7.11. Show that the image of a proper map between manifolds is closed. In fact, this remains true for proper maps between topological spaces, provided ♦ that the target space is locally compact and Hausdorff. Here is an exercise to show that for maps between noncompact manifolds, the properness assumption is essential for the definition of mapping degree. Exercise V.7.12. Describe a smooth map f : S 1 × R → S 1 × R such that for any k ∈ Z there is a regular value q ∈ S 1 × R of f with #f −1 (q) = k. Show that if one requires f to be an immersion, one can still realise all k ∈ N0 . ♦ Remark V.7.13. The mapping degree can also be defined for continuous maps. In the case of compact, connected m-dimensional manifolds M, N , the mapping degree mod 2 of a continuous map f : (M, ∂M ) → (N, ∂N ) is determined by the induced homomorphism f∗ : Hm (M, ∂M ; Z2 ) → Hm (N, ∂N ; Z2 ) being the trivial or the nontrivial homomorphism Z2 → Z2 . In the noncompact case, one can use compactly supported cohomology instead.

V.7.3

The degree of the evaluation map

Choose an almost complex structure J ∈ Jκ such that M := MJ is a manifold. According to Theorem II.2.7, the evaluation map ev :

M×D (u, z)

−→ 7−→

Z u(z)

is proper. As summarised in Section II.1.6, the image of ev is indeed contained 2n−2 in the closed cylinder Z, and outside a compact subset D2 × DK we only find points in the image of ev coming from a unique flat disc. We may assume that K has been chosen large enough such that, for some ε > 0, the region
2n−2 2n−2 D 2 × DK \ BK−ε is foliated completely by flat discs, and no nonstandard disc passes through this 2n−2 region. In particular,
for any u ∈ M we have either u(D) ⊂ D2 × DK or 2n−2 2 u(D) ∩ D × DK = ∅.

V.7. The moduli space M and the evaluation map

173

We now work with a ‘truncated’ moduli space  2n−2 M∨ := u ∈ M : u(D) ⊂ D2 × DK . Equivalently, we can describe M∨ as the projection of
2n−2 ⊂M×D ev−1 D2 × DK to M. By the properness of the evaluation map, M∨ is compact. We have
2n−2 , Int(M∨ ) × D = ev−1 D2 × BK which shows that Int(M∨ ) is an open subset of M, and hence a manifold of dimension 2n − 2. A neighbourhood of the boundary ∂M∨ in M∨ consists of flat 2n−2 2n−2 \ BK−ε discs ust , parametrised by s + it with s + it ∈ DK . This means that M∨ is a manifold with boundary. We now consider the evaluation map

2n−2 2n−2 ) . (V.13) , ∂(D2 × BK ev : M∨ × D, ∂(M∨ × D) −→ D2 × BK ocker & J¨ anich 1973] for this differential topoAfter smoothing corners (see [Br¨ logical technique), we may regard domain and target as smooth manifolds with boundary. In Lemma II.2.6 we have shown that the evaluation map is continuous. This suffices if one works with the homological definition of mapping degree. The replacement for Note V.7.8 is provided by the next exercise. Exercise V.7.14. Let M, N be compact, connected m-dimensional manifolds with boundary, and f : (M, ∂M ) → (N, ∂N ) a continuous map. Show that if there is a point q ∈ Int(N ) not in the image of f , then deg2 (f ) = 0. Hint: If y ∈ Int(N ) is a point not in the image of f , the induced homomorphism f∗ : Hm (M, ∂M ; Z2 ) → Hm (N, ∂N ; Z2 ) factorises through the homology group Hm (N \ {y}, ∂N ; Z2 ). Show this latter group to be zero using the relative Mayer– ♦ Vietoris sequence [Hatcher 2002, p. 152]. If we want to use the differential topological definition of mapping degree, we need to establish that ev is at least of class C 1 . Exercise V.7.15. Recall that a tangent vector to the Fr´echet manifold C ∞ (D, R2n ) at some point u is simply a map ξ ∈ C ∞ (D, R2n ), where ξ(z) ought to be regarded as an element of the tangent space Tu(z) R2n . Show that the Gateaux derivative of the evaluation map ev : C ∞ (D, R2n ) × D −→ R2n is given by T(u,z) ev(ξ, v) = ξ(z) + Tz u(v). Show further that ev is of class C 1 in the sense of Definition IV.1.1. By similar arguments one sees that ev is in fact smooth.

♦

Chapter V. Transversality

174

Remark V.7.16. Sard’s theorem remains true for maps f : M → N of finite degree of differentiability C k , provided that  k > max dim M − dim N, 0 . Thus, for maps between equidimensional manifolds, C 1 is sufficient. We now apply these considerations to the evaluation map ev in (V.13). From 2n−2 the behaviour of the holomorphic discs near D2 × ∂BK , we deduce that the mod 2 degree of ev equals 1. This concludes the proof of Theorem I.8.13. In the proof of Corollary I.8.14 we use the observation from Note V.7.8 or Exercise V.7.14. The proof of Gromov’s nonsqueezing theorem (Theorem I.3.1) is now complete.

Finis

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Index a priori estimate, 75, 107 and Fredholm property of the Cauchy–Riemann operator, 127 fails for the C k -norms, 78 semantics, 42 with boundary, 119 action-energy inequality, 24 follows from the classical isoperimetric inequality, 61 implies the classical isoperimetric inequality, 25 almost complex structure, 8 can only exist on even-dim. manifolds, 8 compatible – defines a Riemannian metric, 8 compatible – is orthogonal w.r.t. the associated Riemannian metric, 4, 9 compatible with a symplectic form, 8 even dimension is not sufficient for existence, 8 generic, 36, 144 area element, 19 area of a smooth surface, 27 Arzel`a–Ascoli theorem, 52 Atkinson characterisation of Fredholm operators, 133

automorphism group of D, 30 acts simply transitively on pairs (z0 , eiθ ), 34 acts simply transitively on triples of boundary points, 30 of B, 34 of the upper half-plane, 34 ˆ 29 automorphism of a subset of C, B, ambient space for the definition of the moduli space M, 107 is a smooth separable Banach manifold, 107 is the closure of C, 108 Baire space, 142 complete metric spaces are –s, 142 Banach algebra, 105 Lp -space is not a –, 80 Sobolev space W k,p (k ≥ 1) is a –, 105 Banach manifold, 94 coordinate changes, 98–99 manifold charts, 97–98 spaces of continuous maps are –s, 95–99 tangent space, 94, 97, 107 Banach space finite-dimensional subspace is closed, 132 subspace of finite (co-)dimension has a closed complement, 132

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Geiges, K. Zehmisch, A Course on Holomorphic Discs, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-36064-0

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182 bootstrapping for proving elliptic regularity, 117– 119 needs Sobolev norms, 123 from | . |1,p to | . |k,p , 57 from C 1 to C k logic of reasoning, 82 needs Sobolev norms, 78, 123 from the (1, p)-norm to the (k, p)norm, 82–87 invented by M¨ unchhausen and should be called pigtailing, 65 boundary lemma of E. Hopf, 47 bounded linear operator, see linear operator bubbling, 53–56 bubble disc at a boundary point, 55 bubble sphere at a boundary point, 54 at an interior point, 53 double bubble, 55 for solutions of an inhomogeneous Cauchy–Riemann equation, 56 Calder´on–Zygmund inequality, 73 Carleman similarity principle, 146 generalisations, 150 history, 144 Cartan’s formula, 5 Cauchy principal value, 69 Cauchy–Riemann equation(s), 10 are equivalent to ux + J0 uy = 0, 10 inhomogeneous, 68 bubbling, 56 solution operator, 69 Cauchy–Riemann operator linear – ∂, 66 commutes with derivatives, 75 with factor 1/2 dropped, 68

INDEX nonlinear – ∂ J , 90 extension to B, 108 Fredholm index, 127 Gateaux derivative, 108 is a Fredholm map, 127 is of class C 1 , 108 is of class C ∞ , 113 linearisation, 99–100 linearisation equals ∂ for flat discs, 100 linearisation is surjective for flat discs, 127 regularity, 113 relevance of the a priori estimate for the Fredholm property, 127 universal – ∂ • , 161 linearisation, 163 linearisation is surjective, 163 Cayley transformation based at Jb , 159 ˆ 29 of C, on matrices, 31 chain rule for weak derivatives, 105–106 C k -boundary, 47 C k -norm, 50 classification of 1-manifolds, 20, 170 ∞ Cloc -convergence, 54 compact linear operator, see linear operator compatibility J0 is compatible with the standard symplectic form on Cn , 9 of almost complex structure with a symplectic form, 8 of linear complex structure with the standard symplectic form, 28 complex anti-linear map, 43 matrix, 31 complex structure, see linear complex structure conjugate set, 42 contact geometry is the Dark Lady, 17

INDEX continuous linear operator, see linear operator convolution, 73 (D1)–(D3), key properties of the pseudoholomorphic disc in the proof of the nonsqueezing theorem, 10 degree, see mapping degree mod 2 differentiability different notions of continuous – in infinite dimensions, 93 Fr´echet –, 92 Gateaux –, 91 higher orders of – in Banach and Fr´echet spaces, 91 of class C 1 , 91 implies Fr´echet –, 92 weak, 101 rules of differentiation, 104–106 Dirichlet energy is invariant under conformal reparametrisations, 38 of a holomorphic curve, 19 of a J-holomorphic curve, 37 w.r.t. ω-compatible J equals the symplectic energy iff curve is J-holomorphic, 37 Dirichlet integral, see Dirichlet energy divergence as ‘source strength’, 3 domain, 13 with C k -boundary, 47 dual coefficient, 75 elliptic regularity, 113–123 semantics, 89–90 Weyl lemma, 166 embedding of a Sobolev space, 103 energy of a loop, 24 action-energy inequality, 24 equivalent norms, 56 (EV1)–(EV3), properties of the evaluation map, 49–50

183 evaluation map, 36 boundedness properties, 49 has deg2 = 1 and hence is surjective, 174 is continuous, 51 is proper, 52 follows from having uniform C k bounds on Mnst , 53 is smooth, 173 examples a priori estimate fails for the C k norms, 78–79 bubbling, 53–56 compact linear operator, 131 formula for (T − S)−1 for T an invertible operator and kSk small, 134 Fredholm property is essential for the Sard–Smale theorem, 143 function without a weak derivative, 102 implicit function theorem fails for Fr´echet spaces, 91–92 J0 is compatible with the standard symplectic form on Cn , 9 Lipschitz domain, 76 M¨ obius transformation, 54, 58 metrics h . , . iJ and h . , . i define equivalent norms, 56 minimum principle fails for subharmonic functions, 17 monotonicity lemma, 19 non-Hamiltonian symplectomorphism, 6 norm of a holomorphic function is subharmonic, 15 product estimate fails for Lp -norm, 80 weakly continuous Gateaux differentiability does not imply Fr´echet differentiability, 93

184

INDEX

exponential map differential of –, 96 on a finite-dimensional Riemannian manifold, 94, 96 on Ct0 , 95

geodesically complete Riemannian manifold, 94 Gromov’s nonsqueezing theorem, see nonsqueezing theorem

fibre derivative, 99 first-countable topological space, 51 flat disc, 38 any disc u ∈ M contained in the J0 -region is flat, 38–39 Floer norm, 156 properties, 157 truncated, 156 Floer space of almost complex structures, 160 Fr´echet differentiable, see differentiability, Fr´echet Fr´echet metric, 50–51 convergence w.r.t. – is uniform convergence of all derivatives, 50 Fr´echet space, 50 Fredholm map, 142 index, 142 is locally proper, 143 Fredholm operator, 126 Atkinson characterisation, 133 dual is likewise Fredholm, 134 Fredholm plus compact is Fredholm, 135 Fredholm property is essential for the Sard–Smale theorem, 143 index, 126 is constant on connected components, 134 semi-Fredholm property, 136 subset of Fredholm operators is open, 134

Hamilton equations, 2 Hamiltonian diffeomorphism, 6 non-Hamiltonian symplectomorphism, 6 Hamiltonian function, 2 Hamiltonian vector field definition via gradient of the Hamiltonian function, 3 definition with the help of a symplectic form, 4 harmonic map, 11 H¨ older inequality, 75 holomorphic curves are minimal surfaces, 1, 27 holomorphic maps are harmonic, 11 homotopic maps, 170 homotopy condition (M2), 35 Hopf boundary lemma, 47

Gateaux differentiable, see differentiability, Gateaux Gateaux, correct spelling, 91 ‘generic’, meaning of, 144

implicit function theorem fails for Fr´echet spaces, 91–92 for Banach spaces, 141 index of a Fredholm map, 142 of a Fredholm operator, 126 is constant on connected components, 134 of Du , 127 injective point, 151 Inj(u) ⊂ D is dense, 151 Inj(u) ⊂ D is open, 151 neighbourhood of ∂D consists of –s, 151 inversion in the unit circle, 43

INDEX isoperimetric inequality, 21 classical, 24 follows from the action-energy inequality, 25 implies the action-energy inequality, 61 J-holomorphic curve, 11 (self-)intersections, 152, 155 –s are minimal surfaces, 38 conformal reparametrisation of a – is J-holomorphic, 11 has a complex linear differential, 10–11 injective point, see injective point J0 -holomorphic curves are holomorphic, 11 local normal form, 152–153 nonconstant – has positive symplectic energy, 37 jet, 118 partial 1-jet, 118 Lagrangian boundary condition (M1), 35 Lagrangian submanifold, 35 Leibniz rule for weak derivatives, 104 level of a holomorphic disc u ∈ M, 35 linear complex structure, 28 compatible with the standard symplectic form, 28 linear operator bounded or continuous, 126 compact, 103, 131 composition with bounded operator is compact, 131 examples, 131 sum is compact, 131 subset of Fredholm operators is open, 134 subset of invertible bounded operators is open, 134

185 Liouville’s theorem on preservation of the symplectic form, 5 on preservation of volume, 3 Lipschitz domain, 76 local normal form for J-holomorphic curves, 152–153 locally proper map, 143 Lp -norm, 66 product estimate fails for –, 80 (M1)–(M3), defining conditions for M, 35 1-manifolds, classification, 20, 170 mapping degree mod 2, 171 deg2 = 1 implies surjectivity, 171, 173 depends only on the smooth homotopy class, 172 for continuous maps, 172 for manifolds with boundary, 172 for proper maps, 172 maximum principle strong – for subharmonic functions, 16 derived from the boundary lemma, 49 weak – for subharmonic functions, 16 used in the proof of the boundary lemma, 48 mean value inequality for subharmonic functions, 16 mean value property for harmonic functions, 17, 166 for holomorphic functions, 15 mean value theorem, Lp -version, 115 Meyers and Serrin, see Sobolev space W k,p (G) minimality property of holomorphic curves, 27 of J-holomorphic curves, 38

186 minimum principle fails for subharmonic functions, 17 for harmonic functions, 17 moduli space M definition, 35 homotopy condition crucial for transversality arguments, 35 in dimension 4, 36 is a closed subset of C ∞ (D, R2n ), 53 is a manifold of dimension 2n − 2 for generic J, 168 strategy for proving it to be a manifold, 144 three-point condition fixes parametrisation of u ∈ M, 35 topology, 113–114 u ∈ M has a neighbourhood of ∂D consisting of injective points, 151 u ∈ M has embedded boundary, 49 u ∈ M has symplectic energy π, 37 u ∈ M is an embedding near ∂D, 36, 145 u ∈ M maps Int(D) to Z, 49 M¨obius transformation, 29 automorphism group of the upper half-plane, 34 automorphism group of B, 34 biholomorphism between unit disc and upper half-plane, 54, 58 Cayley transformation, 29 properties, 30 monotonicity lemma for proper holomorphic maps, 14 Morera’s theorem, 43 nondegeneracy of a 2-form Ω, 4 forces manifold to be even-dimensional, 4 is equivalent to Ωn being a volume form, 4

INDEX nonsqueezing theorem, 7 idea of the proof, 9–13 physical interpretation, 7 nonstandard discs, 41, 44 bound on boundaries, 45 lie in a bounded region, 47 i.e. Mnst is uniformly C 0 -bounded, 52 uniform C k -bound, 82 norms C k -norm, 50 equivalent –, 56 Floer norm, 156 Lp -norm, 66 `p -norm, 66 operator norm, 51 Sobolev (k, p)-norm, 66 truncated Floer norm, 156 notation B, open unit disc, 33 Br2n , open ball, 7 Br (p), ball centred at p, 45 ˆ extended complex plane, C, Riemann sphere, 29 Crit, set of critical points, 146 D, closed unit disc, 10 Dr , closed ball, 9 Dα u, higher derivatives, 50 ∂x , unit tangent vector in the xdirection, 2 ∂, linear Cauchy–Riemann operator, 66, 68 ∂ J , nonlinear Cauchy–Riemann operator, 90 ∂ • , universal Cauchy–Riemann operator, 161 ∂h f , directional derivative, 91 ∂v F (q, v), partial Jacobian, 109 df, dz f , differential of a real-valued function f (at z), 9

INDEX notation (cont.) dλn , n-dimensional Lebesgue measure, 69 Du , linearisation of the nonlinear Cauchy–Riemann operator, 99 D(u,J) , linearisation of the universal Cauchy–Riemann operator, 163 EndC (Cn ), C-linear endomorphisms, 147 EndR (Cn ), R-linear endomorphisms, 129 exp, exponential map, 94 Exp, exponential map with base point, 96 function spaces C ∞ (D, R2n ), 10 Cc∞ (G), 65, 101 C ` (G), 103 C0k (Ω, V ), 156 Cκ∞ (Ω), 156 L1loc (G), 101 W k,p (G), 102 W k,p (G), 104 W0k,p (G), 106 GL, general linear group, 129 H, closed upper half-plane, 30, 42 ˆ closure of H in C, ˆ 54 H, Inj, set of injective points, 151 Jκ , Floer space, 160 Km×n , vector space of (m × n)-matrices over K, 28 N, natural numbers {1, 2, . . .}, 19 N0 , non-negative integers, 50 |∇u|2 , Dirichlet integrand for holomorphic curve, 18 |∇u|2J , Dirichlet integrand for J-holomorphic curve, 37 norms | . |J , 37 | . |k (seminorm), 92 | . |p , 66 | . |u , 18

187 k . kC k , 50 k . kLp (G) , 66 k . kκ,m , 156 k . kκ , 156 k . kop , 51 k . kk,p,G , 66 R+ , real numbers > 0, 61 R+ 0 , real numbers ≥ 0, 20 Riemannian metrics h . , . i, 4 h . , . iJ , 8, 9 h . , . iu , 18 scalar products h . , . i, 4 h . , . iS 1 , 61 h . , . iR2n , 18 spaces of maps B ⊂ C 0 , W 1,p -discs satisfying (M1)–(M3), 107 Bt ⊂ B, maps of level t, 107 C, smooth discs satisfying (M1)– (M3), 90 C k , C k -discs satisfying (M1)– (M3), 94 Ctk ⊂ C k , discs of level t, 94 Γ( . ), continuous sections of a bundle, 97 L(E, F), continuous linear maps E → F, 91 M, MJ , moduli space, 35 Mnst ⊂ M, subset of nonstandard discs, 44 ∗, convolution, 73 subscripts denote partial derivatives, 10 τh , shift operator, 115 T M , tangent bundle, 8 T Φ, Tp Φ, differential of Φ (at p), 9 Tp M , tangent space, 4 Tp∗ M , cotangent space, 4 u is unfortunate, 10 U , closure of a set U , 9 U , conjugate set (in the context of Schwarz reflection), 42

188 notation (cont.) uh , difference quotient, 115 ust , flat disc, 38 Z, open unit cylinder, 7 open mapping theorem, 127 operator norm, notation, 31 partial 1-jet, 118 Peter Paul inequality, 121 pigtailing, see bootstrapping Poincar´e inequality, 67 analogue for the Lp -norm, 75 primitive 1-form, 23 proper map, 13 between manifolds has closed image, 172 locally –, 143 pseudoholomorphic curve, see J-holomorphic curve (R1)–(R2), defining properties of a regular value, 141 regular parametrisation of a curve, 22 regular value in finite dimensions, 140 of a map between Banach manifolds, 141 regularity of solutions to the nonlinear Cauchy–Riemann equation, see elliptic regularity residual set, 142 countable intersection of –s is residual, 142 Riemann removable singularities theorem classical, 60–61 for J-holomorphic curves, 61 Riemann sphere, 29 Sard’s theorem, 20, 142

INDEX Sard–Smale theorem, 142 for maps of finite degree of differentiability, 174 Schwarz lemma, 33 Schwarz reflection in the real line, 42 in the unit circle, 43 M¨ obius transformation relates the two incarnations, 44 second-countable topological space, 142 Seeley’s extension result for smooth functions, 10 semi-Fredholm estimate, 136 semi-Fredholm property, 136 separable metric space, 103 separability is equivalent to countable base of topology, 143 subspace is likewise separable, 108 sequential compactness is the same as compactness in metric spaces, 51 sequential continuity is the same as continuity in first countable spaces, 51 shift operator, 115–117 smooth functions on sets with boundary, 10 smoothing corners, 173 Sobolev embedding theorem, 103 Sobolev estimates, 106–107 Sobolev inequality, 76, 103 Sobolev (k, p)-norm, 66 estimate on products, 79, 105 Sobolev space W k,p (G), 102 embeds into C k−1 (G), 103 equals H k,p (G) (Meyers and Serrin), 102 is a Banach algebra, 105 is a separable Banach space, 103 meaning of ‘embedding’ into C ` (G), 103 rules of differentiation, 104–106

INDEX Sobolev space W0k,p (G) characterisation, 106 definition, 106 spherical mean, 15 standard (almost) complex structure J0 on R2n , 2 metric on R2n equals h . , . iJ0 , 9 symplectic form ω on R2n , 4 Stone–Weierstraß theorem, 136, 157 subharmonic function, 14 mean value inequality, 16 norm of a holomorphic function is a –, 15 strong maximum principle, 16 weak maximum principle, 16 symplectic action of a loop, 23 action-energy inequality, 24 equals the enclosed area for simple closed planar curves, 25 symplectic embedding, 6 symplectic energy a.k.a. symplectic area, 19 is invariant under reparametrisations, 38 of u ∈ M equals π, 37 of a holomorphic curve equals its Dirichlet energy, 19 of a smooth map D → M , 11 of nonconstant J-holomorphic curve is positive, 37 symplectic form, 4 standard – on R2n , 4 symplectic manifold, 4 symplectic matrix, 29 symplectomorphism, 6 non-Hamiltonian –, 6 (T0)–(T1), defining properties of the tangent space Tu Ct0 , 94 tangent space of Banach manifold, 94, 97, 107 test functions, 101 three-point condition (M3), 35

189 ‘to rob Peter to pay Paul’, 120 totally geodesic submanifold, 94 transversality map transverse to a submanifold, 140 semantics, 140 strategy, 144 uncertainty principle nonsqueezing theorem as classical version of the –, 7 uniformly equicontinuous sequence of continuous maps, 52 universal moduli space U , 161 is a Banach manifold, 167 (V1)–(V3), properties of the purported limit in the bubbling-off argument, 57–58 weak derivative, 101 coincides with actual derivative for C 1 -functions, 101 function without a –, 102 rules of differentiation, 104–106 Weyl lemma, 166 Wirtinger derivatives, 77 (Y0)–(Y2), properties of elements in TJ J0 , 161