Embedding Problems in Symplectic Geometry

de Gruyter Expositions in Mathematics 40 Editors O. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Acade...

Author: Felix Schlenk

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de Gruyter Expositions in Mathematics 40

Editors O. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R. O. Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Semigroups, K. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) 2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues 3 The Stefan Problem, A. M. Meirmanov 4 Finite Soluble Groups, K. Doerk, T. O. Hawkes 5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin 6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin 7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, M. V. Zaicev 8 Nilpotent Groups and their Automorphisms, E. I. Khukhro 9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug 10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini 11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao 12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub 13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov, B. A. Plamenevsky 14 Subgroup Lattices of Groups, R. Schmidt 15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H. Tiep 16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig 19 Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov 20 Semigroups in Algebra, Geometry and Analysis, K. H. Hofmann, J. D. Lawson, E. B. Vinberg (Eds.) 21 Compact Projective Planes, H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel 22 An Introduction to Lorentz Surfaces, T. Weinstein 23 Lectures in Real Geometry, F. Broglia (Ed.) 24 Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V. Pukhnachov, S. I. Shmarev 25 Character Theory of Finite Groups, B. Huppert 26 Positivity in Lie Theory: Open Problems, J. Hilgert, J. D. Lawson, K.-H. Neeb, E. B. Vinberg (Eds.) ˇ ech Compactification, N. Hindman, D. Strauss 27 Algebra in the Stone-C 28 Holomorphy and Convexity in Lie Theory, K.-H. Neeb 29 Monoids, Acts and Categories, M. Kilp, U. Knauer, A. V. Mikhalev 30 Relative Homological Algebra, Edgar E. Enochs, Overtoun M. G. Jenda 31 Nonlinear Wave Equations Perturbed by Viscous Terms, Viktor P. Maslov, Petr P. Mosolov 32 Conformal Geometry of Discrete Groups and Manifolds, Boris N. Apanasov 33 Compositions of Quadratic Forms, Daniel B. Shapiro 34 Extension of Holomorphic Functions, Marek Jarnicki, Peter Pflug 35 Loops in Group Theory and Lie Theory, Pe´ter T. Nagy, Karl Strambach 36 Automatic Sequences, Friedrich von Haeseler 37 Error Calculus for Finance and Physics, Nicolas Bouleau 38 Simple Lie Algebras over Fields of Positive Characteristic, I. Structure Theory, Helmut Strade 39 Trigonometric Sums in Number Theory and Analysis, Gennady I. Arkhipov, Vladimir N. Chubarikov, Anatoly A. Karatsuba

Embedding Problems in Symplectic Geometry by

Felix Schlenk

≥

Walter de Gruyter · Berlin · New York

Author Felix Schlenk Mathematisches Institut Universität Leipzig Augustusplatz 10/11 04109 Leipzig Germany e-mail: [email protected]

Mathematics Subject Classification 2000: 53-02; 53C15, 53D35, 37J05, 51M15, 52C17, 57R17, 57R40, 58F05, 70Hxx Key words: symplectic embeddings, symplectic geometry, symplectic packings, symplectic capacities, geometric constructions, Hamiltonian systems, rigidity and flexibility

앝 Printed on acid-free paper which falls within the guidelines 앪 of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Schlenk, Felix, 1970⫺ Embedding problems in symplectic geometry / by Felix Schlenk. p. cm ⫺ (De Gruyter expositions in mathematics ; 40) Includes bibliographical references and index. ISBN 3-11-017876-1 (cloth : acid-free paper) 1. Symplectic geometry. 2. Embeddings (Mathematics) I. Title. II. Series. QA665.S35 2005 516.316⫺dc22 2005000895

ISBN 3-11-017876-1 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at ⬍http://dnb.ddb.de⬎. 쑔 Copyright 2005 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the author’s TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Cover design: Thomas Bonnie, Hamburg.

To my parents

Preface

Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time-1-maps of Hamiltonian flows. The spectacular rigidity phenomena for symplectic mappings discovered in the last two decades demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. The most geometric expression of symplectic rigidity are obstructions to certain symplectic embeddings. For instance, Gromov’s Nonsqueezing Theorem states that there does not exist a symplectic embedding of the 2n-dimensional ball B 2n (r) of radius r into the infinite cylinder B 2 (1)×R2n−2 if r > 1. On the other hand, not much was known about the existence of interesting symplectic embeddings. The aim of this book is to describe several elementary and explicit symplectic embedding constructions, such as “symplectic folding”, “symplectic wrapping” and “symplectic lifting”. These constructions are used to solve some specific symplectic embedding problems, and they prompt many new questions on symplectic embeddings. We feel that the embedding constructions described in this book are more important than the results we prove by them. Hopefully, they shall prove useful for solving other problems in symplectic geometry and will lead to further understanding of the still mysterious nature of symplectic mappings. The exposition is self-contained, and the only prerequisites are a basic knowledge of differential forms and smooth manifolds. The book is addressed to mathematicians interested in geometry or dynamics. Maybe, it will also be useful to physicists working in a field related to symplectic geometry. Acknowledgements. This book grew out of my PhD thesis written at ETH Zürich from 1996 to 2000. I am very grateful to my advisor Edi Zehnder for his support, his patience, and his continuous interest in my work. His insight in mathematics and his criticism prevented me more than once from further pursuing a wrong idea. He always found an encouraging word, and he never lost his humour, even not in bad times. Last but not least, his great skill in presenting mathematical results has finally influenced, I hope, my own style. Many ideas of this book grew out of discussions with Paul Biran, David Hermann, Helmut Hofer, Daniel Hug, Tom Ilmanen, Wlodek Kuperberg, Urs Lang, François Laudenbach, Thomas Mautsch, Dusa Mc Duff and Leonid Polterovich. I am in particular indebted to Dusa Mc Duff who explained to me symplectic folding, a technique basic for the whole book. Doing symplectic geometry at ETH has been greatly facilitated through the existence of the symplectic group. When I started my thesis in 1996, this group con-

viii

Preface

sisted of Casim Abbas, Michel Andenmatten, Kai Cieliebak, Hansjörg Geiges, Helmut Hofer, Markus Kriener, Torsten Linnemann, Laurent Moatty, Matthias Schwarz, Karl Friedrich Siburg, Edi Zehnder and myself, and when I finished my thesis, the group consisted of Meike Akveld, Urs Frauenfelder, Ralph Gautschi, Janko Latschev, Thomas Mautsch, Dietmar Salamon, Joa Weber, Katrin Wehrheim, Edi Zehnder and myself. I in particular wish to thank Dietmar Salamon, who helped creating a great atmosphere in the symplectic group; his enthusiasm for mathematics has been a continuous source of motivation for me. This book is the visible fruit of my early studies in mathematics. A more important fruit are the friendships with Rolf Heeb, Laurent Lazzarini, Christian Rüede and Ivo Stalder. Sana, Selin kedim, o zamanki sonsuz sabır ve sevgin için te¸sekkür ederim. Some final work on this book has been done in autumn 2004 at Leipzig University. I wish to thank the Mathematisches Institut for its hospitality, and Anna Wienhard and Peter Albers for carefully reading the introduction. Last but not least, I thank Jutta Mann, Irene Zimmermann and Manfred Karbe for editing my book with so much patience and care. Leipzig, December 2004

Felix Schlenk

Contents

Preface

vii

1

1 1 4 11

Introduction 1.1 From classical mechanics to symplectic geometry 1.2 Symplectic embedding obstructions 1.3 Symplectic embedding constructions

2 Proof of Theorem 1 2.1 Comparison of the relations ≤i 2.2 Rigidity for ellipsoids 2.3 Rigidity for polydiscs ?

23 23 24 28

3

Proof of Theorem 2 3.1 Reformulation of Theorem 2 3.2 The folding construction 3.3 End of the proof

31 31 39 47

4

Multiple symplectic folding in four dimensions 4.1 Modification of the folding construction 4.2 Multiple folding 4.3 Embeddings into balls 4.4 Embeddings into cubes

52 52 53 57 73

5

Symplectic folding in higher dimensions 5.1 Four types of folding 5.2 Embedding polydiscs into cubes 5.3 Embedding ellipsoids into balls

82 82 84 90

6

Proof of Theorem 3 6.1 Proof of lima→∞ paP (M, ω) = 1 6.2 Proof of lima→∞ paE (M, ω) = 1 6.3 Asymptotic embedding invariants

107 107 123 147

x 7

Contents

Symplectic wrapping 7.1 The wrapping construction 7.2 Folding versus wrapping

149 149 157

8 Proof of Theorem 4 8.1 A more general statement 8.2 A further motivation for Problem ζ 8.3 Proof by symplectic folding 8.4 Proof by symplectic lifting

162 162 165 168 177

9

188 189

Packing symplectic manifolds by hand 9.1 Motivations for the symplectic packing problem 9.2 The packing numbers of the 4-ball and CP2 and of ruled symplectic 4-manifolds 9.3 Explicit maximal packings in four dimensions 9.4 Maximal packings in higher dimensions

194 198 213

Appendix A The Extension after Restriction Principle B Flexibility for volume preserving embeddings C Symplectic capacities and the invariants cB and cC D Computer programs E Some other symplectic embedding problems

215 215 219 224 235 238

References

241

Index

247

Chapter 1

Introduction

In the first section of this introduction we recall how symplectomorphisms of R2n arise in classical mechanics, and introduce such notions as “Hamiltonian”, “symplectic” and “volume preserving”. In the second section we briefly tell how symplectic rigidity phenomena were discovered, and then state two paradigms of symplectic non-embedding theorems as well as a symplectic non-embedding result proved in Chapter 2. From Section 1.3 on we describe various symplectic embedding theorems and, in particular, the results proved in this book.

1.1

From classical mechanics to symplectic geometry

Consider a particle of mass 1 in some Rn subject to a force field F . According to Newton’s second law of motion, the acceleration of the particle is equal to the force acting upon it, x¨ = F . In many classical problems, such as those in celestial mechanics, the force field F is a potential field which depends only on the position of the particle and on time, so that x(t) ¨ = ∇U (x(t), t). Introducing the auxiliary variables y = x, ˙ this second order system becomes the first order system of twice as many equations x(t) ˙ = y(t),

y(t) ˙ = ∇U (x(t), t).

(1.1.1)

Besides for special potentials U , it is a hopeless task to solve (1.1.1) explicitly. One can, however, obtain some quantitative insight as follows. The structure of the system (1.1.1) is not very beautiful. Notice, though, that (1.1.1) is “skew-coupled” in the sense that the derivative of x depends on y only and vice versa. We capitalize on this by considering the function H (x, y, t) =

y2 − U (x, t), 2

(1.1.2)

2

1 Introduction

which represents the total energy (i.e., the sum of kinetic and potential energy) of our particle. With this notation, the Newtonian system (1.1.1) becomes the Hamiltonian system  ∂H (x, y, t),  x(t) ˙ =   ∂y (1.1.3)  ∂H   (x, y, t). y(t) ˙ =− ∂x Notice the beautiful skew-symmetry of this system. In order to write it in a more compact form, we consider the constant exact differential 2-form ω0 =

n .

dxi ∧ dyi

(1.1.4)

i=1

on R2n . It is called the standard symplectic form on R2n . It’s n’th exterior product is ω0n = ω0 ∧ · · · ∧ ω0 = n! 0 , where the volume form 0

= dx1 ∧ dy1 ∧ · · · ∧ dxn ∧ dyn

agrees with the Euclidean volume form dx1 ∧ · · · ∧ dxn ∧ dy1 ∧ · · · ∧ dyn up to the n(n−1) factor (−1) 2 . It follows that ω0 is a non-degenerate 2-form, so that the equation ω0 (XH (z, t), ·) = dH (z, t)

(1.1.5)

of 1-forms has a unique solution XH (z, t) for each z = (x, y) ∈ R2n and each t ∈ R. The time-dependent vector field XH is called Hamiltonian vector field of H . Notice ∂H , − now that XH = ∂H ∂y ∂x , so that the Hamiltonian system (1.1.3) takes the compact form (1.1.6) z˙ (t) = XH (z(t), t). Under suitable assumptions on the potential U this ordinary differential equation can be solved for all initial values z(0) = z ∈ R2n and for all times t. The resulting flow t } defined by {ϕH   t d t ϕ (z) = XH ϕH (z), t ,  dt H (1.1.7)   0 2n ϕH (z) = z, z ∈ R , t is called a Hamiltonian is called the Hamiltonian flow of H . Each diffeomorphism ϕH t diffeomorphism. More generally, any time-t-map ϕH obtained in this way via a smooth function H : R2n ×R → R, not necessarily of the form (1.1.2), is called a Hamiltonian diffeomorphism. The above formal manipulations were primarily motivated by aesthetic considerations. As the following facts show, there are important pay-offs, however. t . Fact 1. If H is time-independent, then H is preserved by the flow ϕH

1.1 From classical mechanics to symplectic geometry

3

Proof. Since H is time-independent and in view of definitions (1.1.7) and (1.1.5), t t t t d t t d dt H ϕH = dH ϕH dt ϕH = dH ϕH XH ϕH = ω0 XH , XH ϕH , 2

which vanishes because ω0 is skew-symmetric.

Historically, this fact was the main reason for working in the Hamiltonian formalism. For us, two other pay-offs will be more important. Fact 2. Hamiltonian diffeomorphisms preserve the symplectic form ω0 . Proof. Using Cartan’s formula LXH = dιXH + ιXH d, definition (1.1.5) and dω0 = 0, we compute LXH ω0 = dιXH ω0 + ιXH dω0 = ddH + 0 = 0 for all t. t ∗ t ∗ t ∗ d ϕH ω0 = ϕH LXH ω0 = 0 for all t, so that ϕH ω0 = ω0 . Therefore, dt

2

A diffeomorphism ϕ of R2n is called symplectic diffeomorphism or symplectomorphism if it preserves the symplectic form ω0 , ϕ ∗ ω0 = ω0 . In classical mechanics, symplectomorphisms play the role of those coordinate transformations which preserve the class of Hamiltonian vector fields, and are thus called canonical transformations. Symplectic geometry of ( R2n , ω0 ) is the study of its automorphisms, which are the symplectomorphisms. By Fact 2, the set of Hamiltonian diffeomorphisms is embedded in this geometry. A diffeomorphism ϕ of R2n is called volume preserving if it preserves the volume form 0 , ϕ∗ 0 = 0. Of course, a diffeomorphism is volume preserving if and only if it preserves the Euclidean volume form. Fact 3. Symplectomorphisms preserve the volume form Proof. ϕ ∗

0

= ϕ∗

1

n n! ω0

=

1 n!

(ϕ ∗ ω0 )n =

1 n n! ω0

=

0.

0.

2

These facts go back to the 19th century. Putting Fact 2 and Fact 3 together we find Liouville’s Theorem stating that Hamiltonian diffeomorphisms preserve the volume in phase space. There is no analogue of Liouville’s Theorem for the flow in Rn generated by the Newtonian system (1.1.1). Summarizing, we have Hamiltonian ⇒ symplectic ⇒ volume preserving.

4

1 Introduction

It is well-known that symplectomorphisms of R2n are Hamiltonian diffeomorphisms (see Appendix A), so that “Hamiltonian” and “symplectic” is the same. In dimension 2, “symplectic” and “volume preserving” is also the same. In higher dimensions, however, the difference between “symplectic” and “volume preserving” turns out to be huge and lies at the heart of symplectic geometry.

1.2

Symplectic embedding obstructions

The most striking examples for the difference between “symplectic” and “volume preserving” are obstructions to certain symplectic embeddings. Before describing such obstructions, we briefly tell the story of 1.2.1 The discovery of symplectic rigidity phenomena. We only describe the discovery of the first rigidity phenomena found. For the theories invented during these discoveries and for preceding and subsequent developments and further results we refer to the papers [18], [31] and to the books [32], [39], [62], [63]. Local considerations show that there are much less symplectomorphisms than volume preserving diffeomorphisms of R2n if n ≥ 2: The linear symplectic group has dimension 2n2 + n, while the group of matrices with determinant 1 has dimension 4n2 − 1. Moreover, locally any symplectic map can be represented in terms of a single function, a so-called generating function, see [39, Appendix 1], while one needs 2n−1 functions to describe a volume preserving diffeomorphism locally. Also notice that the Lie algebra of the group of symplectomorphisms can be identified with the set of time-independent Hamiltonian functions, while the Lie algebra of the group of volume preserving diffeomorphisms consists of divergence-free vector fields, which depend on 2n − 1 functions. These local differences do not imply, however, that the set of symplectomorphisms of R2n is also much smaller than the set of volume preserving diffeomorphisms from a global point of view: Every time-dependent compactly supported function on R2n generates a Hamiltonian diffeomorphism, and so one could well believe that whatever can be done by a volume preserving diffeomorphism can “approximately” also be done by a Hamiltonian or symplectic diffeomorphism. This opinion was indeed shared by many physicists until the mid 1980s. Global properties distinguishing Hamiltonian or symplectic diffeomorphisms from volume preserving diffeomorphisms were discovered only around 1980. There are various reasons for this “delay” in the discovery of symplectic rigidity. One reason is that the preservation of volume of Hamiltonian diffeomorphisms and stability problems in celestial mechanics had attracted and absorbed much attention, leading to ergodic theory and KAM theory. Another reason is that many interesting questions in classical mechanics (such as the restricted 3-body problem) lead to problems in dimension 2, where “symplectic” and “volume preserving” is the same. But the main reason for this delay was undoubtedly the difficulty in establishing global symplectic rigidity phenomena. No such phenomenon known today admits an easy proof.

1.2 Symplectic embedding obstructions

5

In the 1960s, Arnold pointed out the special role played by the 2-form ω0 in Fact 2 and made several seminal and fruitful conjectures in symplectic geometry, whose proofs in particular would have demonstrated that both Hamiltonian and symplectic diffeomorphisms are distinguished from volume preserving diffeomorphisms by global properties. A breakthrough came in 1983 when Conley and Zehnder, [18], proved one of Arnold’s conjectures. Denote the standard 2n-dimensional torus R2n /Z2n endowed with the induced symplectic form ω0 by (T 2n , ω0 ). Arnold conjecture for the torus. Every Hamiltonian diffeomorphism of the standard torus (T 2n , ω0 ) must have at least 2n + 1 fixed points. It in particular follows that volume preserving (or symplectic) diffeomorphisms of (T 2n , ω0 ) cannot be C 0 -approximated by Hamiltonian diffeomorphisms in general. Indeed, translations demonstrate that this global fixed point theorem is a truly Hamiltonian result, which does not hold for all volume preserving or symplectic diffeomorphisms. For a Hamiltonian diffeomorphism which is generated by a time-independent Hamiltonian or which is C 1 -close to the identity, the theorem follows from classical Lusternik–Schnirelmann theory. The point of this theorem is that it holds for arbitrary Hamiltonian diffeomorphisms, also for those far from the identity. The first rigidity phenomenon for symplectomorphisms of R2n was found by Gromov and Eliashberg. In the early 1970s, Gromov proved the following alternative. Gromov’s Alternative. The group of symplectomorphisms of R2n is either C 0 -closed in the group of all diffeomorphisms (hardness), or its C 0 -closure is the group of volume preserving diffeomorphisms (softness). Notice that “symplectic” is a C 1 -condition, so that there is no obvious reason for hardness. The soft alternative would have meant that there are no interesting global invariants in symplectic geometry. In the late 1970s, Eliashberg decided Gromov’s Alternative in favour of hardness. C 0 -stability for symplectomorphisms. The group of symplectomorphisms of R2n is C 0 -closed in the group of all diffeomorphisms. It follows that volume preserving diffeomorphisms of R2n cannot be C 0 -approximated by symplectomorphisms in general. For references and an elegant proof we refer to [39, Section 2.2]. An important ingredient of each known proof is a symplectic non-embedding result. 1.2.2 Symplectic non-embedding theorems. A smooth map ϕ : U → R2n defined on an open (not necessarily connected) subset U of R2n is called symplectic if ϕ ∗ ω0 = ω0 . Locally, symplectic maps are embeddings. Indeed, symplectic maps 1 n ω0 and are thus immersions. preserve the volume form 0 = n!

6

1 Introduction

Flexibility for symplectic immersions. For every open set V in R2n there exists a symplectic immersion of R2n into V . Proof. Since translations are symplectic, we can assume that V contains the origin. Let D be an open disc in R2 centred at the origin whose radius r is so small that D × · · · × D ⊂ V . We shall construct a symplectic immersion ϕ of R2 into D. The product ϕ × · · · × ϕ will then symplectically immerse R2n into V . Step 1. Choose a diffeomorphism f : R →]0, r 2 /2[. Then the map y 2 2 , (1.2.1) R →]0, r /2[×R, (x, y) → f (x), f (x) is a symplectomorphism. Step 2. The map ]0, r 2 /2[×R → D,

(x, y) →

√ √ 2x cos y, 2x sin y ,

is a symplectic immersion.

2

A symplectic map ϕ : U → R2n is called a symplectic embedding if it is injective. In view of the above result, we shall only consider symplectic embeddings from now on. A domain V in R2n is a non-empty connected open subset of R2n . The basic problem addressed in this book is Basic Problem. Consider an open set U in R2n and a domain V in R2n . Does there exist a symplectic embedding of U into V ? A reader with a more physical or dynamical background may rather ask for embeddings of U into V induced by Hamiltonian diffeomorphisms of R2n . This is not the same problem in general: Example. Let U be the annulus

U = (x, y) ∈ R2 | 1 < x 2 + y 2 < 2 of area π and let V be the disc of area a. As is easy to see (or by Proposition 1 below), U symplectically embeds into V if and only if a ≥ π . On the other hand, U 2 symplectically embeds into V via a Hamiltonian diffeomorphism √ of R only if a ≥ 2π since such an embedding must map the whole disc of radius 2 into V . 3 For a large class of domains U in R2n , however, finding a symplectic or a Hamiltonian embedding is almost the same problem. A domain U in R2n is called starshaped if U contains a point p such that for every point z ∈ U the straight line between p and z is contained in U .


7

Extension after Restriction Principle. Assume that ϕ : U → R2n is a symplectic embedding of a bounded starshaped domain U ⊂ R2n . Then for any subset A ⊂ U whose closure in R2n is contained in U there exists a Hamiltonian diffeomorphism A of R2n such that A |A = ϕ|A . A proof of this well-known fact can be found in Appendix A. In most of the results discussed or proved in this book, U will be a bounded starshaped (and in fact convex) domain – or a union of finitely many balls, for which the Extension after Restriction Principle also applies, see Appendix E. Since symplectic embeddings preserve the volume form 0 and are injective, they , preserve the total volume Vol(U ) = U 0 . A necessary condition for the existence of a symplectic embedding of U into V is therefore Vol(U ) ≤ Vol(V ).

(1.2.2)

For volume preserving embeddings, this necessary condition is also sufficient. Proposition 1. An open set U in R2n embeds into a domain V in R2n by a volume preserving embedding if and only if Vol(U ) ≤ Vol(V ). Notice that we did not assume that Vol(U ) is finite. A proof of Proposition 1 can be found in Appendix B. Since in dimension 2 an embedding is symplectic if and only if it is volume preserving, the Basic Problem is completely solved in this dimension by Proposition 1. In higher dimensions, however, strong obstructions to symplectic embeddings which are different from the volume condition (1.2.2) appear. Consider the open 2n-dimensional ball of radius r 7 .n 6 B 2n π r 2 = (x, y) ∈ R2n xi2 + yi2 < r 2 i=1

and the open 2n-dimensional symplectic cylinder

Z 2n (π) = (x, y) ∈ R2n | x12 + y12 < 1 . While the ball B 2n (a) has finite volume for each a, the symplectic cylinder Z 2n (π ) has infinite volume, of course. The following theorem proved by Gromov in his seminal work [31] is the most geometric expression of symplectic rigidity. Gromov’s Nonsqueezing Theorem. The ball B 2n (a) symplectically embeds into the cylinder Z 2n (π ) if and only if a ≤ π. Remarks. 1. Proposition 1 shows that for n ≥ 2 the whole R2n embeds into Z 2n (π ) by a volume preserving embedding. Explicit such embeddings are obtained by making use of maps of the form (1.2.1). The linear volume preserving diffeomorphism (x, y) → ( x1 , −1 x2 , x3 , . . . , xn , y1 , −1 y2 , y3 , . . . , yn )

8

1 Introduction

of R2n embeds the ball of radius −1 into Z 2n (π ). 2. The “symplectic cylinder” Z 2n (π) in the Nonsqueezing Theorem cannot be replaced by the “Lagrangian cylinder”

(x, y) ∈ R2n | x12 + x22 < 1 . √ Indeed, for a = 2/2 the n-fold product of the map (1.2.1) symplectically embeds the whole R2n into this cylinder. The linear symplectomorphism (x, y) → ( x, −1 y)

(1.2.3)

of R2n embeds the ball of radius −1 into this cylinder. 3. Combined with Gromov’s Alternative, the Nonsqueezing Theorem implies the C 0 -Stability Theorem at once. In [20], Ekeland and Hofer observed that the C 0 Stability Theorem easily follows from the Nonsqueezing Theorem alone, see also [39, Section 2.2]. 4. For far reaching generalizations of Gromov’s Nonsqueezing Theorem we refer to Remark 9.3.7 in Chapter 9. 3 Gromov deduced his Nonsqueezing Theorem from his compactness theorem for pseudo-holomorphic spheres. In his proof, the obstruction , to a symplectic embedding 2n 2n of B (a) into Z (π) for a > π is the symplectic area S ω0 of a holomorphic curve S in B 2n (a) passing through the centre, which is at least a > π. Using Gromov’s compactness theorem for pseudo-holomorphic discs with Lagrangian boundary conditions, Sikorav found another amazing Nonsqueezing Theorem. Let S 1 be the unit circle in R2 (x, y), and consider the torus T n = S 1 × · · · × S 1 in R2n . Sikorav proved in [79] that there does not exist a symplectomorphism of R2n which maps T n into Z 2n (π). Notice that the volume of T n in R2n vanishes, and that T n does , not bound any open set! In Sikorav’s proof, the obstruction is the symplectic area D ω0 of a closed holomorphic disc D ⊂ R2n with boundary on T n , which is at least π . Sikorav’s result combined with the Extension after Restriction Principle implies the following remarkable version of the Nonsqueezing Theorem, which is due to Hermann, [37]. Symplectic Hedgehog Theorem. For n ≥ 2, no starshaped domain in R2n containing the torus T n symplectically embeds into the cylinder Z 2n (π ). A more martial reader may prefer calling it Symplectic Flail Theorem. Notice that there are starshaped domains containing T n of arbitrarily small volume! We finally describe a symplectic non-embedding result which will lead to the search for interesting symplectic embedding constructions. Given an open subset U of R2n and a number λ > 0 we set λ U = {λz | z ∈ U }.


9

Our Basic Problem can be reformulated as Problem UV. Consider an open set U in R2n and a domain V in R2n . What is the smallest λ such that U symplectically embeds into λV ? In the two theorems above, the symplectic embedding realizing the smallest λ was simply the identity embedding. In these theorems, the set U was a round ball B 2n (a) with a > π and a starshaped domain containing the “round” torus T n , the set V was the “long and thin” cylinder Z 2n (π), and the outcome was that these “round” sets U cannot be symplectically “squeezed” into the “long but thinner” set V . In order to formulate a symplectic embedding problem in which we have a chance to find interesting symplectic embeddings, we therefore take now U “long and thin” and V “round”. Our hope is then that U can be symplectically “folded” or “wrapped” into V . To fix the ideas, we take V to be a ball and U to be an ellipsoid. Using complex notation zi = (xi , yi ), we define the open symplectic ellipsoid with radii √ ai /π as 6 .n E(a1 , . . . , an ) = (z1 , . . . , zn ) ∈ Cn

i=1

7 π |zi |2 0 there exists an > 0 such that the 2n-dimensional ellipsoid E(, . . . , , a) symplectically embeds into B 2n (π ). Proof. This is an immediate consequence of Gromov’s h-principle for symplectic embeddings of codimension at least 2. Indeed, choose a smooth embedding ϕ0 of the closed disc B 2 (a) into B 2n (π). According to [32, p. 335] or [24, Theorem 12.1.1], arbitrarily C 0 -close to ϕ0 there exists a symplectic embedding ϕ1 : B 2 (a) → B 2n (π ), meaning that ϕ1∗ ω0 = ω0 . Using the Symplectic Neighbourhood Theorem, we find > 0 such that ϕ1 extends to a symplectic embedding of B 2n−2 () × B 2 (a) and in particular to a symplectic embedding of E(, . . . , , a). 2 Since the C 0 -small perturbation ϕ1 of the smooth embedding ϕ0 provided by the h-principle is not explicit, this embedding method gives no quantitative information on the number > 0. In a large part of this book we shall be concerned with providing quantitative information on . We first investigate the zone of transition between rigidity and flexibility in Problem EB. Our hope is still to “fold” or “wrap” a “long and thin” ellipsoid into a smaller ball in a symplectic way. To find such constructions, we start with giving a list of Elementary symplectic embeddings 1. Linear symplectomorphisms. The group Sp(n; R) of linear symplectomorphisms of R2n contains transformations of the form (1.2.3) and, more generally, of the form −1 y (1.3.1) (x, y) → Ax, AT where A is any non-singular (n × n)-matrix. It also contains the unitary group U(n). Translations are also symplectic, of course. 2. Products of area preserving embeddings. Every area and orientation preserving embedding of a domain in R2 into another domain in R2 is symplectic, and by Proposition 1 there are plenty of such embeddings. An example are the “inverse symplectic polar coordinates” √ √ 2x cos y, 2x sin y (1.3.2) (x, y) → embedding 0, a/2π × ]0, 2π [ into B 2 (a), which we met before. As we shall see in Section 3.1, symplectic embeddings of domains in R2 can be described in an almost

1.3 Symplectic embedding constructions

13

explicit way. Taking products, we obtain almost explicit symplectic embeddings of domains in R2n . 3. Lifts. For convenience we write (u, v, x, y) = (x1 , y1 , x2 , y2 ). Using definition (1.1.3) we find that the Hamiltonian vector field of the Hamiltonian function (u, v, x, y) → −x is (0, 0, 0, 1), so that the time-1-map of the Hamiltonian flow is the translation (u, v, x, y) → (u, v, x, y + 1). Choose a smooth function f : R → [0, 1] such that f (s) = 0 if s ≤ 0 and f (s) = 1 if s ≥ 1. The vector field of the Hamiltonian function (u, v, x, y) → −f (u)x is Hamiltonian 0, f (u)x, 0, f (u) , so that its time-1-map (1.3.3) (u, v, x, y) → u, v + f (u)x, x, y + f (u) fixes the half space {u ≤ 0} and lifts the space {u ≥ 1} by 1 in the y-direction. Choosing f such that f (s) = 0 if s ≤ 0 or s ≥ 3

and

f (s) = 1 if s ∈ [1, 2]

and looking at the time-1-map generated by the Hamiltonian function (u, v, x, y) → −f (u)f (v)x we find “true” lifts (called elevators in the US, I guess).

(1.3.4) 3

At first glance, these elementary symplectic embeddings look useless for Problem EB. Indeed, none of them embeds the ellipsoid E(a1 , . . . , an ) into a ball B 2n (A) with A < an . However, these elementary symplectic embeddings will serve as building blocks for all our embedding constructions: Each of the symplectic embedding constructions described in the sequel will be a composition of elementary symplectic embeddings as above! The first quantitative embedding result addressing Problem EB was proved by Traynor in [81] by means of a symplectic wrapping construction. Given λ > 0 the ellipsoid λE(a1 , a2 ) symplectically embeds into the ball λB 4 (A) if and only if E(a1 , a2 ) symplectically embeds into B 4 (A). We can thus assume without loss of generality that a1 = π . Traynor’s Wrapping Theorem. There exists a symplectic embedding E π, k(k − 1)π → B 4 (kπ + ) for every integer k ≥ 2 and every > 0. The symplectic wrapping construction invented by Traynor is a composition of linear symplectomorphisms and products of area preserving embeddings. It first uses

14

1 Introduction

a product of area preserving embeddings to view an ellipsoid as a Lagrangian product × 2 of a simplex and a square in R2+ (x) × R2+ (y), and then uses a map of the form (1.3.1) to wrap this product around the torus T 2 (y) = R2 (y)/2π Z2 in R2+ (x) × T 2 (y). The point is then that the product of the area preserving map (1.3.2) extends to a symplectic embedding of R2+ (x) × T 2 (y) into R2 (x) × R2 (y). Details and an extension of the symplectic wrapping construction to higher dimensions are given in Section 6.1. The contribution to Problem EB made by Traynor’s Wrapping Theorem is encoded in the piecewise linear function wEB on [π, ∞[ drawn in Figure 1.1 below, in which we again assume a1 = π and write a = a2 . We in particular see that wEB (a) < a only for a > 3π , so that Traynor’s Wrapping Theorem does not tell us whether Theorem 1 is sharp. On the other hand, the obstructions to symplectic embeddings found in Section 1.2.2 confirm our hopes that some kind of folding can be used to show that Theorem 1 is sharp: Arguing heuristically, we consider the two (symplectic) areas s1 (U ) and s2 (U ) of the projections of a domain U in R4 to the coordinate planes R2 (x1 , y1 ) and R2 (x2 , y2 ). The obstructions to symplectic embeddings found in Section 1.2.2 were symplectic areas of surfaces different from these projections, but numerically they are equal to s1 in both the Nonsqueezing Theorem and the Symplectic Hedgehog Theorem and equal to s2 in Theorem 1. Consider now an ellipsoid E = E(a1 , a2 ). When we “fold E appropriately” to E , the smaller projection will double, s1 (E ) = 2a1 , while the larger projection should decrease, s2 (E ) < a2 . If a2 ≤ 2a1 , then s1 (E ) = 2a1 ≥ a2 = s1 (B 4 (a2 )), so that E does not fit into a ball B 4 (A) with A < a2 , as predicted by Theorem 1. If a2 > 2a1 , then s1 (E ) = 2a1 < a2 and s2 (E ) < a2 , however, so that we can hope that folding can be achieved in such a way that E fits into a ball B 4 (A) with A < a2 . This can indeed be done in a symplectic way. Theorem 2. Assume an > 2a1 . Then there exists a symplectic embedding of the ellipsoid E(a1 , . . . , a1 , an ) into the ball B 2n (an − δ) for every δ ∈ 0, a2n − a1 . The reader might ask why we look at “skinny” ellipsoids with an−1 = a1 in Theorem 2. The reason is that for “flat” ellipsoids an analogous embedding result does not hold in general. For instance, the third Ekeland–Hofer capacity c3 implies that for n ≥ 3 the “flat” 2n-dimensional ellipsoid E(a, 3a, . . . , 3a) does not symplectically embed into the ball B 2n (A) if A < 3a. The second Ekeland–Hofer capacity c2 implies that the “mixed” ellipsoid E(a, 2a, 3a) does not symplectically embed into the ball B 6 (A) if A < 2a, but we do not know the answer to Question 1. Does the ellipsoid E(a, 2a, 3a) symplectically embed into B 6 (A) for some A < 3a? Symplectic folding was invented by Lalonde and Mc Duff in [48] in order to prove the General Nonsqueezing Theorem stated in Remark 9.3.7 as well as an inequality between Gromov width and displacement energy implying that the Hofer metric


15

on the group of compactly supported Hamiltonian diffeomorphisms is always nondegenerate. In the same work [48] Lalonde and Mc Duff also observed that symplectic folding can be used to prove Theorem 2 in the case n = 2. A refinement of their symplectic folding construction in dimension 4 will prove Theorem 2 in all dimensions. The symplectic folding construction is a composition of products of area preserving embeddings and a lift. Viewing an ellipsoid E(a1 , a2 ) as fibred over the larger disc B 2 (a2 ), this construction first separates the smaller fibres from the larger ones by a suitable area preserving embedding of B 2 (a2 ) into R2 , then lifts the smaller fibres by the lift (1.3.3), and finally turns these lifted fibres over the larger fibres via another area preserving embedding. An idea of the construction can be obtained from Figure 3.12 on page 50 and from Figure 4.2 on page 53. Theorem 2 can be substantially improved by folding more than once. An idea of multiple symplectic folding is given by Figure 4.3 on page 54. In describing the results for Problem EB, we now restrict ourselves to dimension 4 for the sake of clarity. As before we can assume a1 = π and write a = a2 . The optimal values A for the embedding problems E(π, a) → B 4 (A) are encoded in the “characteristic function” χEB on [π, ∞[ defined by

χEB (a) = inf A | E(π, a) symplectically embeds into B 4 (A) . We illustrate the results with the help of Figure 1.1. In view of Theorem 1 we have χEB (a) = a for a ∈ [π, 2π]. For a > 2π, the second Ekeland–Hofer capacity c2 still implies that χEB (a) is vacuous if a ≥ 4π , since ≥ 2π. This information √ the volume condition Vol E(π, a) ≤ Vol B 4 (χEB (a)) translates to χEB (a) ≥ π a. The estimate χEB (a) ≤ a/2 + π stated in Theorem 2 is obtained by folding once. It will turn out that for a > 2π and for each k ≥ 1, folding k + 1 times embeds E(π, a) into a strictly smaller ball than folding k times. The function fEB on ]2π, ∞[ defined by

fEB (a) = inf A | E(π, a) embeds into B 4 (A) by multiple symplectic folding is therefore obtained by folding “infinitely many times”. The graph of the function fEB is computed by a computer program. The function wEB encoding Traynor’s Wrapping Theorem is alternatingly larger and smaller than fEB . We are particularly interested in the behaviour of χEB (a) as a → 2π + and as a → ∞. We shall prove that lim sup →0+

3 fEB (2π + ) − 2π ≤ , 7

and so the same estimate holds for χEB . Question 2. How does χEB (a) look like near a = 2π ? In particular, lim sup →0+

3 χEB (2π + ) − 2π < ? 7

16

1 Introduction

We have fEB (a) < wEB (a) for all a ∈ ]2π, 5.1622π ]. The computer program for fEB yields the particular values fEB (3π ) ≈ 2.3801π

and

fEB (4π ) ≈ 2.6916π.

We do not expect that χEB (3π ) = fEB (3π ) and χEB (4π ) = fEB (4π ). Question 3. Is it true that χEB (3π ) = χEB (4π) = 2π ? √ The difference wEB (a) − π a between wEB and√ the volume condition is bounded √ 3)π. We shall also prove that fEB (a) − π a is bounded. It follows that by (3 − √ χEB (a) − πa is bounded. We in particular have Vol (E(π, a)) = 1. a→∞ Vol B 4 (χEB (a)) lim

(1.3.5)

This means that the embedding obstructions encountered for small a more and more disappear as a → ∞. A A = a2 + π

A=a 6π

χEB (a) ?

wEB (a) 5π fEB (a) 4π A=

√

πa

3π

2π

c2

a 2π

4π

6π

8π

12π

15π

20π

24π

Figure 1.1. What is known about the embedding problem E(π, a) → B 4 (A).

Denote by D(a) = B 2 (a) the open disc in R2 of area a centred at the origin. The open symplectic polydisc P (a1 , . . . , an ) in R2n is defined as P (a1 , . . . , an ) = D(a1 ) × · · · × D(an ).


17

The “n-cube” P 2n (a, . . . , a) will be denoted by C 2n (a). Up to now, our model sets were ellipsoids, which we tried to symplectically embed into small balls. Starting from Sikorav’s Nonsqueezing Theorem for the torus T n and noticing that (the closure of) C 2n (π ) is the convex hull of T n , we could equally well have taken polydiscs and cubes. Problem PC. What is the smallest cube C 2n (A) into which P (a1 , . . . , an ) symplectically embeds? For this problem, no interesting obstructions are known, however. The reason is that symplectic capacities only see the size min {a1 , . . . , an } of the smallest disc of a polydisc and thus do not provide any obstruction for symplectic embeddings of polydiscs into cubes stronger than the volume condition. In particular, it is unknown whether the analogue of Theorem 1 for polydiscs holds true. On the other hand, both symplectic folding and symplectic wrapping can be used to construct interesting symplectic embeddings of polydiscs into cubes. Somewhat more generally, we shall study symplectic embeddings of both ellipsoids and polydiscs into balls and cubes. While embedding an open set U into a minimal ball is related to minimizing its diameter (a 1-dimensional, metric quantity), embedding U into a minimal cube amounts to minimizing the areas of its projections to the symplectic coordinate planes (a 2-dimensional, “more symplectic” quantity). We refer to Appendix C for details on this. 1.3.2 Flexibility for skinny shapes. Let us come back to Problem UV, which we reformulate as Problem UV. Consider an open set U in R2n and a domain V in R2n . What is the largest λ such that λU symplectically embeds into V ? As before, we shall eventually specialize U to an ellipsoid or a polydisc, but this time we take V to be an arbitrary domain in R2n of finite volume. In fact, we shall look at symplectic embeddings into arbitrary connected symplectic manifolds of finite volume. A reader not familiar with smooth manifolds may skip the subsequent generalities on symplectic manifolds and take (M, ω) in Theorem 3 below to be a domain in R2n of finite volume. As will become clear in Chapter 6 not much is lost thereby. A differential 2-form ω on a smooth manifold M is called symplectic if ω is non-degenerate and closed. The pair (M, ω) is then called a symplectic manifold. The non-degeneracy of ω implies that M is even-dimensional, dim M = 2n, and 1 n ω is a volume form on M, so that M is orientable. The non-degeneracy that = n! together with the closedness of ω imply that (M, ω) is locally isomorphic to ( R2n , ω0 ) with ω0 as in (1.1.4):

18

1 Introduction

Darboux’s Theorem. For every point p ∈ M there exists a coordinate chart ϕU : U → R2n such that ϕU (p) = 0 and ϕU∗ ω0 = ω. Therefore, a symplectic manifold is a smooth 2n-dimensional manifold admitting an atlas {(U, ϕU )} such that all coordinate changes ϕV ϕU−1 : ϕU (U ∩ V ) → ϕV (U ∩ V ) are symplectic. Examples of symplectic manifolds are open subsets of ( R2n , ω0 ), the torus R2n /Z2n endowed with the induced symplectic form, surfaces equipped with an area form, Kähler manifolds like complex projective space CPn endowed with their Kähler form, and cotangent bundles with their canonical symplectic form. Many more examples are obtained by taking products and via the symplectic blow-up operation. We refer to the book [62] for more information on symplectic manifolds. As before, we endow each open subset U of R2n with the standard symplectic form ω0 . A smooth embedding ϕ : U → (M, ω) is called symplectic if ϕ ∗ ω = ω0 . Problem UV generalizes to Problem UM. Consider an open set U in R2n and a connected 2n-dimensional symplectic manifold (M, ω). What is the largest number λ such that λU symplectically embeds into (M, ω)? A smooth embedding ϕ : U → (M, ω) is called volume preserving if ϕ∗

=

0

1 n 1 n where as before 0 = n! ω0 and = n! ω . Of course, every symplectic embedding is volume preserving. A necessary condition for a symplectic embedding of U into (M, ω) is therefore Vol (U ) ≤ Vol (M, ω) , 1 where we set Vol (M, ω) = n! M ωn . For volume preserving embeddings, this obvious condition is the only one in view of the following generalization of Proposition 1, a proof of which can again be found in Appendix B.

Proposition 2. An open set U in R2n embeds into (M, ω) by a volume preserving embedding if and only if Vol(U ) ≤ Vol(M, ω). For symplectic embeddings of “round” shapes U ⊂ R2n into (M, ω), however, there often are strong obstructions beyond the volume condition. We have already seen this in Section 1.2.2 in case that (M, ω) is a cylinder or a ball, and many more examples can be found in Chapters 4 and 9. To give one other example, we consider


19

2 2 2 the product , (M, ω) = (S × S , σ ⊕ kσ ), where σ is4an area form on the 2-sphere S of area S 2 σ = π and where k ≥ 1. Then the ball B (a) symplectically embeds into (M, ω) if and only if a ≤ π . For skinny shapes, though, the situation for symplectic embeddings is not too different from the one for volume preserving embeddings: We choose U to be a 2n-dimensional ellipsoid E (π, . . . , π, a) or a 2n-dimensional polydisc P (π, . . . , π, a), consider a connected symplectic manifold , 2n-dimensional 1 n , and define for each a ≥ π the (M, ω) of finite volume Vol (M, ω) = n! ω M numbers Vol λE (π, . . . , π, a) E pa (M, ω) = sup , Vol (M, ω) λ Vol λP (π, . . . , π, a) P , pa (M, ω) = sup Vol (M, ω) λ

where the supremum is taken over all those λ for which λE(π, . . . , π, a) respectively λP (π, . . . , π, a) symplectically embeds into (M, ω). Theorem 3. Assume that (M, ω) is a connected symplectic manifold of finite volume. Then lim paE (M, ω) = 1 and lim paP (M, ω) = 1. a→∞

a→∞

This means that the obstructions encountered for symplectic embeddings of round shapes more and more disappear as we pass to skinny shapes. Notice that if (M, ω) is a 4-dimensional ball, the first statement in Theorem 3 is equivalent to the identity (1.3.5), which also followed from Traynor’s Wrapping Theorem. Symplectic folding can be used to prove the full statement of Theorem 3. The second statement in Theorem 3 will be proved along the following lines. First, fill almost all of M with cubes. Using multiple symplectic folding, these cubes can then almost be filled with symplectically embedded thin polydiscs, see Figure 5.2 on page 85. Using the remaining space in M, these embeddings can finally be glued to a symplectic embedding of a very long and thin polydisc, see Figure 6.1 on page 108. The proof of the first statement is more involved and uses a non-elementary result of Mc Duff and Polterovich on filling a cube by balls. 1.3.3 A vanishing theorem. The Basic Problem and its variations asked for symplectic embeddings which were not required to have any additional properties. We now look at symplectic embeddings ϕ of the ball B 2n (a) into the symplectic cylinder Z 2n (π ) whose image ϕ(B 2n (a)) ⊂ Z 2n (π) has a specific property. By Gromov’s Nonsqueezing Theorem there does not exist a symplectic embedding of B 2n (a) into Z 2n (π ) if a > π. So fix a ∈ ]0, π ]. The simply connected hull Tˆ of a subset T of R2 is the union of its closure T and the bounded components of R2 \ T . We denote by µ the Lebesgue measure on R2 , and we abbreviate µ(T ˆ ) = µ(Tˆ ). For the unit circle

20

1 Introduction

S 1 in R2 we have µ(S 1 ) = 0 < π = µ(S ˆ 1 ). As is well-known, the Nonsqueezing Theorem is equivalent to each of the identities a = inf µ p ϕ B 2n (a) , ϕ a = inf µˆ p ϕ B 2n (a) , ϕ

where ϕ varies over all symplectomorphisms of R2n which embed B 2n (a) into Z 2n (π ) and where p : Z 2n (π) → B 2 (π) is the projection, see [22] and Appendix C.2. Following McDuff, [59], we consider sections of the image ϕ(B 2n (a)) instead of its projection, and define

ζ (a) := inf sup µ p ϕ(B 2n (a)) ∩ Dx , ϕ x

ˆζ (a) := inf sup µˆ p ϕ(B 2n (a)) ∩ Dx , ϕ

x

where ϕ again varies over all symplectomorphisms of R2n which embed B 2n (a) into Z 2n (π ), and where Dx ⊂ Z 2n (π) denotes the disc Dx = B 2 (π) × {x}, Clearly,

x ∈ R2n−2 .

ζ (a) ≤ ζˆ (a) ≤ a.

It is also well-known that the Nonsqueezing Theorem is equivalent to the identity ζˆ (π ) = π.

(1.3.6)

Indeed, the Nonsqueezing Theorem implies that for every symplectomorphism ϕ of R2n which embeds B 2n (π) into Z 2n (π) there exists an x in R2n−2 such that ϕ(B 2n (π )) ∩ Dx contains the unit circle S 1 × {x}. On her search for symplectic rigidity phenomena beyond the Nonsqueezing Theorem, Mc Duff therefore posed the following problem. Problem ζ . Find a non-trivial lower bound for the function ζ (a). In particular, is it true that ζ (a) → π as a → π ? A further motivation for this problem comes from convex geometry and from the fact that on bounded convex subsets of ( R2n , ω0 ) normalized symplectic capacities agree up to a constant. It was known to Polterovich that ζ (a)/a → 0 as a → 0. The following result answers the question in Problem ζ in the negative and completely solves Problem ζ . Theorem 4. ζ (a) = 0 for all a ∈ ]0, π ] and ζˆ (a) = 0 for all a ∈ ]0, π [. The second assertion in Theorem 4 can be proved by symplectic folding. In order to prove the full theorem, we shall iterate the symplectic lifting construction briefly described in Section 1.3.1. An idea of the two proofs is given in Figure 8.1 on page 171 and in Figure 8.9 on page 182.


21

1.3.4 Symplectic packings. We finally look at the symplectic packing problem. Given a connected 2n-dimensional symplectic manifold (M, ω) of finite volume and given a natural number k, this problem asks Problem kBM. What is the largest number a for which the disjoint union of k equal balls B 2n (a) symplectically embeds into (M, ω)? Equivalently, one studies the k’th symplectic packing number k Vol B 2n (a) pk (M, ω) = sup Vol (M, ω) a 1 where the supremum is taken over all those a for which ki=1 B 2n (a) symplectically embeds into (M, ω). Obstructions to full packings of a ball were already found by Gromov in [31], where he proved that pk (B 2n (π ), ω0 ) ≤ 2kn for 2 ≤ k ≤ 2n . Later on, spectacular progress in the symplectic packing problem was made by McDuff and Polterovich in [61] and by Biran in [7], [8], who obtained symplectic packings via the symplectic blow-up operation. These works established many further packing obstructions for small values of k. For large k, however, it was shown in [61] that lim pk (M, ω) = 1

k→∞

for every connected symplectic manifold (M, ω) of finite volume, and it was shown in [7], [8] that for many symplectic 4-manifolds (M, ω) there exists a number k0 such that pk (M, ω) = 1 for all k ≥ k0 . This transition from rigidity for symplectic packings by few balls to flexibility for packings by many balls is reminiscent to the transition from rigidity to flexibility for symplectic embeddings of ellipsoids discussed in Sections 1.3.1 and 1.3.2. Essentially all presently known packing numbers were obtained in [61], [7], [8]. The symplectic packings found in these works are not explicit, however. For some symplectic manifolds as balls and products of surfaces and for some values of k, explicit maximal symplectic packings were constructed by Karshon [41], Traynor [81], Kruglikov [45], and Maley, Mastrangeli and Traynor [54]. In the last chapter, we shall describe a very simple and explicit construction realizing the packing numbers pk (M, ω) for those symplectic 4-manifolds (M, ω) and numbers k considered in [41], [81], [45], [54] as well as for ruled symplectic 4-manifolds and small values of k. For example, we shall see that maximal packings of the standard 4-ball by 5 or 6 balls and of the product S 2 × S 2 of 2-spheres of equal area by 5 balls can be described by Figure 1.2. These symplectic packings are simply obtained via products α1 ×α2 of suitable area preserving diffeomorphisms between a disc and a rectangle. Taking n-fold products we shall also construct a full packing of the standard 2n-ball by l n balls for each l ∈ N in a most simple way.

22

1 Introduction

Figure 1.2. Maximal symplectic packings of the 4-ball by 5 or 6 balls and of S 2 × S 2 by 5 balls.

Contrary to the symplectic embedding results described before, none of our symplectic packing results is new in view of the packings in [61], [7], [8]. We were just curious how maximal packings might look like, and we hope the reader will enjoy the pictures in Chapter 9, too. Moreover, in the range of k for which the explicit constructions in [41], [81], [45], [54] and our constructions fail to give maximal packings, they give a feeling that the balls in the packings from [61], [7], [8] must be “wild”. The book is organized as follows: In Chapter 2 we prove Theorem 1 and several other rigidity results for ellipsoids. In Chapter 3 we prove Theorem 2 by symplectic folding. In Chapter 4 we use multiple symplectic folding to obtain rather satisfactory results for symplectic embeddings of 4-dimensional ellipsoids and polydiscs into 4-dimensional balls and cubes. In Chapter 5 we look at higher dimensions. We will concentrate on embedding skinny ellipsoids into balls and skinny polydiscs into cubes. The results in this chapter form half of the proof of Theorem 3, which is completed in Chapter 6. In Chapter 6 we shall also notice that for certain symplectic manifolds our embedding methods can be used to improve Theorem 3. In Chapter 7 we recall the symplectic wrapping method invented by Traynor, and compare the results obtained by symplectic folding and wrapping. In Chapter 8 we review the motivations for Problem ζ and prove Theorem 4 and its generalizations by symplectic folding and by symplectic lifting. In Chapter 9 we give various motivations for the symplectic packing problem, collect the known symplectic packing numbers, and pack balls and ruled symplectic 4-manifolds by hand. In Appendix A we give the well-known proof of the Extension after Restriction Principle and discuss an extension of this principle to unbounded domains. In Appendix B we prove Proposition 2. In Appendix C we clarify the relations between the invariants defined by Problem UB and Problem UC and other symplectic invariants. Appendix D provides computer programs necessary to compute the optimal embeddings of 4-dimensional ellipsoids into a 4-ball and a 4-cube which can be obtained by multiple symplectic folding. In Appendix E we describe some symplectic embedding problems not studied in this book; while some of them are almost solved, others are widely open. Throughout this book we work in the C ∞ -category, i.e., all manifolds and diffeomorphisms are assumed to be C ∞ -smooth, and so are all symplectic forms and maps.

Chapter 2

Proof of Theorem 1

This chapter contains the rigidity results proved in this book. We start with giving a feeling for the difference between linear and non-linear symplectic embeddings. We then look at ellipsoids and use Ekeland–Hofer capacities to prove a generalization of Theorem 1. We finally notice that the polydisc analogues of our rigidity results for ellipsoids are either wrong or unknown. We denote by O(n) the set of bounded domains in R2n diffeomorphic + to a ball. Each U ∈ O(n) is endowed with the standard symplectic structure ω0 = ni=1 dxi ∧ dyi . , 1 n Orienting U by the volume form 0 = n! ω0 we write |U | = U 0 for the usual volume of U . Let D(n) be the group of all symplectomorphisms of R2n and let Sp(n; R) be its subgroup of linear symplectomorphisms of R2n . Define the following relations on O(n): U ≤1 V ⇐⇒ There exists a ϕ ∈ Sp(n; R) with ϕ(U ) ⊂ V . U ≤2 V ⇐⇒ There exists a ϕ ∈ D(n) with ϕ(U ) ⊂ V . U ≤3 V ⇐⇒ There exists a symplectic embedding ϕ : U → V .

2.1

Comparison of the relations ≤i

Clearly, ≤1 ⇒ ≤2 ⇒ ≤3 . It is, however, well known that the relations ≤l are different. Proposition 2.1.1. The relations ≤l are all different. Proof. In order to show that the relations ≤1 and ≤2 are different it suffices to find an area and orientation preserving diffeomorphism ϕ of R2 , ω0 mapping the unit disc D(π) to a set which is not convex. The existence of such a ϕ follows, for instance, from Lemma 3.1.5 below. For the products U = D(π ) × · · · × D(π ) and V = ϕ (D(π)) × · · · × ϕ (D(π)) we then have U ≤1 V and U ≤2 V . The construction of sets U and V ∈ O(n) with U ≤3 V but U ≤2 V relies on the following simple observation. Suppose that U ≤2 V and in addition that |U | = |V |. If ϕ is a map realizing U ≤2 V , no point of R2n \ U can then be mapped to V , and we conclude that ϕ is a homeomorphism from the boundary ∂U of U to the boundary ∂V of V . Following Traynor, [81], we consider now the slit disc SD(π) = D(π) \ {(x, y) | x ≥ 0, y = 0},

24

2 Proof of Theorem 1

and we set U = C 2n (π) = D(π) × · · · × D(π ) and V = SD(π ) × · · · × SD(π ). By Proposition 1 of the introduction, D(π) ≤3 SD(π ), see also Lemma 3.1.5 below. Therefore, U ≤3 V , and clearly |U | = |V |. But ∂U and ∂V are not homeomorphic. 2 We wish to mention that for n ≥ 2 more interesting examples showing that ≤2 and ≤3 are different were found by Eliashberg and Hofer, [23], and by Cieliebak, [15]. In order to describe their examples, we assume that ∂U is smooth. Recall that the characteristic line bundle LU of ∂U is defined as LU = {(x, ξ ) ∈ T ∂U | ω0 (ξ, η) = 0 for all η ∈ Tx ∂U } .

(2.1.1)

The unparametrized integral curves of LU are called characteristics and form the characteristic foliation of ∂U . Moreover, ∂U is said to be of contact type if on a neighbourhood of ∂U there exists a smooth vector field X which is transverse to ∂U ∂ on R2n and meets LX ω0 = dιX ω0 = ω0 . E.g., the radial vector field X = 21 ∂r meets LX ω0 = ω0 , and so all starshaped domains in R2n are of contact type. If U ≤2 V , then the characteristic foliations of ∂U and ∂V are isomorphic, and ∂U is of contact type if and only if ∂V is of contact type. Theorem 1.1 in [23] and its proof show that there exist convex U, V ∈ O(n) with smooth boundaries such that U and V are symplectomorphic and C ∞ -close to the ball B 2n (π ), but the characteristic foliation of ∂U contains an isolated closed characteristic while the one of ∂V does not. And Corollary A in [15] and its proof imply that given any U ∈ O(n), n ≥ 2, with smooth boundary ∂U of contact type, there exists a symplectomorphic and C 0 -close set V ∈ O(n) whose boundary is not of contact type. We in particular see that even for U being a ball, U ≤3 V does not imply U ≤2 V .

2.2

Rigidity for ellipsoids

Proposition 2.1.1 shows that in order to detect some rigidity via the relations ≤l we must pass to a small subcategory of sets: Let E (n) be the collection of symplectic ellipsoids defined in Section 1.2.2, E (n) = {E(a) = E(a1 , . . . , an )},

a = (a1 , . . . , an ),

and write l for the restrictions of the relations ≤l to E (n). Notice again that 1 ⇒ 2 ⇒ 3 .

(2.2.1)

The equivalence (2.2.5) below and the theorems in Section 1.3.1 combined with the Extension after Restriction Principle from Section 1.2.2 show that the relations 1 and 2 are different. The relations 2 and 3 are very similar: Since ellipsoids are starshaped, the Extension after Restriction Principle implies E(a) 3 E(a ) ⇒ E(δa) 2 E(a )

for all δ ∈ ]0, 1[.

(2.2.2)

25

2.2 Rigidity for ellipsoids

It is, however, not known whether 2 and 3 are the same: While Theorem 2.2.4 proves this under an additional condition, the folding construction of Section 3.2 suggests that 2 and 3 are different in general. But let us first prove a general and common rigidity property of these relations: Proposition 2.2.1. The relations l are partial orderings on E (n) . Proof. The relations are clearly reflexive and transitive, so we are left with identitivity, i.e., E(a) l E(a ) and E(a ) l E(a) ⇒ E(a) = E(a ). Of course, the identitivity of 3 implies the one of 2 which, in turn, implies the one of 1 . To prove the identitivity of 3 we use Ekeland–Hofer capacities introduced in [21]. Definition 2.2.2. An extrinsic symplectic capacity on (R2n , ω0 ) is a map c associating with each subset S of R2n a number c(S) ∈ [0, ∞] in such a way that the following axioms are satisfied. A1. Monotonicity: c(S) ≤ c(T ) if there exists ϕ ∈ D(n) such that ϕ(S) ⊂ T . A2. Conformality : c(λS) = λ2 c(S) for all λ ∈ R \ {0}. A3. Nontriviality: 0 < c(B 2n (π)) and c(Z 2n (π )) < ∞. The Ekeland–Hofer capacities form a countable family {cj }, j ≥ 1, of extrinsic symplectic capacities on R2n . For a symplectic ellipsoid E = E(a1 , . . . , an ) these invariants are given by the identity of sets {c1 (E) ≤ c2 (E) ≤ . . .} = {kai | k = 1, 2, . . . ; i = 1, . . . , n} ,

(2.2.3)

see [21, Proposition 4]. Observe that for any l = 1, 2, 3 and λ > 0 E(a) l E(a ) ⇒ E(λa) l E(λa ).

(2.2.4)

This is seen by conjugating the given map ϕ with the dilatation by λ−1 . Recalling (2.2.2) we conclude that for any δ1 , δ2 ∈ ]0, 1[ the postulated relations E(a) 3 E(a ) 3 E(a) imply

E(δ2 δ1 a) 2 E(δ1 a ) 2 E(a).

Now the monotonicity property (A1) of the capacities and the set of relations in (2.2.3) 2 immediately imply that a = a . This completes the proof of Proposition 2.2.1.

26


Remark 2.2.3. In the above proof we derived the identitivity of 1 and 2 from the one of 3 . We find it instructive to give direct proofs. It is well known from linear symplectic algebra [39, p. 40] that E(a) 1 E(a ) ⇐⇒ ai ≤ ai

for all i,

(2.2.5)

in particular 1 is identitive. In order to give an elementary proof of the identitivity of 2 we look at the characteristic foliation on the boundary and at the actions A(γ ) of closed characteristics. To compute the characteristic foliation of ∂E(a) recall that E(a) = H −1 (1), where + 2 H (z) = ni=1 π|zaii | . Using definition (1.1.3) we find XH (z) = −2π J

1

1 a1 z1 , . . . , an zn

√ where J = −1 ∈ C = R2 (x, y) is the standard complex structure. The characteristic on ∂E(a) through z = z(0) can therefore by parametrized as z(t) = (z1 (t), . . . , zn (t)), where zi (t) = e−2π J t/ai zi (0),

i = 1, . . . , n.

If the n numbers a1 , . . . , an are linearly independent over Z, then the only periodic orbits are (0, . . . , 0, zi (t), 0, . . . , 0) with zi (t) = e−2π J t/ai zi (0)

and

π |zi (0)|2 = ai .

In general, the traces of the closed characteristics on ∂E(a) form the disjoint union ∂E(a 1 ) ∪ · · · ∪ ∂E(a d ) where a 1 ∪· · ·∪a d is the partition of a = (a1 , . . . , an ) into maximal linearly dependent , subsets. Recall that the action of a closed characteristic γ is defined as A(γ ) = γ λ, by a(γ ) the smallest subset of a such that γ ⊂ ∂E (a(γ )), where dλ = ω0 . Denoting + and choosing λ = ni=1 xi dyi , we readily compute that A(γ ) is the least common multiple of the elements in a(γ ), A(γ ) = lcm (a(γ )) .

(2.2.6)

Assume now that E(a) 2 E(b) and E(b) 2 E(a). Then there exists a symplectomorphism ϕ of R2n such that ϕ (E(a)) = E(b), and we see as in the proof of Proposition 2.1.1 that ϕ (∂E(a)) = ∂E(b). It follows easily from the definition (2.1.1) of LE(a) and LE(b) that ϕ maps the characteristic foliation on ∂E(a) to the one of ∂E(b). Moreover, the actions of closed characteristics are preserved. Indeed, if γ is a closed characteristic on ∂E(a), we choose a smooth closed disc D ⊂ R2n with boundary γ and find / / / / / λ= ω0 = ϕ ∗ ω0 = ω0 = λ, ϕ(γ )

ϕ(D)

D

D

γ

2.2 Rigidity for ellipsoids

27

so that A (ϕ(γ )) = A(γ ). Denoting the simple action spectrum of E(a) by σ (E(a)) = {A (γ ) | γ is a closed characteristic on ∂E(a)} we in particular have σ (E(a)) = σ (E(b)). If a1 , . . . , an are linearly independent over Z, then ∂E(a) carries only the n closed characteristics ∂E(ai ) with action ai , so that {a1 , . . . , an } = σ (E(a)) = σ (E(b)) = {b1 , . . . , bn } , proving E(a) = E(b). The proof of the general case is not much harder: Of course, a1 = min {σ (E(a))} = min {σ (E(b))} = b1 . Arguing by induction we assume that ai = bi for i = 1, . . . , k − 1. Suppose that ak < bk . We then consider the subsets (a | ak ) ⊂ ∂E(a) and (b | ak ) ⊂ ∂E(b) formed by those closed characteristics whose action divides ak . By the above discussion, ϕ ( (a | ak )) = (b | ak ), so that (a | ak ) and (b | ak ) must be homeomorphic. On the other hand, let {ai1 , . . . , ail } be the set of those ai in {a1 , . . . , ak−1 } which divide ak . We then read off from (2.2.6) that (b | ak ) = ∂E(ai1 , . . . , ail ). Similarly, (a | ak ) = ∂E(ai1 , . . . , ail , ak , . . . , ak+mk −1 ) where mk is the multiplicity of ak in a. In particular, dim (b | ak ) < dim (a | ak ). This contradiction shows that ak ≥ bk . Interchanging a and b we also find ak ≤ bk , so that ak = bk . This completes the 3 induction, and the identitivity of 2 is proved in an elementary way. Recall that 2 does not imply 1 in general. However, a suitable pinching condition guarantees that “linear” and “non linear” coincide: Theorem 2.2.4. Let κ ∈

b

2,b

. Then the following statements are equivalent:

(i) B 2n (κ) 1 E(a) 1 E(a ) 1 B 2n (b), (ii) B 2n (κ) 2 E(a) 2 E(a ) 2 B 2n (b), (iii) B 2n (κ) 3 E(a) 3 E(a ) 3 B 2n (b). We should mention that for n = 2, Theorem 2.2.4 was proved in [26]. Their proof uses a deep result of McDuff, [55], stating that the space of symplectic embeddings of a closed ball into a larger ball is connected, and then uses the isotopy invariance of symplectic homology. However, Ekeland–Hofer capacities provide an easy proof as we shall see. The crucial observation is that capacities have – in contrast to symplectic homology – the monotonicity property. Proof of Theorem 2.2.4. In view of (2.2.1) it is enough to show the implication (iii) ⇒ (i). We start with showing the implication (ii) ⇒ (i). By assumption, B 2n (κ) 2 E(a) 2 B 2n (b).

28


Hence, by the monotonicity of the first Ekeland–Hofer capacity c1 we obtain κ ≤ a1 ≤ b,

(2.2.7)

κ ≤ cn (E(a)) ≤ b.

(2.2.8)

and by the monotonicity of cn

The estimates (2.2.7) and κ > b/2 imply 2a1 > b, whence the only Ekeland–Hofer capacities of E(a) possibly smaller than b are a1 , . . . , an . It follows therefore from (2.2.8) that an = cn (E(a)), whence ci (E(a)) = ai for i = 1, . . . , n. Similarly we find ci (E(a )) = ai for i = 1, . . . , n, and from E(a) 2 E(a ) we conclude ai ≤ ai . (iii) ⇒ (i) now follows by a similar reasoning as in the proof of the identitivity of 3 . Indeed, starting from B 2n (κ) 3 E(a) 3 E(a ) 3 B 2n (b), the implication (2.2.2) shows that for any δ1 , δ2 , δ3 ∈ ]0, 1[ B 2n (δ3 δ2 δ1 κ) 2 E(δ2 δ1 a) 2 E(δ1 a ) 2 B 2n (b). Choosing δ1 , δ2 , δ3 so large that δ3 δ2 δ1 κ > b/2 we can apply the already proved implication to see B 2n (δ3 δ2 δ1 κ) 1 E(δ2 δ1 a) 1 E(δ1 a) 1 B 2n (b), and since δ1 , δ2 , δ3 can be chosen arbitrarily close to 1, the statement (i) follows in view of (2.2.5). This completes the proof of Theorem 2.2.4. 2 In Section 1.2.2 we gave a direct proof of Theorem 1. Here, we show how Theorem 1 follows from Theorem 2.2.4. In the notation of this section, Theorem 1 reads Theorem 2.2.5. Assume that E(a1 , . . . , an ) 3 B 2n (A) for some A < an . Then an > 2a1 . Proof. Arguing by contradiction we assume E(a1 , . . . , an ) 3 B 2n (A) for some A < an and an ≤ 2a1 . A volume comparison shows a1 < A. Hence, a1 ∈ A2 , A . Therefore, B 2n (a1 ) 3 E(a1 , . . . , an ) 3 B 2n (A), Theorem 2.2.4 and the equivalence (2.2.5) imply that an ≤ A. This contradiction shows an > 2a1 , as claimed. 2

2.3

Rigidity for polydiscs ?

The rigidity results for symplectic embeddings of ellipsoids into ellipsoids found in the previous section were proved with the help of Ekeland–Hofer capacities. Recall

29

2.3 Rigidity for polydiscs ?

that P (a1 , . . . , an ) denotes the open symplectic polydisc. We may again assume a1 ≤ a2 ≤ · · · ≤ an . The Ekeland–Hofer capacities of a polydisc are given by cj (P (a1 , . . . , an )) = j a1 ,

j = 1, 2, . . . ,

(2.3.1)

[21, Proposition 5], and so they only see the smallest area a1 . Many of the polydisc analogues of the rigidity results for ellipsoids are therefore either wrong or much harder to prove. It is for instance not true anymore that P (a1 , . . . , an ) embeds into P (A1 , . . . , An ) by a linear symplectomorphism if and only if ai ≤ Ai for all i, as the following example shows. √ Lemma 2.3.1. Assume r > 1 + 2. Then there exists A < πr 2 such that the polydisc P 2n (π, . . . , π, πr 2 ) embeds into the cube C 2n (A) = P 2n (A, . . . , A) by a linear symplectomorphism. Proof. It is enough to prove the lemma for n = 2. Consider the linear symplectomorphism given by 1 (z1 , z2 ) → (z1 , z2 ) = √ (z1 + z2 , z1 − z2 ). 2 For (z1 , z2 ) ∈ P (π, πr 2 ) and i = 1, 2 we have 2 1 1 r2 2 2 z ≤ |z | | | |z | |z |z + r. + + 2 < + 1 2 1 2 i 2 2 2 The right hand side of (2.3.2) is strictly smaller than r 2 provided that r > 1 +

(2.3.2) √

2. 2

Moreover, it is not known whether the full analogue of Proposition 2.2.1 for polydiscs instead of ellipsoids holds true. Let P (n) be the collection of polydiscs P (n) = {P (a1 , . . . , an )} and write l for the restrictions of the relations ≤l to P (n), l = 1, 2, 3. Again 2 and 3 are very similar, and again all the relations l are clearly reflexive and transitive. Furthermore, the smooth part of the boundary ∂P (a1 , . . . , an ) is foliated by closed characteristics with actions a1 , . . . , an , so that the identitivity of 2 and hence the one of 1 follows at once. The identitivity of 2 also follows from a result proved in [26] by using symplectic homology: Symplectomorphic polydiscs are equal. For n = 2, the identitivity of 3 follows from the monotonicity of any symplectic capacity, which show that the smaller discs are equal, and from the equality of the volumes, which then shows that also the larger discs are equal. For n ≥ 3, however, we do not know whether the relation 3 is identitive. In particular, we have no answer to the following question.

30


Question 2.3.2. Assume that there exist symplectic embeddings P (a1 , a2 , a3 ) → P (a1 , a2 , a3 ) and P (a1 , a2 , a3 ) → P (a1 , a2 , a3 ). Is it then true that a2 = a2 and a3 = a3 ? We also do not know whether the polydisc-analogue of Theorem 1 or of Theorem 2.2.4 holds true. The symplectic embedding results proved in the subsequent chapters will suggest, however, that the polydisc-analogue of Theorem 1 holds true, see Conjecture 7.2.4.

Chapter 3

Proof of Theorem 2

In this chapter we prove Theorem 2 by symplectic folding. After reducing Theorem 2 to a symplectic embedding problem in dimension 4, we construct essentially explicit symplectomorphisms between 2-dimensional simply connected domains. This construction is important for the symplectic folding construction, which is described in detail in Section 3.2. While symplectic folding will be the main tool until Chapter 8, the construction of explicit 2-dimensional symplectomorphisms will be basic also for the symplectic packing constructions given in Chapter 9. This chapter almost coincides with the paper [73].

3.1

Reformulation of Theorem 2

Recall from the introduction that the ellipsoid E(a1 , . . . , an ) is defined by 6 .n E(a1 , . . . , an ) = (z1 , . . . , zn ) ∈ Cn

7 π |zi |2 2π. Then E (π, . . . , π, a) symplectically embeds into 2n B 2 + π + for every > 0.

The symplectic folding construction of Lalonde and Mc Duff considers a 4-ellipsoid as a fibration of discs of varying size over a disc and applies the flexibility of volume preserving maps to both the base and the fibres. It is therefore purely four dimensional in nature. We will refine the method in such a way that it allows us to prove Theorem 3.1.1 for every n ≥ 2. We shall conclude Theorem 3.1.1 from the following proposition in dimension 4. Proposition 3.1.2. Assume a > 2π . Given > 0 there exists a symplectic embedding

a +π + : E(a, π) → B 4 2 satisfying π |(z1 , z2 )|2
0 the unique solution of h(r, t0 (r)) = 2π. Substituting ϕ = h(r, t) into formula (3.1.2) we obtain, using d det β (reih(r,t) ) · dt h(r, t) = 1, that / 2π =

t0 (r)

dt = t0 (r).

0

Hence h(r, 2π) = 2π. Therefore, the two functions in t, h(r, t +2π )−2π and h(r, t), solve the same initial value problem (3.1.3), and so the claim (3.1.4) follows. The desired diffeomorphism is now defined by α : U \ {0} → V \ {p},

reiϕ → β(reih(r,ϕ) ).

It is area preserving. Indeed, representing α as the composition reiϕ → (r, ϕ) → (r, h(r, ϕ)) → reih(r,ϕ) → β(reih(r,ϕ) )

34


we obtain for the determinant of the Jacobian 1 ∂h · (r, ϕ) · r · det β (reih(r,ϕ) ) = 1, r ∂ϕ where we again have used (3.1.3). Finally, α is a translation in a punctured neighbourhood of the origin and thus smoothly extends to the origin. This finishes the proof of Lemma 3.1.5. 2 Consider a bounded domain U ⊂ C and a continuous function f : U → R>0 . The set F (U, f ) in C2 defined by

F (U, f ) = (z1 , z2 ) ∈ C2 | z1 ∈ U, π |z2 |2 < f (z1 ) is the trivial fibration over U having as fibre over z1 the disc of capacity f (z1 ). Given two such fibrations F (U, f ) and F (V , g), a symplectic embedding ϕ : U → V defines a symplectic embedding ϕ × id : F (U, f ) → F (V , g) if and only if f (z1 ) ≤ g(ϕ(z1 )) for all z1 ∈ U . Examples 3.1.7. 1. The ellipsoid E(a, b) can be represented as π |z1 |2 . E(a, b) = F D(a), f (z1 ) = b 1 − a 2. Define the open trapezoid T (a, b) by T (a, b) = F (R(a), g), where R(a) = { z1 = (u, v) | 0 < u < a, 0 < v < 1 } is a rectangle and g(z1 ) = g(u) = b(1 − u/a). We set T 4 (a) = T (a, a). The example is inspired by [49, p. 54]. It will be very useful to think of T (a, b) as depicted in Figure 3.1. 3 fibre capacity b

a

u

Figure 3.1. The trapezoid T (a, b).

In order to reformulate Proposition 3.1.2 we shall prove the following lemma which later on allows us to work with more convenient “shapes”.

35

3.1 Reformulation of Theorem 2

Lemma 3.1.8. Assume > 0. Then (i) E(a, b) symplectically embeds into T (a + , b + ), (ii) T 4 (a) symplectically embeds into B 4 (a + ). Proof. (i) Set = a 2 /(ab +a +b). We are going to use Lemma 3.1.5 to construct an area preserving diffeomorphism α : D(a) → R(a) such that for the first coordinate in the image R(a), u(α(z1 )) ≤ π |z1 |2 +

for all z1 ∈ D(a),

(3.1.5)

see Figures 3.2 and 3.3. v 1

1 2

L1 L0

L1

2

4a 3 4 2 4

u a

Figure 3.2. Constructing the embedding α.

In an “optimal world” we would choose the loops Lˆ u , 0 < u < a, in the image R(a) as the boundaries 0), (u, 1). If 1), (u, of the rectangles with corners (0, 0), (0, ˆ we would then have u α(z ˆ 1 ) ≤ π |z1 |2 for the family Lˆ = Lˆ u induced a map α, all (z1 , z2 ) ∈ R(a). The non admissible family Lˆ can be perturbed to an admissible family L in such a way that the induced map α satisfies the estimate (3.1.5). Indeed, choose the translation disc appearing in the proof of Lemma 3.1.5 as the disc of radius /8 centred at (u0 , v0 ) = 2 , 21 . For r < /8 the loops L(r) are therefore the circles centred at (u0 , v0 ). In the following, all rectangles considered have edges parallel to the coordinate axes. We may thus describe a rectangle by specifying its lower left and upper right corner. Let L0 be the boundary of the rectangle with corners 4 , 4a and 3 , and let , 1 − L be the boundary of R(a). We define a family of loops Ls 1 4 4a by linearly interpolating between L0 and L1 , i.e., Ls is the boundary of the rectangle with corners (1 − s) , (1 − s) and us , 1 − + s , s ∈ [0, 1], 4 4a 4a 4a

36


where us = 34 + s a − 34 . Since us < a, the area enclosed by Ls is estimated from below by 3 1−2 > us − . (3.1.6) us − 4 4a 4 Let {Ls }, s ∈ [0, 1[, be the smooth family of smooth loops obtained from {Ls } by smoothing the corners as indicated in Figure 3.2. By choosing the smooth corners of 1 Ls more and more rectangular as s → 1, we can arrange that the set 0<s 0, there exists a symplectic embedding

a + π + , (z1 , z2 ) → (z1 , z2 ), : T (a, π ) → T 4 2 z1 = (u, v) and z1 = (u , v ), satisfying u + π|z2 |2
2π , where c = 1 − /π. As in the proof of Lemma 3.1.8 we can construct a symplectic embedding α × id : E(ca, cπ ) → T (ca + , cπ + ) = T (ca + , π) satisfying the estimate u(α(z1 )) ≤ π |z1 |2 +

a( )2 caπ + a + π

for all z1 ∈ D(ca)

(3.1.10)

and another symplectic embedding ca

ca

ω × id : T 4 + π + → B 4 + π + 2 2 2 satisfying π |ω(z1 )|2 ≤ u +

for all z1 = (u, v) ∈ R

ca 2

+ π + .

(3.1.11)

Since ca + > 2π, Proposition 3.1.9 applied to ca + replacing a and /2 replacing guarantees a symplectic embedding ca

: T (ca + , π) → T 4 + π + , 2 (z1 , z2 ) → (1 (z1 , z2 ), 2 (z1 , z2 )), satisfying u (1 (α(z1 ), z2 )) + π |2 (α(z1 ), z2 )|2
2π . Now choose > 0 so small √ √ that π+3 < π + . We denote the dilatation by c in R4 also by c, and define c √ −1 √ ˆ : E(a, π) → R4 by = c c. Then symplectically embeds E(a, π) a π+2 a 4 4 into B 2 + c ⊂ B 2 +π + , and since π |z1 |2 < a for all (z1 , z2 ) ∈ E(a, π) and by the choice of , 2 √ π √ ˆ c z1 , c z2 c 1 ca π 2 |z1 |2 + 3 + + π c|z2 |2 < c 2 a

π |(z1 , z2 )|2 =

a 3 1 π 2 |z1 |2 + + + π |z2 |2 2 c c a a π 2 |z1 |2 < ++ + π |z2 |2 2 a =

for all (z1 , z2 ) ∈ E(a, π). This proves the required estimate (3.1.9), and so the proof of Corollary 3.1.10 is complete. 2 It remains to prove Proposition 3.1.9. This is done in the following two sections.

3.2 The folding construction The idea in the construction of an embedding as in Proposition 3.1.9 is to separate the small fibres from the large ones and then to fold the two parts on top of each other. As in the previous section we denote the coordinates in the base and the fibre by z1 = (u, v) and z2 = (x, y), respectively.

40


Step 1. Following [49, Lemma 2.1] we first separate the “low” regions over R(a) from the “high” ones. We may do this using Lemma 3.1.5. We prefer, however, to give an explicit construction. Let δ > 0 be small. Set F = F (U, f ), where U and f are described in Figure 3.4, and write 6 7 a P1 = U ∩ u ≤ + δ , 2 6 7 a+π + 11δ , P2 = U ∩ u ≥ 2 L = U \ (P1 ∪ P2 ).

v U 1 P1

δ

P2

L

u f π π 2

u a +δ 2

a+π + 11δ 2

a + π2 + 12δ

Figure 3.4. Separating the low fibres from the large fibres.

Hence, U is the disjoint union U = P1

2

L

2

P2 .

Choose a smooth function h : [0, a + δ] → ]0, 1] as in Figure 3.5, i.e. (i) h(w) = 1 for w ∈ 0, a2 , (ii) h (w) < 0 for w ∈ a2 , a2 + δ 2 , (iii) h a2 + δ 2 = δ, (iv) h(w) = h(a − w) for all w ∈ [0, a + δ].

41

3.2 The folding construction h(w)

1

δ w a +δ 2

a a + δ2 2 2

Figure 3.5. The function h.

By (ii), (iii) and (iv) we have that / a +δ 2 2 1 dw < δ a h(w) 2

/ and

a 2 +δ a 2 2 +δ−δ

1 dw < δ. h(w)

(3.2.1)

We may thus further require that (v) h(w) < δ for w ∈ a2 + δ 2 , a2 + δ − δ 2 , (vi)

,

a 2 +δ a 2

1 h(w)

dw =

π 2

+ 12δ.

Consider the map

/

β : R(a) → R , 2

(u, v) → 0

u

1 dw, h(u)v . h(w)

Clearly, β is a symplectic embedding. We see from (i), (iv) and (vi) that

π β u< a = id and β u> a +δ = id + + 11δ, 0 . { } { } 2 2

(3.2.2)

2

These identities and the estimates (3.2.1) and (v) imply that β embeds R(a) into U (cf. Figure 3.6, where the black region in R(a) is mapped to the black region in U , and so on). Finally, by construction, β × id symplectically embeds T (a, π ) into F . Step 2. We next map the fibres into a convenient shape. Using Lemma 3.1.5 in a similar way as it was used in the proof of Lemma π 3.1.8 we find a symplectomorphism σ mapping D(π) to the rectangle Re and D 2 to the rectangle with smooth corners Ri as specified in Figure 3.7. We require that for z2 ∈ D π2 π

π|z2 |2 + 2δ > y(σ (z2 )) − − − 2δ , 2

42


v

v

1 β u a a +δ 2 2

u a 2

a

a+π + 12δ 2

a + π2 + 11δ

Figure 3.6. The embedding β : R(a) → U .

i.e. y(σ (z2 )) < π |z2 |2 −

π 2

for z2 ∈ D

π 2

(3.2.3)

.

Write for the resulting bundle (id × σ )F of rectangles with smooth corners 2 2 (id × σ )F = S = S(P1 ) S(L) S(P2 ). In order to fold S(P2 ) over S(P1 ) we first move S(P2 ) along the y-axis and then turn it in the z1 -direction over S(P1 ). z2 y − 21

σ

1 2

−δ Ri Re

x

− π2 − 2δ −π

Figure 3.7. Preparing the fibres.

Step 3. In order to move S(P2 ) along the y-axis we follow again [48, p. 355]. Let c : R → [0, 1 − 2δ] be a smooth cut off function as in Figure 3.8: 0, t ≤ a2 + 2δ and t ≥ a+π 2 + 10δ, c(t) = 1 − 2δ, a2 + 3δ ≤ t ≤ a+π 2 + 9δ,

43

3.2 The folding construction c(t)

1 − 2δ

t a + 2δ 2

a+π + 10δ 2

Figure 3.8. The cut off c.

Set I (t) =

,t 0

c(s) ds and define the diffeomorphism ϕ : R4 → R4 by 1 , y + I (u) . ϕ(u, x, v, y) = u, x, v + c(u) x + 2

We then find for the derivative

I 0 dϕ(u, x, v, y) = 2 A I2

with A =

(3.2.4)

∗ c(u) , c(u) 0

whence ϕ is a symplectomorphism in view of the criterion in [39, p. 5]. Moreover, + 10δ , we find defining the number I∞ by I∞ = I a+π 2 ϕ u≤ a +2δ = id { 2 } and assuming that δ
λ3 l ≤ δ /3. a 0 so small a Similar to Figure 3.5 we choose a smooth function h : 2 + δ, 2 + 2δ → ]0, 1] such that (i) h(u) = 1 for u near a2 + δ and u near a2 + 2δ, λ3 l a 3δ (ii) h(u) = λδ3 for u ∈ a2 + 3δ 2,2 + 2 + δ , (iii)

,

a 3δ 2+ 2 a 2 +δ

1 h(w)

dw = δ

,

and

a 2 +2δ a 3δ λ3 l 2+ 2 + δ

1 h(w)

dw =

δ 2

.

+ δ, u0 + l + 2δ × [0, δ] defined by / u 1 a +δ+ dw, h(u)v (u, v) → a 2 h(w) 2 +δ

The embedding γδ : B →

a

2

and illustrated in Figure 3.10 is symplectic. δ

δ γδ δ 2 λ3 l

u

λ3 a +δ 2

u0

s u0 + l u0 + l + 2δ

Figure 3.10. The map γδ .

We now map the image of γδ to a domain B in the (u , v )-plane as painted in Figure 3.9: By the choice of l we may require that the part of the “outer” boundary of

46


B between (u0 , 0) and (u1 , 1), which contains (u1 , 0), is smooth, has length l, and is parametrized by ζ (s), where the parameter s ∈ I := [u0 , u0 + l] is arc length and ζ (s) = (s, 0) ζ (s) = (u1 + u0 + l − s, 1)

on [u0 , u1 ] , δ on u0 + l − , u0 + l . 4

(3.2.10)

Denote the inward pointing unit normal vector field along ζ by ν. We choose λ1 > 0 so small that η : I × [0, λ1 ] → R2 , (s, t) → ζ (s) + t ν(s) is an embedding. In order to make the map area preserving, we consider the initial value problem ∂f ∂t (s, t) = 1/ det dη(s, f (s, t)) (3.2.11) f (s, 0) = 0 in which s ∈ I is a parameter. The existence and uniqueness theorem for ordinary differential equations with parameters yields a smooth solution f on I × [0, λ2 ] for some λ2 > 0. Then f (s, t) < λ1 for all (s, t) ∈ I × [0, λ2 ]. This and the second equation in (3.2.11) imply that the composition η

γζ : (s, t) → (s, f (s, t)) → (u , v ) is a diffeomorphism of I ×[0, λ2 ] onto half of a tubular neighbourhood of ζ . Moreover, by the first equation in (3.2.11), det γζ (s, t) =

∂f (s, t) det dη(s, f (s, t)) = 1, ∂t

i.e., γζ is area preserving. In view of the identities (3.2.10) for ζ , the map γζ is the identity in R2 for s near u0 and t ∈ [0, λ2 ], and γζ is an isometry for s near u0 + l and t ∈ [0, λ2 ]. We now choose the parameter λ3 > 0 in the construction of γδ smaller than λ2 . Restrict γζ to the gray region I ×]0, λ3 [ in the image of γδ , and let γ ζ be the smooth extension of γζ to the image of γδ which is the identity on {u ≤ u0 } and an isometry on {u ≥ u0 + l}. By (i), the composition γ ζ γδ is the identity near u = a/2 + δ and an isometry near u = a/2 + 2δ. It thus smoothly fits with the map γ |U \B already defined at the beginning of this step. Step 5. We finally adjust the fibres. In view of the constructions in Step 2 and Step 3, the projection of the image ϕ(S) to the z2 -plane is contained in a tower shaped domain T (cf. Figure 3.11), and by the second inequality in (3.2.6) we have T ⊂ { (x, y) | y < π2 + 4δ }. Using once more our Lemma 3.1.5 we construct a symplectomorphism τ from a neighbourhood of T to a disc such that the preimages

47

3.3 End of the proof

of the concentric circles in the image are as in Figure 3.11. We require that for z2 = (x, y), π π for y ≥ − − 2δ, (3.2.12) π |τ (z2 )|2 < y + + 3δ 2 2 π (3.2.13) π|τ (z2 )|2 < π |σ −1 (z2 )|2 + + 8δ for z2 ∈ Re , 2 where σ : D(π) → Re is the diffeomorphism constructed in Step 2. y

π + 4δ 2

T x

− π2 − 2δ

−π

Figure 3.11. Mapping the tower to a disc.

Step 1 to Step 5 are the ingredients of our folding construction. The folding map : T (a, π) → R4 is defined as the composition of maps = (id ×τ ) (γ × id) ϕ (id × σ ) (β × id) = (γ × τ ) ϕ (β × σ ). (3.2.14)

3.3

End of the proof

Recall that it remains Proposition 3.1.9. So let > 0 be as in Proposition 3.1.9 1 toprove

, 15 . We define the desired map as in (3.2.14). It remains to and set δ = min 15 verify that meets the required estimate (3.1.8). So let z = (z1 , z2 ) = (u, v, x, y) ∈ T (a, π) and write (z) = (u , v , z2 ). By the choice of δ it suffices to show that for all (u, v, z2 ) ∈ T (a, π ) πu a + π |z2 |2 − π|z2 |2 < + 15δ. (3.3.1) u − a 2

48


We distinguish 1 1three cases according to the locus of the image β(z1 ) in the set U = P1 L P2 (see Figure 3.4 and Figure 3.6). We denote the u-coordinate of β(z1 ) = β(u, v) by u (β(u, v)). Case 1. β(z1 ) ∈ P1 . The first identity in (3.2.5) implies ϕ|S(P1 ) = id, and Step 4 implies γ |S(P1 ) = id. Therefore, u = u (β(u, v)). Moreover, u (β(u, v)) a2 . Summarizing, we have u < u + δ. Using again ϕ|S(P1 ) = id we find σ (z2 ) ∈ Re and z2 = τ (σ (z2 )). Hence, the estimate (3.2.13) for the map τ yields π|z2 |2 = π |τ (σ (z2 ))|2 < π |σ −1 (σ (z2 ))|2 +

π π + 8δ = π |z2 |2 + + 8δ. 2 2

Finally, we have u ≤ a2 + δ. Indeed, if u > a2 + δ, then the second identity in (3.2.2) implies β(u, v) ∈ P2 . Altogether we can estimate π πu π + π |z2 |2 − π|z2 |2 < u 1 − + δ + + 8δ u − a a 2 π π a 1− + + 10δ < 2 a 2 a = + 10δ. 2 Case 2. β(z1 ) ∈ P2 . We have ϕ|S(P2 ) = id +(0, 0, 0, I∞ ) by the second identity in (3.2.5), and so, in view of the identity (3.2.8), u = u (γ (β(z1 ))) = a + π2 + 12δ − u (β(u, v)). Moreover, u (β(u, v)) > u + π2 + 10δ. Indeed, the definition of β shows that if u (β(u, v)) ≥ a+π then u (β(u, v)) = u + π2 + 11δ, and if 2 + 12δ, a+π a+π u (β(u, v)) ∈ 2 + 11δ, 2 + 12δ , then u < a2 + δ. Summarizing, we have u < a − u + 2δ. Step 2 shows σ (z2 ) ∈ Ri , and so y σ (z2 )+(0, I∞ ) ≥ − π2 −2δ. Hence, the estimates (3.2.12), (3.2.3) and (3.2.6) imply 2 π|z2 |2 = π τ σ (z2 ) + (0, I∞ ) π < y(σ (z2 )) + I∞ + + 3δ 2

π π π 2 + + 5δ + + 3δ < π|z2 | − 2 2 2 π 2 = π|z2 | + + 8δ. 2

49

3.3 End of the proof

Finally, we have u ≥ a2 . Indeed, if u < a2 , then the first identity in (3.2.2) implies β(u, v) ∈ P1 . Altogether we can estimate π πu π u − + π |z2 |2 − π |z2 |2 < a − u 1 + + 2δ + + 8δ a a 2 π π a 1+ + + 10δ ≤a− 2 a 2 a = + 10δ. 2 Case 3. β(z1 ) ∈ L. Using the definition of ϕ, the estimate (3.2.9) implies u < −u (β(u, v)) +

π + a + 13δ. 2

Since π|z2 |2 < π2 , we have σ (z2 ) ∈ Ri , cf. Figure 3.7. In particular, y σ (z2 ) + (0, I (u (β(u, v)))) ≥ − π2 − 2δ. Hence, the estimates (3.2.12) and (3.2.3) and the estimate I (t) < (1 − 2δ)(t − ( a2 + 2δ)) read off from Figure 3.8 yield 2 π|z2 |2 = π τ x(σ (z2 )), y(σ (z2 )) + I (u (β(u, v))) π < y(σ (z2 )) + I (u (β(u, v))) + + 3δ 2

π π a 2 + (1 − 2δ) u (β(u, v)) − − 2δ + + 3δ < π|z2 | − 2 2 2 a 2 = π |z2 | + u (β(u, v)) − − 2δ − 2δu (β(u, v)) + δa + 4δ 2 + 3δ. 2 Finally, we have u (β(u, v)) > a2 + δ by the definition of L, and u ≥ identity in (3.2.2). Altogether we can estimate u −

a 2

by the first

πa πu π + π |z2 |2 − π|z2 |2 < −u (β(u, v)) + + a + 13δ − a 2 aa 2 a +δ + u (β(u, v)) − − 2δ − 2δ 2 2 + δa + 4δ 2 + 3δ a = + 14δ + 2δ 2 2 a < + 15δ, 2

where in the last step we have used that 2δ 2 < δ which follows from δ
a2 } to the light triangle and the triangle {π |z2 |2 > π2 } to the dark triangle. Notice that the wording “folding” is not completely appropriate for what is going on: The triangle {u > a2 } is not really “turned over”, but rather first “lifted” by ϕ and then “turned around” by γ × id . We did not find a better wording, though. 3

A

π

u A

a

Figure 3.13. How the parts of the ellipsoid are mapped.

Chapter 4

Multiple symplectic folding in four dimensions

In four dimensions we shall exploit the great flexibility of symplectic maps which only depend on the fibre coordinates to provide rather satisfactory embedding results for simple shapes. We first discuss a modification of the symplectic folding construction described in Section 3.2, then explain multiple folding, and finally calculate the optimal embeddings of ellipsoids and polydiscs into balls and cubes which can be obtained by these methods. In order not to disturb the exposition unnecessarily with the arbitrarily small δ-terms (arising from “rounding off corners” and so on) we shall skip them in the sequel. Since all the sets under consideration will be bounded and all constructions will involve only finitely many steps, we will not lose control of the δ-terms.

4.1

Modification of the folding construction

The map σ in Step 2 of the folding construction given in Section 3.2 was dictated by the estimate (3.1.8) used to obtain the 2n-dimensional result, n > 2. As a consequence, the map ϕ of Step 3 had to disjoin the z2 -projection of S(P2 ) from the one of S(P1 ), and we ended up with the unused white sandwiched triangle in Figure 3.13. In order to use this room for n = 2, in which case the estimate (3.1.8) is not necessary, we modify the folding construction as follows: Replace the map σ of Step 2 by the map σ given by Figure 4.1. If we define ϕ as in (3.2.4), the z2 -projection of the image of ϕ will almost coincide with the image of σ . Choose now γ as in Step 4 and define the final map τ on a neighbourhood of the image of σ such that it restricts to σ −1 on the image of σ . If all the δ’s were chosen appropriately, the composite map defined as in (3.2.14) will be one-to-one, and the image of will be contained in T 4 (a/2 + π + ) for some small . The map can be visualized as in Figure 4.2. Essentially, restricts to theidentity on the floor F1 and maps the white triangle with vertices a2 , 0 , (a, 0), a2 , π2 to the floor F2 .

53

4.2 Multiple folding y z2

π 2

σ

x − 21

1 2

Figure 4.1. The modified map σ .

A

π π 2

F2 F1

u a 2

A

a

Figure 4.2. Folding in four dimensions.

4.2

Multiple folding

Neither Theorem 2 nor Traynor’s Wrapping Theorem tells us whether the ellipsoid E(π, 4π) symplectically embeds into the ball B 4 (A) for some A ≤ 3π (cf. Figure 7.1, p. 159). Multiple symplectic folding, which is explained in this section, will provide a symplectic embedding of E(π, 4π ) into B 4 (2.692 π ). To understand the general construction it is enough to look at a 3-fold. Up to the final fibre adjusting map τ , the folding map is then the composition of maps explained in Figure 4.3, in which the pictures are to be understood as in Figure 3.12: The horizontal direction refers to the base and the vertical direction indicates the fibres. Here are the details: Fix a > π and view the ellipsoid E(a, π) as the trapezoid T (a, π ), which fibres over the rectangle R(a) = {(u, v) | 0 0 satisfying + 4 i=1 ui = a; the ui will be specified in 4.3.1 for embedding E(a, π) into a ball and

54

4 Multiple symplectic folding in four dimensions β × id

id ×σ

ϕ

γ1 × id

F3 S3

S2 F2

S1

F4 γ3 × id

γ2 × id

F1

Figure 4.3. Multiple folding.

in 4.4.1 for embedding E(a, π) into a cube. Then define the heights li , i = 1, 2, 3, by li = π −

i π. uj , a

i = 1, 2, 3.

j =1

Step 1 (Separating smaller fibres from larger ones). Let U and f be as in Figure 4.4. Proceeding as in Step 1 of the folding construction in Section 3.2 we find a symplectic embedding β : R(a) → U such that (β × id)(T (a, π )) ⊂ F (U, f ). Step 2 (Preparing the fibres). The map σ is explained in Figure 4.5. More precisely, σ maps the central black disc to the black disc D, and up to some neglected δ-terms we have  l1 − l2 + l3 − π |z2 |2 for most z2 ∈ D (l3 ) \ D,      l1 − l2 + π |z2 |2 for most z2 ∈ D (l2 ) \ D (l3 ) , y(σ (z2 )) = 2   l1 − π |z2 | for most z2 ∈ D (l1 ) \ D (l2 ) ,     π |z2 |2 for most z2 ∈ D (π ) \ D (l1 ) . In general, when folding n times, σ maps the circles in the n + 1 annuli around a small central disc alternately to rectangular loops with essentially constant maximal but decreasing minimal y-coordinate and to rectangular loops with essentially constant minimal but increasing maximal y-coordinate.

55

4.2 Multiple folding v U 1 P1

P3

P2

P4 u

L1

L2

L3

f π l1 l2 l3 u1

u2

l1

l2 u3 l3

u

u4

Figure 4.4. F (U, f ). y z2 l1

σ

l1 − l2 + l3 l1 − l2 x − 21

1 2

Figure 4.5. The map σ .

Step 3 (Lifting the+ fibres). Choose cut off ,functions ci over Li , i = 1, 2, 3, and t 3 abbreviate c(t) = i=1 ci (t) and I (t) = 0 c(s) ds. The symplectic embedding 4 ϕ : (β × σ )(T (a, π)) → R is defined as in (3.2.4) by

1 ϕ(u, x, v, y) = u, x, v + c(u) x + , y + I (u) . 2 Step 4 (Folding). Step 4 in Section 3.2 now requires three steps.

56

4 Multiple symplectic folding in four dimensions

1. The folding map γ1 is essentially the map γ of Section 3.2: On the part of the base denoted by P1 it is the identity, for u1 ≤ u ≤ u1 + l1 it looks like the map in Figure 3.9, and for u > u1 + l1 it is an isometry. Observe that by construction, the slope of the stairs S2 is 1, while the slope of the upper edge of the floor F1 is π/a < 1. The sets S2 and F1 are thus disjoint. map, but 2. The map γ2 × id is not really a global product 1 1 restricts to a product on certain pieces of its domain: It is the identity on F1 S1 F2 , and it is the product γ2 × id on the remaining domain, where γ2 is explained in Figure 4.6: It is the identity on the gray part of its domain, maps the black square to the black part of its image, and is an isometry on {u ≤ 0}. The map γ2 is constructed the same way as the map γ in Section 3.2.

u 0 γ2

u

Figure 4.6. Folding on the left.

3. The map γ3 × id, which turns F4 over F3 , is analogous to the map γ1 × id: It is an isometry on F4 , looks like the map given by Figure 3.9 on S3 , and restricts to the identity everywhere else. Step 5 (Adjusting the fibres). The z2 -projection of the image of ϕ is a tower shaped domain T . The final map τ is a symplectomorphism from a small neighbourhood of

57

4.3 Embeddings into balls

T to a disc. It is enough to choose τ in such a way that up to some neglected δ-term we have for any z2 = (x, y), z2 = (x , y ) ∈ T y < y ⇒ |τ (z2 )|2 < |τ (z2 )|2 . This finishes the 3-fold folding construction. The n-fold folding construction is analogous. Here, we have u1 , . . . , un+1 , heights li = π −

i π. uj , a

i = 1, . . . , n,

(4.2.1)

j =1

floors F1 , . . . , Fn+1 , and stairs S1 , . . . , Sn . The thin stairs require some locus in space specified in the following lemma, which is proved in Steps 4.1 and 4.2 above. Folding Lemma 4.2.1. Let Si be the stairs connecting the floors Fi and Fi+1 of minimal respectively maximal height li , i = 1, . . . , n. (i) If i is odd, then the stairs Si are contained in a trapezoid with horizontal lower edge of length li , left edge of length 2li , and right edge of length li , cf. Figure 4.7 (i); moreover, no smaller trapezoid contains Si . (ii) If i is even, then the stairs Si are contained in a trapezoid with horizontal upper edge of length li , left edge of length li , and right edge of length 2li , cf. Figure 4.7 (ii); moreover, no smaller trapezoid contains Si . li li

Fi+1 l i

Si

Si Fi

li F i+1 li

li

li

Fi

li

(i) Folding on the right.

(ii) Folding on the left.

Figure 4.7. The locus of the stairs.

4.3

Embeddings into balls

In this section we use multiple symplectic folding to construct good embeddings of four dimensional ellipsoids and polydiscs into four dimensional balls.

58


4.3.1 Embedding ellipsoids into balls. Recall from Theorem 1 that if a ≤ 2π then the ellipsoid E(π, a), which is symplectomorphic to E(a, π), does not symplectically embed into B 4 (A) if A < a. We therefore fix a > 2π. We again think of E(a, π) as T (a, π) and of B 4 (A) as T 4 (A) = T (A, A). In order to find the smallest trapezoid T 4 (A) into which T (a, π) embeds via multiple symplectic folding we have to choose the ui ’s, which appeared in Section 4.2, optimally. Our strategy to do so is as follows. We shall describe a procedure which associates with each u1 ∈ ]0, a[ the number 2π (4.3.1) u1 , A(a, u1 ) = 2π + 1 − a and either a finite sequence u2 , u3 , . . . , uN+1 and the attribute admissible, or an empty or finite sequence u2 , u3 , . . . , uN and the attribute non-admissible. In both cases the number N = N (u1 ) will depend only on u1 , and the procedure will describe a symplectic embedding of T (a, π) into T 4 (A(a, u1 )) by (N -fold) symplectic folding if and only if u1 is admissible. In view of a > 2π and equation (4.3.1) we have to look for the smallest admissible u1 . As we shall see, u1 is non-admissible if u1 ≤ aπ/(a + π ), and u1 is admissible if u1 > a/2. Moreover, we will show that if u1 is admissible, then u1 is admissible for any u1 > u1 . It follows that there is a unique u0 = u0 (a) such that u1 is admissible if u1 > u0 and u1 is non-admissible if u1 < u0 . Therefore, our procedure yields a symplectic embedding of T (a, π ) into T 4 (A(a, u0 ) + ) for any > 0. Finally, we will explain why our procedure is optimal, i.e., multiple symplectic folding does not yield an embedding of T (a, π ) into T 4 (A) if A ≤ A(a, u0 ). We start with describing our procedure. Fix u1 ∈ ]0, a[ and fold at u1 . The minimal height of the first floor F1 is then l1 = π − (π/a)u1 . As suggested in Figure 4.8, we define A(a, u1 ) by the condition that the second floor F2 touches the “upper right boundary” of T 4 (A(a, u1 )), i.e., 2π u1 . A(a, u1 ) = u1 + 2 l1 = 2π + 1 − a We now successively try to choose ui , i ≥ 2, in such a way that ui is maximal and the stairs Si are contained in T 4 (A(a, u1 )). Define the remaining length r1 by r1 = a −u1 . The image of T (a, π) after folding at u1 is contained in T 4 (A(a, u1 )) if and only if r1 < u1 , i.e., u1 > a/2. If r1 < u1 , we set u2 = r1 , and u1 is admissible. Indeed, the data u1 , u2 then describe a symplectic embedding of T (a, π ) into T 4 (A(a, u1 )) obtained by folding once. Since u1 > a/2, these embeddings are not better than the one constructed in Section 3.2. If r1 ≥ u1 , we are forced to fold a second time. Assume that we fold at u1 − u2 > 0, i.e., the second floor F2 has length u2 . Then the height of F2 at u1 − u2 is l2 = l1 − (π/a)u2 . If l1 ≥ u1 , then l2 ≥ u1 − (π/a)u2 > u1 − u2 , and so the Folding Lemma 4.2.1 (ii) shows that the stairs S2 are not contained in {u > 0}. A necessary condition that S2 is contained in T 4 (A(a, u1 )) is therefore l1 < u1 . In view of l1 = π − (π/a)u1 this condition is equivalent to the condition on u1 u1 >

aπ . a+π

(4.3.2)

59


A(a, u1 )

l l2 2 π

l2

l1 l1 u1

u1 + l1

u A(a, u1 )

a

Figure 4.8. A 12-fold.

If this condition is not met, then u1 is non-admissible. If l1 < u1 , then the Folding Lemma 4.2.1 (ii) shows that for u2 small enough we have S2 ⊂ {u > 0}, and that the maximal such u2 is characterized by the equation l2 + u2 = u1 , which by (4.2.1) translates into the formula u2 =

a+π aπ u1 − . a−π a−π

(4.3.3)

We define u2 by (4.3.3). Then S2 ⊂ {u > 0}, but it is still possible that S2 is not contained in T 4 (A(a, u1 )), in which case u1 is non-admissible. Assume that S2 ⊂ T 4 (A(a, u1 )). We denote the length a−u1 −u2 = r1 −u2 of the new remainder by r2 . If r2 < u2 , we set u3 = r2 , and u1 is admissible. Indeed, the data u1 , u2 , u3 then describe a symplectic embedding of T (a, π ) into T 4 (A(a, u1 )) obtained by folding twice. If r2 ≥ u2 , we are forced to fold a third time. Assume that we fold at l2 +u3 , i.e., the third floor F3 has length u3 , and its height at l2 + u3 is l3 = l2 − (π/a)u3 . If u2 ≤ 2 l2 , then u2 − u3 ≤ 2 l2 − u3 < 2 l2 − (2π/a)u3 = 2 l3 , and so the Folding Lemma 4.2.1 (i) shows that the stairs S3 are not contained in T 4 (A(a, u1 )). A necessary condition that S3 is contained in T 4 (A(a, u1 )) is therefore u2 > 2 l2 , and if this condition is not met, u1 is non-admissible. If u2 > 2 l2 , the Folding Lemma 4.2.1 (i) shows that S3 ⊂ T 4 (A(a, u1 )) whenever u2 − u3 > 2 l2 − (2π/a)u3 = 2 l3 . Since a > 2π, the maximal such u3 is characterized by the equation u2 − u3 = 2 l2 − (2π/a)u3 , i.e., u3 = We define u3 by (4.3.4).

a (u2 − 2 l2 ). a − 2π

(4.3.4)

60


Assume now that we folded already i times, where i is even, that u2 , . . . , ui have been+ chosen maximal, and that u1 is still possibly admissible. Denote the length a − ji =1 uj of the remainder by ri . If ri < ui , we set ui+1 = ri , and u1 is admissible. Indeed, u1 , u2 , . . . , ui+1 then describe a symplectic embedding of T (a, π ) into T 4 (A(a, u1 )) obtained by folding i times. If ri ≥ ui , we are forced to fold again. As in the case i = 2, the Folding Lemma 4.2.1 (i) shows that if ui ≤ 2 li , then u1 is non-admissible, and that if ui > 2 li , then the maximal ui+1 for which the stairs Si+1 are contained in T 4 (A(a, u1 )) is a (ui − 2 li ). (4.3.5) a − 2π + We define ui+1 by (4.3.5). Denote the length a − ji+1 =1 uj = ri − ui+1 of the new remainder by ri+1 . If ri+1 < ui+1 + li , we set ui+2 = ri+1 , and u1 is admissible. Indeed, u1 , u2 , . . . , ui+2 then describe a symplectic embedding of T (a, π ) into T 4 (A(a, u1 )) obtained by folding i + 1 times. If ri+1 ≥ ui+1 + li , we are forced to fold again. The Folding Lemma 4.2.1 (ii) shows that for ui+2 small enough we have Si+2 ⊂ {u > 0}, and that the maximal such ui+2 is characterized by the equation ui+2 + li+2 = ui+1 + li , which by (4.2.1) translates into the formula ui+1 =

ui+2 =

a+π ui+1 . a−π

(4.3.6)

We define ui+2 by (4.3.6). Then Si+2 ⊂ {u > 0}, but possibly Si+2 is not contained in T 4 (A(a, u1 )), in which case u1 in non-admissible. If Si+2 ⊂ T 4 (A(a, u1 )), we go on as before. Proceeding this way we try to decide for each u1 ∈ ]0, a[ whether it is admissible or not. The value u1 is admissible if and only if our procedure leads to an embedding of T (a, π) into T 4 (A(a, u1 )) after a finite number N(u1 ) of folds, and u1 is nonadmissible if and only if the procedure leads to an embedding obstruction after finitely many folds or if the procedure does not terminate. As we have already seen, u1 is non-admissible if u1 ≤ aπ/(a + π ), and u1 is admissible if u1 > a/2. Also recall that we have to look for the smallest admissible u1 . The following lemma implies that there exists a unique u0 = u0 (a) in the interval aπ a I (a) := , (4.3.7) a+π 2 such that u1 is admissible if u1 > u0 and u1 is non-admissible if u1 < u0 . Lemma 4.3.1. Assume that u1 , u1 ∈ ]0, a[ and that u1 < u1 . If u1 is admissible, then u1 is admissible, and N (u1 ) ≤ N (u1 ). Proof. We abbreviate N = N (u1 ) and N = N(u1 ). Assume that the embedding procedure associated with u1 is described by u1 , . . . , uN +1 . We shall prove the lemma

61


by going through our procedure and checking that the folding conditions for u1 are met whenever they are met for u1 . If u1 is admissible and N = 1, we are done. Otherwise, u1 < u1 < a/2, and so N ≥ 2. Then u1 > u1 > aπ/(a + π ). Observe that the next condition S2 ⊂ T 4 (A(a, u1 )) is equivalent to l2 < u2 . Equation (4.3.3) shows u2 > u2 , and now equation (4.2.1) shows l2 < l2 . Therefore, l2 < l2 < u2 < u2 , and so S2 ⊂ T 4 (A(a, u1 )). If u1 is admissible and N = 2, we are done. Otherwise, u2 < u2 < r2 < r2 , and so N ≥ 3. In view of (4.2.1) we then have u2 > u2 > 2 l2 > 2 l2 , and equation (4.3.4) shows u3 > u3 . If u1 is admissible and N = 3, we are done. Otherwise, r3 > u3 +l2 , i.e., a −π > (1−π/a)(u1 +u2 )+2u3 , whence a − π > (1 − π/a)(u1 + u2 ) + 2u3 , i.e., r3 > u3 + l2 , and so N > 3. We proceed in this way using equations (4.3.5) and (4.3.6) and observing that the condition Si ⊂ T 4 (A(a, u1 )) for i even is equivalent to li < ui . The lemma then follows. 2 Define the function fEB (a) on ]2π, ∞[ by 2π u0 (a). fEB (a) = 2π + 1 − a

(4.3.8)

Lemma 4.3.1 shows that the ellipsoid E(π, a) symplectically embeds into the ball B 4 (fEB (a) + ) for any > 0. We next explain why our procedure is optimal in the sense that we cannot embed E(π, a) into a ball smaller than B 4 (fEB (a)) by multiple symplectic folding. Indeed, observe that our procedure can equivalently be described as follows: To A ∈ ]2π, a[ associate the number u1 = u1 (a, A) defined by A = 2π + (1 − 2π/a)u1 , and then choose ui , i ≥ 2, maximal. In other words, for each A we successively choose u1 , u2 , . . . maximal with respect to the condition of staying inside T 4 (A). The only possible way of improving our procedure is therefore to choose some of the ui smaller. So let A < fEB (a), let u1 , u2 , . . . , uN be the sequence associated with A by our procedure, and try to circumvent the embedding obstruction arising after folding N times by choosing ui ≤ ui . Then, however, the proof of Lemma 4.3.1 shows that li ≥ li and ri ≥ ri , and so the modified procedure leads to an embedding obstruction after N ≤ N folds. Lemma 4.3.2. We have N (u1 ) → ∞ as u1 u0 . The value u0 is non-admissible, and the procedure associated with u0 does not terminate. Proof. Arguing by contradiction, assume that the first statement of the lemma is wrong. By Lemma 4.3.1, there exists N such that N(u1 ) = N for all u1 > u0 with u1 − u0 small enough. In the sequel we assume that u1 > u0 and that u1 − u0 is so small that N (u1 ) = N . By the proof of Lemma 4.3.1, the functions ui (u1 ), i = 2, . . . , N − 1 are decreasing as u1 u0 , and the functions li (u1 ), i = 1, . . . , N, and uN (u1 ) are increasing as u1 u0 and bounded. We set u0i = lim ui (u1 ), i = 2, . . . , N, u1 u0

and

li0 = lim li (u1 ), i = 1, . . . , N. u1 u0

62


Assume first that during the first N folds there is no embedding obstruction at u0 . Case 1. N = 1. If u0 > u02 , then T (a, π) embeds into T 4 (A(a, u0 )) by folding once, and if u0 = u02 , then the Folding Lemma 4.2.1 (i) shows that T (a, π ) embeds into T 4 (A(a, u0 )) by folding twice. 0 Case 2. N ≥ 3 and N odd. If u0N + lN−1 > u0N +1 , then T (a, π ) embeds into 0 0 T 4 (A(a, u0 )) by folding N times, and if u0N + lN −1 = uN +1 , then the Folding 4 Lemma 4.2.1 (i) shows that T (a, π) embeds into T (A(a, u0 )) by folding N + 1 times.

Case 3. N even. If u0N > u0N+1 , then T (a, π) embeds into T 4 (A(a, u0 )) by folding N times, and if u0N = u0N+1 , then the Folding Lemma 4.2.1 (ii) and a > 2π imply that T (a, π ) embeds into T 4 (A(a, u0 )) by folding N + 1 times. Therefore, u0 is admissible with N (u0 ) = N or N(u0 ) = N + 1. Since all conditions in our procedure are open conditions, it follows that if u1 < u0 and u0 − u1 is small enough, then u1 is admissible with N(u1 ) = N or N(u1 ) = N + 1. This contradicts the definition of u0 . So assume that there is an embedding obstruction at u0 appearing at the latest at the N’th fold. We shall conclude the proof of Lemma 4.3.2 by showing that an embedding obstruction at u0 appearing at the latest at the N ’th fold implies an embedding obstruction at all u1 near u0 . Case I. u0 ≤ aπ/(a + π ). Then u0 = aπ/(a + π ), i.e., l1 (u0 ) = l10 = u0 . Therefore, u02 = 0 and so N ≥ 2 and l20 = l10 = u0 . If N = 2, we find u3 (u1 ) > u2 (u1 ) for u1 near u0 , a contradiction. If N > 2, we find u2 (u1 ) < 2 l2 (u1 ) for u1 near u0 , i.e., there is an embedding obstruction for u1 near u0 ; this is another contradiction. Case II. Si (u0 ) ⊂ T 4 (A(a, u0 )) for some even i, i.e., li0 ≥ u0i for some even i. If i = N, we find uN+1 (u1 ) > uN (u1 ) for u1 near u0 , a contradiction. If i < N, we find ui (u1 ) < 2 li (u1 ) for u1 near u0 , i.e., there is an embedding obstruction for u1 near u0 ; this is another contradiction. Case III. ui (u0 ) ≤ 2 li (u0 ) for some even i. Then u0i = 2 li0 . If i = N, we find in view of a > 2π that uN +1 (u1 ) > uN (u1 ) for u1 near u0 , a contradiction. 0 If i = N − 1, then u0N+1 ≤ lN−1 , and so we find uN +1 (u1 ) < uN −1 (u1 ) for u1 near u0 , contradicting N > i. 0 0 , whence l 0 + l 0 + If i < N − 1, we find u0i+1 = u0i+2 = 0 and li0 = li+1 = li+2 i i+1 0 li+2 > u0i . This contradicts Si+2 (u1 ) ⊂ T 4 (A(a, u1 )) for u1 near u0 . We conclude that N (u1 ) → ∞ as u1 u0 . The value u0 is therefore nonadmissible, and as we have seen in Case I, Case II and Case III above, the procedure associated with u0 cannot lead to an embedding obstruction. Lemma 4.3.2 is thus proved. 2


63

The computer program in Appendix D.1 computing u0 (a) is based on the following lemma, which implies that the value u0 is the only value for which our procedure does not terminate. Lemma 4.3.3. Assume that u1 < u0 . Then the procedure associated with u1 leads to an embedding obstruction after finitely many folds. Proof. By Lemma 4.3.2 our procedure associated with u0 generates the infinite ) and li (u0 ), i = 1, 2, . . . , of lengths and heights of the floors Fi (u0 ). sequences ui (u +0∞ Set h(u0 ) := i=1 li (u0 ). Then h(u0 ) ≤ A(a, u0 ). We claim that h(u0 ) = A(a, u0 ).

(4.3.9)

In other words, the image of T (a, π ) in T 4 (A(a, u0 )) obtained by folding first at u0 and then infinitely many times touches the upper vertex of T 4 (A(a, u0 )), i.e., the sequence Fi (u0 ) of floors creeps into the upper corner of T 4 (A(a, u0 )), cf. Figure 4.8. In order to prove the identity (4.3.9), we argue by contradiction and assume h(u0 ) < A(a, u0 ). Then w := A(a, u0 ) − h(u0 ) > 0. The Folding Lemma 4.2.1 (ii) shows that l2i (u0 ) + u2i (u0 ) > w for all i = 1, 2, . . . . Since h(u0 ) < ∞, there exists (u0 ) < w/2 for+i ≥ j . Then u2i (u0 ) > w − l2i (u0 ) > w/2 for j ∈ N such that l2i+ ∞ i ≥ j , and so a = ∞ i=1 ui (u0 ) > i=1 u2i (u0 ) = ∞. This contradiction proves the identity (4.3.9). Assume now that Lemma 4.3.3 is wrong for some u1 < u0 . Since u1 is nonadmissible, the procedure associated with u1 does not terminate and generates the infinite sequence li (u1 ), i = 1, 2, . . . . The proof of Lemma 4.3.1 shows that li (u1 ) > li (u0 ), i = 1, 2, . . . . Therefore, h(u1 ) :=

∞ .

li (u1 ) >

i=1

∞ .

li (u0 ) = h(u0 ) = A(a, u0 ) > A(a, u1 ).

i=1

The contradiction h(u1 ) > A(a, u1 ) shows that Lemma 4.3.3 holds true.

2

While the computer program in Appendix D.1 computes for each a > 2π and each > 0 the value of fEB (a) up to accuracy , the following lemma gives some qualitative insight into the function fEB (a). Lemma 4.3.4. The function fEB on ]2π, ∞[ is strictly increasing and hence almost everywhere differentiable. Moreover, fEB is Lipschitz continuous with Lipschitz constant at most 1. Proof. Assume 2π < a < a . In view of the procedures associated with the pairs (a, u0 (a )) and (a , u0 (a )), the inequalities l1 (a, u0 (a )) = π −

π π u0 (a ) < π − u0 (a ) = l1 (a , u0 (a )) a a

64


and a − u0 (a ) < a − u0 (a ) imply that u0 (a) ≤ u0 (a ). Therefore, 2π 2π u0 (a) < 2π + 1 − u0 (a ) = fEB (a ). fEB (a) = 2π + 1 − a a Since a < a were arbitrary, we conclude that the function fEB is strictly increasing, and so, as every increasing real function, almost everywhere differentiable. Assume again 2π < a < a . We set δ = a − a and u1 = u0 (a) + δ. Since u0 (a) < a, we have u1 u0 (a) < . (4.3.10) a a It follows that π π l1 (a , u1 ) = π − u1 < π − u0 (a) = l1 (a, u0 (a)). a a In view of the procedures associated with the pairs (a, u0 (a)) and (a , u1 ), this inequality and the equality a −u1 = a −u0 (a) imply that u1 ≥ u0 (a ). Using definition (4.3.8) and inequality (4.3.10), we may therefore estimate 2π 2π u0 (a) fEB (a ) − fEB (a) = 1 − u0 (a ) − 1 − a a 2π 2π u0 (a) ≤ 1 − u1 − 1 − a a u1 u0 (a) − = u1 − u0 (a) − 2π a a < u1 − u0 (a) = a − a. Since a < a were arbitrary, we conclude that the function fEB is Lipschitz continuous with Lipschitz constant at most 1. 2 We do not know whether the function fEB is differentiable on any open interval. We next investigate the behaviour of the function fEB (a) as a → 2π + . Proposition 4.3.5. We have lim sup →0+

fEB (2π + ) − 2π 3 ≤ . 7

Proof. Fix a > 2π . By the proof of Lemma 4.3.1 and by Lemma 4.3.2 there exists a unique value u1,2 = u1,2 (a) such that N (u1,2 ) = 2 and u3 (u1,2 ) = u2 (u1,2 ). Then 2 +aπ . Therefore, u1,2 + 2 u2 (u1,2 ) = a. In view of equation (4.3.3) we find u1,2 = a3a+π A2 (a) := A(a, u1,2 ) = 2π + (a − 2π )

a+π , 3a + π

65


and so lows.

d da A2 (2π )

= 37 . Since fEB (a) ≤ A2 (a) for all a > 2π , the proposition fol2

Remark 4.3.6. The function A2 (a) constructed in the above proof describes the optimal embedding of E(π, a) into a ball obtainable by folding exactly twice. More generally, we can compute the function AN (a) describing the optimal embedding obtainable by folding exactly N times as follows. For each a > 2π and each N = 1, 2, . . . there exists a unique value u1,N = u1,N (a) such that N(u1,N ) = N and such that uN+1 (u1,N ) = uN (u1,N ) + lN−1 (u1,N ) if N is odd and uN +1 (u1,N ) = uN (u1,N ) if N is even. If N is odd, we replace lN−1 (u1,N ) by the expression given in formula (4.2.1). Plugging the expression for uN+1 (u1,N ) into the equation u1,N + u2 (u1,N ) + · · · + uN +1 (u1,N ) = a and then alternately using equations (4.3.5) and (4.3.6), we compute u1,N as a rational function of a. We finally find 2π AN (a) = A(a, u1,N ) = 2π + 1 − u1,N (a). a For instance, 1 A1 (a) = 2π + (a − 2π ) , 2

A2 (a) = 2π + (a − 2π )

and A3 (a) = 2π + (a − 2π )

a+π 3a + π

(a + π )(a + 2π ) . 4(a 2 + aπ + π 2 )

By Lemma 4.3.1, u1,N+1 (a) < u1,N (a) for every N and every a > 2π , and arguing as in the proof of Lemma 4.3.4, we see that the function u1,N (a) is increasing for every N. The family {AN }, N = 1, 2, . . . , is therefore a strictly decreasing family of strictly increasing smooth rational functions on ]2π, ∞[ converging to fEB (a). In view of Dini’s Theorem, the convergence is uniform on bounded sets. One might try to improve the estimate given in Proposition 4.3.5 by showing d 3 d 3 A da N (2π) < 7 for some N . However, da AN (2π ) = 7 for all N ≥ 2. Indeed, for all a > 2π and N > 2 we have u2 (u1,2 (a)) > u2 (u1,N (a)) > 2 l2 (u1,N (a)) > 2 l2 (u1,2 (a)) =

2π u2 (u1,2 (a)), a

and so lima→2π + u2 (u1,N (a)) = lima→2π + u2 (u1,2 (a)). In view of formula (4.3.3), we conclude that lim u1,N (a) = lim u1,2 (a). a→2π +

a→2π +

66


Next, the identity 2π u1,N (a) AN (a) = 2π + 1 − a

for all a > 2π

implies that 2π 2π d d AN (a) = 2 u1,N (a) + 1 − u1,N (a) da a a da

for all a > 2π.

The formal expression for u1,N (a) defines a rational function on R. Since 2π is not d u1,N (a) is bounded near a = 2π . a singularity of u1,N (a), the rational function da Taking the limit a → 2π + we therefore find 1 d AN (2π) = da 2π

lim u1,N (a) =

a→2π +

1 2π

lim u1,2 (a) =

a→2π +

d 3 A2 (2π ) = , da 7 3

as claimed.

We do not know how to analyze the asymptotic behaviour of the function fEB (a) as a → ∞ directly. We shall prove the following proposition at the end of the subsequent section by comparing the optimal multiple folding embedding of E(π, a) into a ball with the optimal multiple folding embedding of the polydisc P (π, a2 + π ) into a ball. Proposition 4.3.7. We have fEB (a) −

√

π a ≤ 2π

for all a > 2π .

The function fEB (a) is further discussed and compared with the result yielded by symplectic wrapping in 7.2.1.1. 4.3.2 Embedding polydiscs into balls. As we have seen in the proof of Lemma 3.1.8, the disc D(a) is symplectomorphic to the rectangle R(a). The polydisc P (π, a) = D(π) × D(a) is therefore symplectomorphic to R(a) × D(π ). Since the fibre D(π) over each point (u, v) ∈ R(a) is the same, the optimal embedding into a ball obtainable by multiple symplectic folding is easier to compute for a polydisc than for an ellipsoid. In contrast to our optimal embeddings of an ellipsoid into a ball, which were obtained by folding “more and more often”, the optimal embedding of a polydisc into a ball obtainable by multiple folding will turn out to be described by a picture as in Figure 4.9. Our embedding result stated in Proposition 4.3.9 below is readily read off from such a picture. We aim, however, to show that this embedding result is the best one obtainable by multiple folding. We therefore proceed in a systematic way. We again think of the ball B 4 (A) as the trapezoid T 4 (A). Fix a ≥ π . Folding R(a) × D(π) first at u1 ∈ ]0, a[ determines T 4 (A(a, u1 )) by the condition that the second floor F2 touches the “upper right boundary” of T 4 (A(a, u1 )). Then A(a, u1 ) =

67


A3

π u u1

A3

a

Figure 4.9. The optimal embedding P (π, a) → B 4 (A) for a = 10π.

u1 +2π. We then successively choose ui , i ≥ 2, maximal with respect to the condition of staying inside T 4 (A(a, u1 )). The Folding Lemma 4.2.1 (ii) shows that a condition for folding a second time, if necessary, is u1 > π, and that then the stairs S2 are contained in T 4 (A(a, u1 )) if and only if u1 > 2π . The only condition on u1 for folding a second time is therefore u1 > 2π. The Folding Lemma 4.2.1 (i) shows that folding a third time, if necessary, is then possible whenever u2 > 2π , i.e., u1 > 3π . For N ≥ 2, the only condition on u1 for folding an N ’th time, if necessary, is u1 > Nπ . If our procedure leads to an embedding obstruction after N folds, then choosing ui ≤ ui leads to an embedding obstruction after N ≤ N folds. It is therefore enough to compare embeddings obtained from our procedure. We say that u1 is admissible if the procedure associated with u1 leads to an embedding of R(a) × D(π) into T 4 (A(a, u1 )). We then write N(u1 ) for the number of folds needed. If u1 < u1 and u1 is admissible, then u1 is admissible and N(u1 ) ≤ N (u1 ). Lemma 4.3.8. Assume that u1 is admissible and that N(u1 ) is even or that N(u1 ) is odd and the last floor FN(u1 )+1 does not touch {u = 0}. Then there exists an admissible u1 such that u1 < u1 and such that N (u1 ) is odd and FN (u1 )+1 touches {u = 0}. Proof. Set N = N (u1 ). Observe that on those admissible u1 ’s for which N(u1 ) = N, the functions u2 (u1 ), . . . , uN (u1 ) are continuous and increasing in u1 , and uN +1 (u1 ) is continuous and decreasing in u1 . Assume first that N is odd and FN+1 does not touch {u = 0}, i.e., u2 < u1 if N = 1 and uN+1 < uN + π if N ≥ 3. If N = 1, shrinking u1 leads to an admissible u1 such that u2 = u1 . If N ≥ 3, shrinking u1 either leads to an admissible u1 such that uN+1 = uN + π , or to u1 = N π . In the second case, however, folding at u1 would already lead to an embedding after N − 1 folds, i.e., u1 < u1 would be admissible with N (u1 ) < N(u1 ), a contradiction.

68


Assume next that N is even. After shrinking u1 , if necessary, we may assume that uN+1 touches the “upper right boundary” of T 4 (A(a, u1 )), i.e., uN +1 + π = uN . We have uN+1 > π, since otherwise N (u1 ) ≤ N − 1. Therefore uN > 2π , and so we can fold another time at u = uN − π and obtain an embedding of R(a) × D(π ) into T 4 (A(a, u1 )) with uN+2 < uN+1 + π. As we have seen above, shrinking u1 leads to an admissible u1 such that uN+2 = uN+1 + π. 2 Assume that a ∈ [π, 2π ] and that u1 is admissible. Then N(u1 ) = 1. By Lemma 4.3.8, we may assume that u1 = a/2. Since A(a, a/2) = a/2 + 2π ≥ a + π, we see that for a ∈ [π, 2π] multiple symplectic folding does not provide a better embedding result than the inclusion P (π, a) → B 4 (π + a). So assume a > 2π . By Lemma 4.3.8 it suffices to analyze those embeddings for which the number of folds is N = 2k − 1 and FN +1 touches {u = 0}. The optimal embedding obtainable by folding once is therefore described by A1 (a) = a/2 + 2π . If N ≥ 3, we read off from Figure 4.9 that a = u1 + u2 + · · · + uN+1 = π + 2(u1 − π ) + 2(u1 − 3π ) + · · · + 2(u1 − Nπ ) + π = 2π + 2ku1 − 2k 2 π provided that u1 > Nπ . Solving for u1 and using the formula A(a, u1 ) = u1 + 2π , we find that the optimal embedding of R(a) × D(π ) into a ball obtainable by folding N = 2k − 1 times is described by Ak (a) =

a − 2π + (k + 2)π 2k

provided that a > 2(k 2 − k + 1)π.

Observe that this formula also holds true for N = 1. Define the function fPB (a) on ]2π, ∞[ by

fPB (a) = min Ak (a) | k = 1, 2, . . . ; a > 2(k 2 − k + 1)π , cf. Figure 4.10. We in particular have proved Proposition 4.3.9. Assume a > 2π. Then the polydisc P (π, a) symplectically embeds into the ball B 4 (fPB (a) + ) for every > 0, where fPB (a) =

a − 2π + (k + 2)π 2k

for the unique integer k for which 2(k 2 − k + 1) < a/π ≤ 2(k 2 + k + 1). √ Remark 4.3.10. Let dPB (a) = fPB (a)− 2π a be the difference between fPB and the its local maxima at ak = 2(k 2 − k + 1)π , volume condition. The function dPB attains √ 2 where dPB (ak ) = (2k + 1)π − 2π k − k + 1. This is an increasing sequence converging to 2π . 3

69


Extend the above function fPB (a) to a function on [π, ∞[ by setting fPB (a) = a + π for a ∈ [π, 2π ]. The problem considered in this section was to understand the characteristic function χPB on [π, ∞[ defined by

χPB (a) = inf A | P (π, a) symplectically embeds into B 4 (A) . The following proposition summarizes what we know about this function. Proposition 4.3.11. The function χPB : [π, ∞[ → R is bounded from below and above by √ max 2π, 2π a ≤ χPB (a) ≤ fPB (a), see Figure 4.10. It is monotone increasing and hence almost everywhere differentiable. Moreover, χPB is Lipschitz continuous with Lipschitz constant at most 2; more precisely, χPB (a ) − χPB (a) ≤

fPB (a) (a − a) for all a ≥ a ≥ π. a

A

A=a+π 7π fPB (a) 5π A=

3π

√ 2πa

c2

2π

a π 2π

6π

10π

Figure 4.10. What is known about χPB (a).

Proof. In view of the monotonicity axiom for the second Ekeland–Hofer capacity, the identities (2.2.3) and χPB (a) ≥ 2π for all (2.3.1) imply √ a ≥ π . The volume condition |P (π, a)| ≤ B 4 (χPB (a)) translates to χPB (a) ≥ 2π a. We conclude that √ max 2π, 2πa ≤ χPB (a). For a ≤ 2π, the estimate χPB (a) ≤ fPB (a) is provided by the inclusion P (π, a) → B 4 (a + π ), and for a > 2π by Proposition 4.3.9. Assume a < a . If ϕ symplectically embeds P (π, a ) into B 4 (A), then ϕ|P (π,a) symplectically embeds P (π, a) into B 4 (A). Therefore χPB is increasing, and so, as

70


every increasing real function, almost everywhere differentiable. Denote the dilatation by a /a by λ. Assume that ψ symplectically embeds P (π, a) into B 4 (A). Then the composition a

ψ λ−1 λ 4 4 a A π, a − → B (A) − → B P (π, a ) −−→ P a a

is a symplectic embedding. Therefore χPB (a ) ≤ aa χPB (a). We conclude that fPB (a) a χPB (a ) − χPB (a) ≤ χPB (a) −1 ≤ (a − a) ≤ 2(a − a) a a 2

as claimed.

For a = 2π, multiple symplectic folding does not provide a better upper bound of χPB (a) than the inclusion P (π, a) → B 4 (π + a), and Ekeland–Hofer capacities do a better lower bound of χPB (a) than the volume condition |P (π, a)| ≤ 4not provide B (χPB (a)). We therefore would like to know the answer to Question 4.3.12. χPB (2π ) < 3π ? We end this section by deriving Proposition 4.3.7 from Proposition 4.3.9. Proof of Proposition 4.3.7. Computer calculations suggest that fEB (a) < fPB all a > 2π . For our purpose, the following result will be sufficient. Lemma 4.3.13. We have fEB (a) < fPB

a 2

a 2

for

+ π for all a > 2π.

Proof. As in 4.3.1 we think of the ellipsoid E(π, a) as the trapezoid T (a, π ) and of the ball B 4 (A) as the trapezoid T 4 (A). We fix a > 2π and let F =

N+1 2 i=1

Fi ∪

N 2

Si

i=1

be the image of the “optimal” embedding of P π, a2 + π into T 4 fPB a2 + π . We recall from Lemma 4.3.8 that the number of folds N is odd and that for N = 3 the set F looks as in Figure 4.9. We define A ∈ ]2π, ∞[ as the unique real number for which

a +π fEB (A) = fPB 2 and we let ∞ ∞ 2 2 Fi ∪ Si F = i=1

i=1

be the image of the “optimal” embedding of the trapezoid T (A, π ) into T 4 (fEB (A)), cf. Figure 4.8.

71


In the sequel we shall compare the volume of F with the volume of F . Since the embeddings of P (π, a2 + π ) and T (A, π) are both “optimal”, the volumes of the 1 1 stairs Si and Si “vanish”. We shall therefore neglect the stairs of both sets. Recall from 4.3.1 that li denotes the minimal height of the floor Fi and that the width and the height of the i’th “triangle” Ti in T 4 (fEB (A)) \ F is 2l2i−1 . Also recall that π > l1 > l2 > · · · . (4.3.11) This and the Folding Lemma 4.2.1 imply that Fi ⊂ F for each odd i ≥ 3, and that F \F is the disjoint union of the thin “triangle” Q1 = F1 \(F1 ∪F2 ), the “rectangles” Qi ⊂ Fi , i even, each contained in a different triangle Tj(i) , and the set Q0 lying in the left end of the floor FN+1 , see Figure 4.11.

fEB (A)

Q4 R4

Q0

Q2

l2

R2

Q1 π

u fEB (A) Figure 4.11. The sets Qi ⊂ F \ F and the sets Rj (i) ⊂ F \ F .

Using the definition (4.2.1) of l1 and the estimate (4.3.2) we find l2 < l1 = π −

π π2 u1 < π − A A+π

and so |Q0 | + |Q1 | ≤ l2 π +

1 π l2 l2 ≤ π 2 . 2 A

(4.3.12)

72


We shall prove that |Q2 | + |Q4 | + · · · + |QN +1 | < |F \ F |.

(4.3.13)

The estimates (4.3.12) and (4.3.13) yield |F | = |F \ F | + |F ∩ F | < π 2 + |F \ F | + |F ∩ F | = π 2 + |F |. Therefore, π2 +

a

πa πA = P π, + π = |F | < π 2 + |F | = π 2 + |T (A, π )| = π 2 + , 2 2 2

i.e., a < A. Since the function fEB is monotone increasing, we conclude that fEB (a) ≤ fPB a2 + π as claimed. In order to prove the estimate (4.3.13) we denote the “triangles” in T 4 (fEB (A))\F of height and width 2π by Ti , i = 1, 2, . . . , and associate with each rectangle Qi in Tj(i) , i = 2, 4, . . . , N + 1, the rectangle Ri ⊂ F ∩ Ti/2 ⊂ F \ F whose width w (Ri ) is equal to the height h(Qi ) of Qi and whose height h (Ri ) is 2π − h(Qi ), cf. Figure 4.11. Since the width w(Qi ) of Qi is 2l2j (i)−1 − w (Ri ) = 2l2j (i)−1 − h (Qi ) we find together with the inequalities (4.3.11) that |Qi | = w(Qi )h(Qi ) = 2l2j (i)−1 − h (Qi ) h(Qi ) < (2π − h (Qi )) h(Qi ) = h (Ri ) w (Ri ) = |Ri | . The estimate (4.3.13) thus follows, and so the proof of Lemma 4.3.13 is complete. 2 Proposition 4.3.7 follows from Lemma 4.3.13 and Proposition 4.3.9. Indeed, in view of Proposition 4.3.9 the function √ d(a) := fPB a2 + π − π a on ]2π, ∞[ has its local maxima at ak = 2(2(k 2 − k + 1) − 1)π,

k = 1, 2, . . . ,

73

4.4 Embeddings into cubes

where d(ak ) = (2k + 1)π −

√

π ak .

The sequence d(ak ) is monotone increasing to 2π. Together with Lemma 4.3.13 we conclude that √ √ fEB (a) − π a ≤ fPB a2 + π − π a ≤ 2π 2

and so the proof of Proposition 4.3.7 is complete.

4.4

Embeddings into cubes

In this section we use multiple symplectic folding to construct symplectic embeddings of four dimensional ellipsoids and polydiscs into four dimensional cubes. While in Section 2.3 the lack of convenient invariants made it impossible to get good rigidity results for embeddings into cubes, multiple symplectic folding provides us with rather satisfactory flexibility results. 4.4.1 Embedding ellipsoids into cubes. Fix a > π . We think of the ellipsoid E(π, a) as T (a, π) and of the cube C 4 (A) as R(A)×D(A). In order to find the smallest A for which T (a, π) embeds into R(A) × D(A) via multiple symplectic folding, we proceed as follows. We fix u1 ∈ ]0, a[ and fold at u1 . By the Folding Lemma 4.2.1 (i),

A(a, u0 )

π

u u0 A(a, u0 )

a

Figure 4.12. The optimal embedding E(π, a) → C 4 (A) for a = 7π.

the stairs S1 are contained in {u < w(a, u1 )}, where π w(a, u1 ) = u1 + l1 = π + 1 − u1 . a

(4.4.1)

We then choose ui , i ≥ 2, maximal with respect to the condition of staying inside {0 < u < w(a, u1 )}. If the remainder r1 = a − u1 is smaller than u1 , we obtain an embedding of T (a, π) into {0 

aπ . a+π

(4.4.2)

If this condition is met, we define u2 by formula (4.3.3). The sequence li , i = 1, 2, . . . , of the widths of the stairs Si is decreasing. Hence, there are no further conditions at the subsequent folds, and if there is an i’th fold, then ui > ui−1 for all i ≥ 3. Under the condition (4.4.2), our procedure therefore embeds T (a, π ) into {0 < u < w(a, u1 )} by folding finitely many times. We denote the number of folds needed by N(u1 ). Recall that li = li (a, u1 ) is the width of the stairs Si as well as the minimal height of the floor Fi as well as the maximal height of Fi+1 aπ, i =1, . . . , N(u1 ). We set li (a, u1 ) = 0 if , a , the functions li (a, u1 ) are decreasing i > N(u1 ). For a > π fixed and u1 ∈ a+π continuous functions of u1 . Therefore, the height of the image h(a, u1 ) =

N(u .1 )

li (a, u1 )

i=1

aπ , a . For u1 ≥ a/2 we is a decreasing continuous function of u1 for u1 ∈ a+π have h(a, u1 ) = π , and for u1 aπ/(a + π ) we have N(u1 ) → ∞ and ui → 0, li (a, u1 ) → l1 (a, u1 ) for all i ≥ 2, and so h(a, u1 ) → ∞ as u1 aπ/(a+π ). On the other hand, w(a, u1 ) = π + (1 − π/a)u1 is a strictly increasing continuous function of u1 such that w(a, aπ0) =π and w(a, a) = a. It follows that w(a, u1 ) = h(a, u1 ) for exactly one u1 ∈ a+π , a , which we call u0 = u0 (a). Define the function fEC (a) on ]π, ∞[ by π u0 (a), fEC (a) = π + 1 − a cf. Figure 4.13. We have shown that the ellipsoid E(π, a) symplectically embeds into the cube C 4 (fEC (a) + ) for any > 0. A computer program for the function fEC (a) is presented in Appendix D.2. Our procedure is optimal in the sense that we cannot embed E(π, a) into a cube smaller than C 4 (fEC (a)) by multiple symplectic folding. This follows from an argument similar to the one given in 4.3.1. Indeed, our procedure can equivalently be described as follows: For each A ∈ ]π, a[ we successively choose u1 , u2 , . . . maximal with respect to the condition of staying inside {0 < u < A}. Then fEC (a) is the smallest A for which we do not run into an embedding obstruction and for which the height of the image is smaller than A. The only way of improving our procedure is therefore to choose some of the ui smaller. So let A < fEC (a), and choose ui ≤ ui . We then either run into an embedding obstruction, or the height of the image of the modified embedding is larger than h(a, u1 ) and hence larger than A.


75

Remarks 4.4.1. 1. We are going to investigate the function fEC (a) in more detail. We have N (u1 ) = 1 if and only if u1 ∈ [a/2, a[. Then h(a, u1 ) = π < π + (1 − π/a)u1 = w(a, u1 ), and so w(a, a/2) = (a + π )/2, and h(a, u1 ) < w(a, u1 ) for all u1 ∈ [a/2, a[. It follows that by folding once we can embed E(π, a) into C 4 a+π 2 + for any > 0, and that folding only once never yields an optimal embedding, i.e., fEC (a) < (a + π )/2 for all a > π. We have N (u1 ) = 2 if and only if u1 < a/2 and l2 +u3 = l2 +(a/π )l2 ≤ w(a, u1 ). Using the formulas (4.4.1), (4.2.1) and (4.3.3) for w, l2 and u2 , we find that the second inequality is equivalent to the condition on u1 a(a 2 + π 2 ) ≤ u1 . 3a 2 + π 2

(4.4.3)

If N (u1 ) = 2, then h(a, u1 ) = 2 l1 + l2 . Plugging the identity u2 = u1 − l2 into the identity a = u1 + u2 + u3 = u1 + u2 + (a/π)l2 and solving for l2 we find h(a, u1 ) = 2π −

2π π(a − 2u1 ) u1 + . a a−π

The equation h = w thus yields u0 (a) =

aπ(2a − π ) a 2 + 2aπ − π 2

(4.4.4)

provided that u0 (a) meets condition (4.4.2), that u0 (a) < a/2, and that u0 (a) meets condition (4.4.3). We compute that u0 (a) meets condition (4.4.2) and that u0 (a) < a/2 whenever a > π, and that u0 (a) meets condition (4.4.3) if and only if π ≤ a ≤ 3π . It follows that (4.4.4) holds for all a ∈ ]π, 3π]. In fact, the identity (4.4.4) also holds true for all those a for which the optimal embedding of T (a, π ) obtainable by multiple folding is a 3-fold for which the height is still h = 2 l1 + l2 , i.e., for which u4 (u0 (a)) ≤ u3 (u0 (a)). The largest a for which (4.4.4) holds true is characterized by the identity u4 (u0 (a)) = u3 (u0 (a)). Using the identity u3 = a+π a−π u2 we compute that the equation a = u0 (a) + u2 (u0 (a)) + 2u3 (u0 (a)) translates into a=

aπ(5a 2 − 2aπ + π 2 ) , (a − π)(a 2 + 2aπ − π 2 )

√ i.e., a = 2 + 5 π. Therefore, fEC (a) =

aπ(3a − π ) 2 a + 2aπ − π 2

√ for π < a ≤ 2 + 5 π.

In general, fEC (a) is a piecewise rational function. Its singularities are those a for which uN(a) (u0 (a)) = uN(a)+1 (u0 (a)), N (a) = 3, 5, 7, . . . , and those a for which the “endpoint” of FN(a)+1 touches the “axis” {u = 0}; here, we set N(a) = N(u0 (a)).

76


Denoting the two sets of singularities by a3 , a5 , a7 , . . . and a4 , a6 , a8 , . . . , we have that the singular set of fEC is the strictly increasing, diverging sequence (ak ), k ≥ 3. √ 2. Let dEC (a) = fEC (a) − π a/2 be the difference between fEC and the volume condition. Set a2 = π . Computer calculations suggest that the function dEC attains exactly one local maximum Mk between a2k and a2k+1 , that dEC attains its local minima mk at a2k+1 , and that dEC strictly increases between a2k+1 and a2k+2 , k ≥ 1. and converging Moreover, they suggest that both (mk ) and (Mk ) are strictly increasing √ to (2/3)π. In particular, we seem to have lima→∞ fEC (a) − π a/2 = (2/3)π. 3 As in the case of the function fEB studied in 4.3.1 we do not know how to analyze the asymptotic behaviour of the function fEC (a) as a → ∞ directly. We shall prove the following proposition at the end of the subsequent section by comparing the optimal multiple folding embedding of E(π, a) into a cube with the optimal multiple folding embedding of the polydisc P (π, a2 + π ) into a cube. Proposition 4.4.2. We have : fEC (a) −

3 πa ≤ π 2 2

for all a > 2π.

Extend the function fEC (a) to a function on [π, ∞[ by setting fEC (π ) = π . The problem considered in this section was to understand the characteristic function χEC on [π, ∞[ defined by

χEC (a) = inf A | E(π, a) symplectically embeds into C 4 (A) . The following proposition summarizes what we know about this function. Proposition 4.4.3. The function χEC : [π, ∞[ → R is bounded from below and above by : πa max π, ≤ χEC (a) ≤ fEC (a), 2 see Figure 4.13. It is monotone increasing and hence almost everywhere differentiable. Moreover, χEC is Lipschitz continuous with Lipschitz constant at most 1; more precisely, χEC (a ) − χEC (a) ≤

fEC (a) (a − a) for all a ≥ a ≥ π. a

Proof. In view of the monotonicity axiom for the first Ekeland–Hofer capacity, the identities (2.2.3) and χEC (a) ≥ π for all a√≥ π , and the volume condi (2.3.1) imply tion |E(π, a)| ≤ C 4 (χEC (a)) translates to χEC (a) ≥ π a/2. The first claim thus follows. The remaining claims follow as in the proof of Proposition 4.3.11. 2

77

4.4 Embeddings into cubes A A=a

4π

A = a+π 2

3π fEC (a) 2π

9

π

πa 2

c1

a 2π

3π

4π

5π

6π

7π

Figure 4.13. What is known about χEC (a).

4.4.2 Embedding polydiscs into cubes. Fix a > π . We think of the polydisc P (π, a) as R(a) × D(π) and of the cube C 4 (A) as R(A) × D(A). In order to find the smallest A for which R(a) × D(π) embeds into R(A) × D(A) via multiple symplectic folding, we proceed as follows. We fix u1 ∈ ]0, a[ and fold at u1 . By the Folding Lemma 4.2.1 (i), the stairs S1 are contained in {u < u1 + π }. We then choose ui , i ≥ 2, maximal with respect to the condition of staying inside {0 π. For N ≥ 2, the only condition on u1 for folding an N’th time, if necessary, is u1 > π .

π u u1 u1 + π

a

Figure 4.14. Folding P (π, a) three times.

78


We say that u1 is admissible if u1 ≥ a/2 or u1 > π. It follows that if u1 is admissible, then our procedure embeds R(a)×D(π ) into R(A(a, u1 ))×D(A(a, u1 )) by a finite number N (u1 ) of folds. Here, A(a, u1 ) = max {u1 + π, (N (u1 ) + 1)π } .

(4.4.5)

Let u2 , . . . , uN(u1 )+1 be the lengths associated with some admissible u1 . Choosing some of the ui , i = 2, . . . , N(u1 ), smaller would lead to an embedding by folding at least N (u1 ) times. It is therefore enough to compare embeddings obtained from our procedure. Assume that a ∈ ]π, 2π ]. Then A(a, u1 ) ≥ 2π for every admissible u1 . It follows that for a ∈ ]π, 2π] multiple symplectic folding does not provide a better embedding result than the inclusion P (π, a) → C 4 (a). So assume a > 2π. Suppose that u1 is admissible and that N := N(u1 ) is even. We claim that if the last floor FN+1 does not touch {u = u1 + π }, then there exists an admissible u1 such that u1 < u1 , N (u1 ) = N and FN +1 touches {u = u1 + π }. Indeed, shrinking u1 either leads to a u1 as claimed or to u1 = π . In the second case, however, we find a ≤ 2π, a contradiction. We may therefore assume that FN +1 touches {u = u1 + π }. A similar argument shows that we may also assume that FN(u1 )+1 touches {u = 0} if N (u1 ) is odd. The optimal embedding

obtainable by folding only once is therefore described by A1 (a) = max a2 + π, 2π = a2 + π, and if the number of folds is N ≥ 2, we read off from Figure 4.14 that a = 2π + (N + 1)(u1 − π ) provided that u1 > π . Since a > 2π , we see that this condition is met. Solving for u1 and using formula (4.4.5) we then find that the optimal embedding of R(a) × D(π ) into a cube obtainable by folding N times is described by

AN (a) = max a+2Nπ , (N + 1)π . N+1 Observe that this formula also holds true for N = 1. Define the function fPC (a) on ]2π, ∞[ by fPC (a) = min{AN (a) | N = 1, 2, . . . }, cf. Figure 7.2. We in particular have proved Proposition 4.4.4. Assume a > 2π . Then the polydisc P (π, a) symplectically embeds into the cube C 4 (fPC (a) + ) for every > 0, where fPC (a) =

(N + 1)π if (N − 1)N + 2 < a+2Nπ N+1

if N 2 + 1
2π. Proof. As in 4.4.1 we think of the ellipsoid E(π, a) as the trapezoid T (a, π ). We fix a > 2π and let N+1 N 2 2 F = Fi ∪ Si i=1

i=1

be the image of the “optimal” embedding of P π, a2 + π into C 4 fPC a2 + π . For N = 3 the set F looks as in Figure 4.14. We define A ∈ ]2π, ∞[ as the unique real number for which a

+π fEC (A) = fPC 2 and we let +1 N2 N 2 F = Fi ∪ Si i=1

i=1

be the image of the “optimal” embedding of T (A, π) into C 4 (fEC (A)), cf. Figure 4.12. As in the proof of Lemma 4.3.13 we shall neglect the stairs of F and F . Recall from 4.4.1 that li denotes the minimal height of the floor Fi and that π > l1 > l2 > · · · . This and the Folding Lemma 4.2.1 imply that F \ F = F1 \ F ∪ FN +1 \ F . The set Q1 := F1 \ F = F1 \ F1 ∪ F2 , which is analogous to the set Q1 in Figure 4.11, has volume 1 π |Q1 | = l2 l2 . 2 A We decompose the set Q0 := FN+1 \ F into the sets Q0 and Q0 = Q0 \ Q0 , where {(u, v, x, y) ∈ Q0 | u < l1 } if N is odd, Q0 := {(u, v, x, y) ∈ Q0 | u > fEC (A) − l1 } if N is even. Using the definition (4.2.1) of l1 and the estimate (4.4.2) we find l2 < l1 = π −

π π2 u1 < π − A A+π

80


and so

Q + |Q1 | ≤ l1 π + 1 l2 π l2 ≤ π 2 . 0 2 A

(4.4.6)

Q < |F \ F |.

(4.4.7)

We shall prove that

0

The estimates (4.4.6) and (4.4.7) and the same argument as in the proof of Lemma 4.3.13 yield a < A. Sincethe function fEC is monotone increasing, we conclude that fEC (a) ≤ fPC a2 + π as claimed. We are left with proving the estimate (4.4.7). The length of the set Q0 is fEC (A) − π − l1 . We assume first that N is even. Recall that lN is the height of the floor FN +1 . Since the length of FN +1 is at most fEC (A) − lN , the height of π Q0 is at most A (fEC (A) − lN ). Therefore, Q ≤ (fEC (A) − π − l1 ) π (fEC (A) − lN ) . 0 A

(4.4.8)

Let R be the union of maximal “rectangles” in F \ F based over { (u, v) | fEC (A) − π ≤ u < fEC (A) − l1 } . If N is odd, R has one component, whose height is at least fEC (A) − lN . If N is even, R has one or two components, whose total height is at least fEC (A) − π − lN . π Together with π − l1 = A (fEC (A) − l1 ) we conclude that |F \ F | > |R| ≥

π (fEC (A) − l1 ) (fEC (A) − π − lN ) . A

(4.4.9)

Since l1 > lN , the right hand side in (4.4.9) is larger than the one in (4.4.8), and so the estimate (4.4.7) follows. Assume now that N is odd. Then the above argument with lN replaced by lN −1 goes through. The estimate (4.4.7) is thus proved, and so the proof of Lemma 4.4.5 is complete. 2 Proposition 4.4.2 follows from Lemma 4.4.5 and Proposition 4.4.4. Indeed, in view of Proposition 4.4.4 the function : a πa d(a) := fPC 2 + π − 2 on ]2π, ∞[ has its local maxima at aN = 2 N 2 − N + 1 π, where

N = 1, 2, . . . ,

8 d(aN ) = N + 1 − N 2 − N + 1 π.


81

The sequence d(aN ) is monotone increasing to 23 π . Together with Lemma 4.4.5 we conclude that : : a πa πa < fPC 2 + π − ≤ 23 π fEC (a) − 2 2 and so the proof of Proposition 4.4.2 is complete.

2

Chapter 5

Symplectic folding in higher dimensions

Even though symplectic folding is a four dimensional process, we can use it to prove interesting symplectic embedding results in higher dimensions as well. The reason is that we can fold into n − 1 different symplectic directions of the (2n − 2)-dimensional fibre over the 2-dimensional symplectic base. We will concentrate on embedding skinny polydiscs into cubes and skinny ellipsoids into balls. The results of this chapter will be used in Chapter 6 to prove Theorem 3.

5.1

Four types of folding

In Chapter 4 we folded on the right and on the left into the y-direction. In the multiple folding procedures considered in this chapter we shall also fold into the (−y)-direction. Hence there will be four types of folding. This section reviews these four types. As usual, we shall neglect those terms in the constructions which can be chosen arbitrarily small. We define F ⊂ R4 by F := { (u, v, x, y) ∈ R4 | u ∈ R, 0 < v < 1, 0 < x < 1, 0 < y < π }. Fix a “folding point” u1 ∈ R and choose a (right-)cut off function cr : R → [0, 1] with support [u1 , u1 + π ] and a (left-)cut off function cl : R → [0, 1] with support [u1 − π, u1 ]. 1. Folding on the right into the y-direction. We fold F on the right at u = u1 into the y-direction by applying the symplectic map φr+ := (γ1 × id) ϕ1 (β1 × id). Here, the maps β1 and γ1 , which are constructed the same way as the maps β and γ in Step 1 and Step 4 of Section 3.2, are explained in the first column of Figure 5.1, and the lifting map ϕ1 is defined by / u ϕ1 (u, v, x, y) = u, x, v + cr (u)x, y + cr (s) ds . u1

83

5.1 Four types of folding 1 0

u β2

β1

β3

β4

1 0

u1

u1 + π

u1 − π

u1

u1

u1 + π

u1 − π

u1

u

1 0

u γ1

γ2

γ3

γ4

1 0

u1

u1 + π

u1 − π

u1

u1

u1 + π

u1 − π

u1

u

Figure 5.1. The maps βi and γi , i = 1, 2, 3, 4.

2. Folding on the left into the y-direction. We fold F on the left at u = u1 into the y-direction by applying the map φl+ := (γ2 × id) ϕ2 (β2 × id). Here, the maps β2 and γ2 are explained in the second column of Figure 5.1, and / u1 ϕ2 (u, v, x, y) = u, x, v − cl (u)x, y + cl (s) ds . u

3. Folding on the right into the (−y)-direction. We fold F on the right at u = u1 into the (−y)-direction by applying the map φr− := (γ3 × id) ϕ3 (β3 × id). Here, the maps β3 and γ3 are explained in the third column of Figure 5.1, and / u ϕ3 (u, v, x, y) = u, x, v − cr (u)x, y − cr (s) ds . u1

4. Folding on the left into the (−y)-direction. We fold F on the left at u = u1 into the (−y)-direction by applying the map φl− := (γ4 × id) ϕ4 (β4 × id). Here, the maps β4 and γ4 are explained in the fourth column of Figure 5.1, and / u1 ϕ4 (u, v, x, y) = u, x, v + cl (u)x, y − cl (s) ds . u

84

5 Symplectic folding in higher dimensions

5.2

Embedding polydiscs into cubes

In this section we shall study symplectic embeddings of skinny polydiscs P 2n (π, . . . , π, a) = D(π) × · · · × D(π ) × D(a) into cubes C 2n (A) for n ≥ 2. As before, we shall work with more convenient shapes. Define the rectangle R(a, b) by R(a, b) = {(x, y) | 0 < x < a, 0 < y < b}. We denote the 2n-dimensional set R(a, 1) × R(1, b) × · · · × R(1, b) by R n (a, b) = R(a, 1) × R(1, b) × · · · × R(1, b). If b = a, we abbreviate R n (a) = R n (a, a). In view of Lemma 3.1.5, the disc D(a) is symplectomorphic to R(a, 1) and the disc D(π ) is symplectomorphic to R(1, π). Therefore, the polydisc P 2n (π, . . . , π, a), which is symplectomorphic to P 2n (a, π, . . . , π), is symplectomorphic to R n (a, π ). Similarly, the cube C 2n (A) is symplectomorphic to R n (A). The symplectic coordinates will be denoted by (u, v, x2 , y2 , . . . , xn , yn ) ≡ (z1 , z2 , . . . , zn ) ∈ R2n where we set again (u, v) = (x1 , y1 ). We abbreviate x = (x2 , . . . , xn ) and y = (y2 , . . . , yn ) as well as 1y = (1, . . . , 1) ∈ Rn−1 (y). We shall associate with each triple a > 2π , N ∈ N, n ≥ 2 a symplectic embedding procedure n (a) : R n (a, π) → R2n . φN In the following description we again neglect the arbitrarily small δ-terms appearing in the actual construction. If n = 2, we proceed as in 4.4.2, cf. Figure 4.14: For a > 2π and N ∈ N we define u1 = u21,N (a) by a = 2π + (N + 1)(u1 − π ).

(5.2.1)

Then u1 > π . We first fold R 2 (a, π) on the right into the y2 -direction by applying the map (z1 , z2 ) → φr+ (z1 , z2 ) at u = u1 . Here, φr+ is the restriction to R 2 (a, π ) of the map φr+ introduced in 5.1.1. 2 (a) terminates at this point. Indeed, in view If N = 1, the embedding procedure φN of definition (5.2.1), “a is used up” and the front face of the second floor of the image

85

5.2 Embedding polydiscs into cubes

touches the hyperplane {u = 0}. If N ≥ 2, the inequality u1 > π implies that we can then fold the second floor

(u, v, x2 , y2 ) ∈ φr+ (R 2 (a, π)) | u < u1 , y2 > π of the image on the left into the y2 -direction by applying the map (z1 , z2 ) → φl+ (z1 , z2 ) to the second floor at u = π . Going on this way, we altogether fold N times into the y2 -direction by alternatingly folding the last floor of the image on the right at u = u1 2 (a) terminates. and on the left at u = π . At this point the embedding procedure φN Indeed, in view of definition (5.2.1), “a is used up” and the front face of the last floor of the image touches the hyperplane {u = u1 + π } if N is even and the hyperplane {u = 0} if N is odd. 3 (a) can be visualized by Figure 5.2. For If n = 3, the embedding procedure φN a > 2π and N ∈ N we define u1 = u31,N (a) by a = 2π + (N + 1)2 (u1 − π ).

(5.2.2)

y3

y2

π

●

π

u1

u1 + π

u

3 (a) : R 3 (a, π) → R2n with N = 4. Figure 5.2. The first 5 folds of an embedding φN

Then u1 > π . Set a = 2π + (N + 1)(u1 − π). The first N folds of the embedding 3 (a) yield a symplectic embedding of R 3 (a, π ) into R6 whose restriction procedure φN

86


to R 3 (a , π) is 2 (a ) × id : R 2 (a , π) × R(1, π) → R4 × R(1, π). φN

We next fold once into the y3 -direction. If N is even, we do this by applying the map (z1 , z2 , z3 ) → (z1 , z2 , z3 )

where (z1 , z3 ) = φr+ (z1 , z3 ) and z2 = z2

to the N + 1’st floor of the image at u = u1 , and if N is odd, we do this by applying the map (z1 , z2 , z3 ) → (z1 , z2 , z3 )

where (z1 , z3 ) = φl+ (z1 , z3 ) and z2 = z2

to the N + 1’st floor of the image at u = π , see Figure 5.2. We then fold the part of the image on which y3 > π exactly N times into the (−y2 )-direction by using restrictions of the maps φr− and φl− and thereby fill a second z1 -z2 -layer. If N = 1, the embedding 3 (a) terminates at this point. Indeed, in view of definition (5.2.2), “a is procedure φN used up” and the front face of the last floor of the image touches the hyperplane {u = 0}. If N ≥ 2, we fold a second time into the y3 -direction, and fill a third z1 -z2 -layer. Going on this way, we altogether fold (N + 1)2 − 1 times, in which we fold N times into the y3 -direction, and thereby fill N + 1 z1 -z2 -layers. At this point 3 (a) terminates. Indeed, in view of definition (5.2.2), “a the embedding procedure φN is used up” and the front face of the last floor of the image touches the hyperplane {u = u1 + π} if N is even and the hyperplane {u = 0} if N is odd. n (a) for n ≥ 4, we proceed by In order to describe the embedding procedure φN induction and assume that we have described the symplectic embeddings n−1 (a ) : R n−1 (a , π) → R2n−2 , φN

a > 2π.

We define u1 = un1,N (a) by a = 2π + (N + 1)n−1 (u1 − π ).

(5.2.3)

Then u1 > π . Set a = 2π + (N + 1)n−2 (u1 − π ). The first (N + 1)n−2 − 1 folds of n (a) yield a symplectic embedding of R n (a, π ) into R2n the embedding procedure φN n whose restriction to R (a , π) is n−1 (a ) × id : R n−1 (a , π) × R(1, π) → R2n−2 × R(1, π). φN

We next fold once into the yn -direction. If N is even, we do this by applying the map (z1 , . . . , zn ) → (z1 , . . . , zn ),

(z1 , zn ) = φr+ (z1 , zn ), zi = zi , i = 2, . . . , n − 1,

to the last floor of the image at u = u1 , and if N is odd, we do this by applying the map (z1 , . . . , zn ) → (z1 , . . . , zn ),

(z1 , zn ) = φl+ (z1 , zn ), zi = zi , i = 2, . . . , n − 1,


87

to the last floor of the image at u = π . We then fill a second z1 -· · · -zn−1 -layer by folding the part of the image on which yn > π exactly (N +1)n−2 −1 times. If N = 1, n (a) terminates at this point in view of definition (5.2.3). the embedding procedure φN If N ≥ 2, we fold a second time into the yn -direction, and fill a third z1 -· · · -zn−1 -layer. Going on this way, we altogether fold (N + 1)n−1 − 1 times, in which we fold N times into the yn -direction, and thereby fill N + 1 z1 -· · · -zn−1 -layers. At this point n (a) terminates in view of definition (5.2.3). the embedding procedure φN The following proposition generalizes Proposition 4.4.4 to arbitrary dimension. Proposition 5.2.1. Assume a > 2π . Then the polydisc P 2n (π, . . . , π, a) symplecti2n (a) + ) for every > 0, where cally embeds into the cube C 2n (fPC (N + 1)π, (N − 1)N n−1 < πa − 2 ≤ (N − 1)(N + 1)n−1 , 2n (a) = fPC a−2π + 2π, (N − 1)(N + 1)n−1 < πa − 2 ≤ N(N + 1)n−1 . (N+1)n−1 Proof. Fix a > 2π and N ∈ N. We define u1 by equation (5.2.3). The previously described embedding procedure yields a symplectic embedding n (a) : R n (a, π) → R n (AN (a) + ) φN

where AN (a) := max{u1 + π, (N + 1)π }. Solving equation (5.2.3) for u1 we find that a − 2π + 2π, (N + 1)π . AN (a) = max (N + 1)n−1 Optimizing the choice of N ∈ N, we conclude that the polydisc P 2n (π, . . . , π, a) 2n (a) + ) for any > 0, where f 2n (a) symplectically embeds into the cube C 2n (fPC PC is defined by 2n (a) = min{AN (a) | N = 1, 2, . . . }. fPC This completes the proof of Proposition 5.2.1.

2

Remarks 5.2.2. 1. Arguing as in 4.4.2 we see that for a ∈ ]π, 2π ] multiple symplectic folding does not provide a better embedding result then the inclusion P 2n (π, . . . , π, a) → C 2n (a), and that the procedure proving Proposition 5.2.1 is optimal in the sense that we cannot embed P 2n (π, . . . , π, a) into a cube smaller than 2n (a)) by multiple symplectic folding. C 2n (fPC 2n (a), n ≥ 3, are compared with the results yielded by La2. The functions fPC grangian folding in 7.2.2.2. 3

In view of the proof of the second statement in Theorem 3, which will be completed in Section 6.1, we also prove

88


Proposition 5.2.3. Fix n ≥ 2. For every a > 3π there exists a natural number N(a) and a symplectic embedding ϕa : R n (a, π) → R n ((N (a) + 1)π ) such that the following assertions hold. (i) If u < π, ϕa (u, v, x, y) = (u, v, x, y), and if u > a − π , ϕa (u, v, x, y) = u − a + (N(a) + 1)π, v, x, y + N(a)π 1y . |ϕa (R n (a, π))| = 1. a→∞ |R n ((N(a) + 1)π )|

(ii) lim

Proof. Fix n ≥ 2 and N ∈ 2N. We set u1 = N π and aˆ N = 2π + (N + 1)n−1 (u1 − π ) = 2π + (N + 1)n−1 (N − 1)π. n (a We recall that in the previous description of the symplectic embedding φN ˆ N ) we have neglected the arbitrarily small δ-terms appearing in the actual construction. Define δN > 0 by π . δN = 3 (N + 1)n−1 − 1

Since N ≥ 2 we find that u1 − 2δN > π + 2δN . In the actual construction associated with aˆ N and N we can therefore achieve the i’th fold as follows. We fold the last floor of the image at u = u1 − 2δN if i is odd and at u = π + 2δN if i is even in such a way that the u-length of the part of the last floor which is mapped to the i’th stairs is equal to δN . After folding (N + 1)n−1 − 1 times we thereby obtain a symplectic embedding n φN (aˆ N , δN ) : R n (aN , π) → R n ((N + 1)π ))

where aN = 2(π + 2δN ) + (N + 1)n−1 (u1 − π − 4δN ) + ((N + 1)n−1 − 1)δN = 2π + (N + 1)n−1 (u1 − π ) − 3 (N + 1)n−1 − 1 δN

(5.2.4)

= π + (N + 1)n−1 (N − 1)π. n (a ˆ N , δN ). In view of the construction of ψN and the inequality We abbreviate ψN = φN π + 2δN < u1 − 2δN we have

ψN (u, v, x, y) = (u, v, x, y)

if u < π + 2δN ,

ψN (u, v, x, y) = (u − aN + (N + 1)π, v, x, y + Nπ 1y ) if u > aN − π − 2δN .

(5.2.5) (5.2.6)

89


Notice that aN < aN +2 for every N ∈ 2N

and

aN → ∞ as N → ∞.

(5.2.7)

The function N : ]π, ∞[ → N, N (a) := min{ N ∈ 2N | aN ≥ a },

(5.2.8)

is therefore well-defined, and N (aN ) = N. Fix a > 3π . Since a ≤ aN (a) , we find a symplectic embedding βa : R(a) → R(aN(a) ) which is the identity on {u < π } and the translation (u, v) → (u + aN(a) − a, v) on {u > a − π }, cf. Figure 3.6 for the case a < aN(a) . We define the symplectic embedding ϕa : R n (a, π) → R n ((N (a) + 1)π ) by ϕa = ψN(a) (βa × id2n−2 ). In view of the formulae (5.2.5) and (5.2.6) and in view of its definition, ϕa meets assertion (i) in Proposition 5.2.3. In order to verify assertion (ii), we first of all observe that n ϕa R (a, π) = R n (a, π) and ϕa R n (a, π ) ⊂ R n ((N (a) + 1)π ) for all a > 3π . Therefore, 1 ≥

|ϕa (R n (a, π))| |R n (a, π )| a = = n n |R ((N(a) + 1)π)| |R ((N(a) + 1)π )| (N(a) + 1)n π

(5.2.9)

for all a > 3π . Assume now that a > a2 . In view of (5.2.7) and the definition (5.2.8) of N (a) we have a ∈ ]aN(a)−2 , aN(a) ]. Using this and the formula (5.2.4) for aN (a)−2 we can further estimate aN(a)−2 (N(a) − 1)n−1 (N (a) − 3) + 1 a > = . (N (a) + 1)n π (N (a) + 1)n π (N(a) + 1)n π

(5.2.10)

The definition (5.2.8) and (5.2.7) imply that N (a) → ∞

as a → ∞.

(5.2.11)

Combining the estimates (5.2.9) and (5.2.10) we therefore conclude that |ϕa (R n (a, π ))| = 1, a→∞ |R n ((N(a) + 1)π )| lim

and so the proof of Proposition 5.2.3 is complete.

2

90


5.3

Embedding ellipsoids into balls

In this section we shall study a problem closely related to symplectically embedding skinny ellipsoids E 2n (π, . . . , π, a) into balls B 2n (A). As in Chapter 3 we start with replacing these sets by symplectomorphic sets which are more convenient to work with. Recall that given a domain U ⊂ R2n and λ > 0 we set λU = {λz ∈ R2n | z ∈ U }. Reorganizing the coordinates, we consider the Lagrangian splitting Rn (x) × Rn (y) of R2n . We set 6 7 .n xi < 1 ⊂ Rn (x), (a1 , . . . , an ) = 0 < x1 , . . . , xn i=1 ai 2(b1 , . . . , bn ) = {0 < yi < bi , 1 ≤ i ≤ n} ⊂ Rn (y), and we abbreviate n (a) = (a, . . . , a) and 2n (b) = 2(b, . . . , b). Lemma 5.3.1. Assume > 0. Then (i) the ellipsoid E(a1 , . . . , an ) symplectically embeds into the Lagrangian product ((1 + )(a1 , . . . , an )) × 2n (1), (i) the Lagrangian product (a1 , . . . , an ) × 2n (1) symplectically embeds into the ellipsoid (1 + )E(a1 , . . . , an ). + Proof. (i) Define by ni=1 ai = . Replacing the parameter a in the proof of Lemma 3.1.8 (i) by ai , 1 ≤ i ≤ n, we obtain area and orientation preserving diffeomorphisms αi : D(ai ) → R(ai ) satisfying xi (αi (zi )) ≤ π|zi |2 +

for all zi ∈ D(ai ), 1 ≤ i ≤ n,

cf. Figure 3.3. For (z1 , . . . , zn ) ∈ E(a1 , . . . , an ) we then find n . xi (αi (zi )) i=1

ai

≤

n . π |zi |2 i=1

ai

+

< 1 + . ai

It follows that the restriction of the symplectic embedding α1 × · · · × αn : D(a1 ) × · · · × D(an ) → R2n to E(a1 , . . . , an ) is as desired. + (ii) Define by ni=1 ai = 2 . Replacing the parameters a and in the proof of Lemma 3.1.8 (ii) by ai and , 1 ≤ i ≤ n, we obtain area and orientation preserving embeddings ωi : R(ai ) → D(ai + ) satisfying π|ωi (zi )|2 ≤ xi +

for all zi = (xi , yi ) ∈ R(ai ), 1 ≤ i ≤ n,

5.3 Embedding ellipsoids into balls

91

cf. Figure 3.3. For (z1 , . . . , zn ) ∈ (a1 , . . . , an ) × 2n (1) we then find n . π |ωi (zi )|2 i=1

ai

≤

n . xi i=1

ai

+

< 1 + 2 < (1 + )2 . ai

It follows that the restriction of the symplectic embedding ω1 × · · · × ωn : R(a1 ) × · · · × R(an ) → R2n to (a1 , . . . , an ) × 2n (1) is as desired.

2

In view of Lemma 5.3.1 we may think of an ellipsoid as a Lagrangian product of a simplex and a cube. In the setting of symplectic folding, however, we want to work with sets which fibre over a symplectic rectangle. We therefore set again (u, v) = (x1 , y1 ) and define the 2n-dimensional trapezoid T n (a, b) by (u, v, x, y) ∈ R2 × Rn−1 (x) × Rn−1 (y) | n . T (a, b) = (u, v) ∈ R(a), (x, y) ∈ 1 − ua n−1 (b) × 2n−1 (1) Then T n (a, b) = (a, b, . . . , b) × 2n (1). If b = a, we abbreviate

T n (a)

=

(5.3.1)

T n (a, a).

Corollary 5.3.2. Assume > 0. Then (i) E 2n (π, . . . , π, a) symplectically embeds into T n (a + , π + ), (ii) T n (a) symplectically embeds into B 2n (a + ). Proof. (i) The ellipsoid E 2n (π, . . . , π, a) is symplectomorphic to the ellipsoid E 2n (a, π, . . . , π), and by Lemma 5.3.1 (i) and the identity (5.3.1) this ellipsoid symplectically embeds into (1 + )(a, π, . . . , π) × 2n (1) = T n (a + a, π + π ) for every > 0. The claim thus follows. (ii) follows from the identity (5.3.1) and Lemma 5.3.1 (ii).

2

By Corollary 5.3.2 the problem of symplectically embedding skinny ellipsoids E 2n (π, . . . , π, a) into balls B 2n (A) is equivalent to the problem of symplectically embedding trapezoids T n (π, a) into trapezoids T n (A). In view of the proof of the first statement in Theorem 3, which will be completed in Section 6.2, we shall, however, consider a somewhat different embedding problem. Instead of embeddings of T n (π, a) we shall study embeddings of the larger set Sa := R(a) × n−1 (π) × 2n−1 (1) ⊂ R2 × Rn−1 (x) × Rn−1 (y) into T n (A).

92


Proposition 5.3.3. Fix n ≥ 2. For every a > 3π there exists a natural number l(a) and a symplectic embedding ϕa : Sa → T n (l(a)2 ) such that the following assertions hold. (i) If u < π, x, ϕa (u, v, x, y) = u, v, (l(a)−1)l(a) π

π (l(a)−1)l(a)

y ,

and if u > a − π, π ϕa (u, v, x, y) = u − a + (l(a) − 1)l(a), v, l(a) π x, l(a) y . |ϕa (Sa )| = 1. (ii) lim n a→∞ T (l(a)2 ) Proof. The proof of Proposition 5.3.3 is more difficult than the proof of the analogous Proposition 5.2.3. The reason is that for n ≥ 4 it is impossible to fill a large (n − 1)simplex with small (n − 1)-simplices. We shall therefore repeatedly rescale the fibres of Sa and fill the cube-factor 2n−1 (1) of the fibres of T n l(a)2 with the small cubefactors of the rescaled fibres. Fix n ≥ 2 and a > 3π. We prove Proposition 5.3.3 in six steps. In the first four steps we construct for each odd number l > 3nπ a symplectic embedding ψl of the unbounded set S∞ := { 0 < u, 0 < v < 1 } × n−1 (π ) × 2n−1 (1) into R2n . The basic idea behind the embeddings ψl is explained in Figure 5.3. In Step 5 we associate to a > 3π an odd number l(a) and use the embeddings ψl to construct symplectic embeddings ϕa : Sa → T n (l(a)2 ) which meet assertion (i). In Step 6 we verify that these embeddings also meet assertion (ii). Step 1. Preparations. Fix l ∈ 2N + 1 with l > 3nπ . We define subsets Pi = Pi (l) of T n (l 2 ) by

Pi := (u, v, x, y) ∈ T n (l 2 ) | (i − 1)l 3nπ, we find that ki − ki+1 > 3, 1 ≤ i ≤ l − 2, and kl−1 > 3. We may therefore define even numbers Ni = Ni (l) by 6 7 ki Ni := max N ∈ 2N N + 1 < ki − ki+1 , 1 ≤ i ≤ l − 2, (5.3.4)

Nl−1 := max { N ∈ 2N | N + 1 < kl−1 } .

(5.3.5)

93


l2 (l − 1)l

u (l − 1)l

l

l2

Figure 5.3. The embedding ψl : S∞ → R2n for l = 7.

In view of definition (5.3.4) we have 0 < di := 1 −

1 ki+1

−

Ni +1 ki

≤

2 ki ,

1 ≤ i ≤ l − 2,

(5.3.6)

and in view of definition (5.3.3) we have ki+1 < ki , 1 ≤ i ≤ l − 2. Then ei :=

2ki+1 ki +ki+1

di ∈ ]0, di [ ,

1 ≤ i ≤ l − 2.

(5.3.7)

The set S∞ is symplectomorphic to the set F1 := { 0 < u, 0 < v < 1 } × n−1 (k1 π ) × 2n−1

1 k1

via the linear symplectomorphism σ : S∞ → F1 ,

(u, v, x, y) → u, v, k1 x, k11 y .

In view of the definitions of k1 and P1 , the fibres n−1 (k1 π ) × 2n−1 (1/k1 ) of F1 are contained in the fibres of P1 , cf. Figure 5.4. Step 2. Multiple folding in P1 . We symplectically embed a part of F1 into P1 by the multiple folding procedure described in Section 5.2. In the following description we shall again neglect the arbitrarily small δ-terms appearing in the actual construction.

94

5 Symplectic folding in higher dimensions l2 k1 π

P1

l2

l

u

Figure 5.4. The subset P1 of T n (l 2 ).

We first fold F1 at u = l − π on the right into the y2 -direction. The lifting map involved in this folding has the form / u c(s) ds (u, x2 , v, y2 ) → u, x2 , v + c(u)x2 , y2 + 0

where the cut off function c : R → [0, 1/(k1 π )] has support in [l − π, l]. We next fold the part of the image on which y2 > 1/k1 at u = π on the left into the y2 -direction. This is possible because l > 2π . The length of the second floor is l − 2π . We fold N1 times alternatingly at u = l − π on the right and at u = π on the left into the y2 -direction. We then fold once into the y3 -direction. Since N1 is even, we do this by folding the part of the image on which y2 > N1 /k1 at u = l − π on the right. Going on this way, we altogether fold (N1 + 1)n−1 − 1 times, in which we fold N1 times into the yn -direction. Denote the multiple folding embedding F1 → R2n thus obtained by µ1 . The image of the projection of µ1 (F1 ) onto Rn−1 (y) is contained in the cube 2n−1 ((N1 + 1)/k1 ), cf. Figure 5.6. Since N1 is even, the infinite end of µ1 (F1 ) points into the u-direction. More precisely, the last floor of µ1 (F1 ) is the subset F1 := ]π, ∞[ × ]0, 1[ × n−1 (k1 π ) × C1 of R2 × Rn−1 (x) × Rn−1 (y) where C1 = (y2 , . . . , yn ) Nk11 < yj
π + δ1 }, cf. Figure 3.6. Define s1 := π + k1 π d1 . In view of the second inequality in (5.3.6) and l > 3nπ we find π + (n − 1)s1 = π + (n − 1)π(1 + k1 d1 ) ≤ π + (n − 1)π(1 + 2) < 3nπ < l. In view of the definition (5.3.8) of δ1 we have s1 k11π = δ1 + k11 + e1 . For j = 2, . . . , n we therefore find a cut off function , ∞cj : R → [0, 1/(k1 π )] with support [π + (j − 2)s1 , π + (j − 1)s1 ] and such that 0 cj (s) ds = 1/k1 + e1 . The symplectic embedding ϕ1 : Im β1 × n−1 (k1 π ) × C1 → R2n ,

(u, v, x, y) → (u , v , x , y )

defined by u = u,

n . v = v+ cj (u)xj , j =2

xj = xj ,

/

yj = yj +

u

cj (s) ds,

j = 2, . . . , n,

0

is the identity on {u < π}, maps {u < l} to P1 and translates {u > l} to the set F1 := { u > l, 0 < v < 1 } × n−1 (k1 π ) × C1 ,

97


cf. Figure 5.5. We restrict the symplectic embedding ϕ1 (β1 × idn−2 ) : ]0, ∞[ × ]0, 1[ × n−1 (k1 π ) × C1 → R2n to the intersection of its domain with µ1 (F1 ) and extend this restriction by the identity to the symplectic embedding λ1 : µ1 (F1 ) → R2n . In view of the construction of the “translation” β1 × id2n−2 the u-length of the part of F1 which λ1 embeds into P1 is δ1 . In view of the identity (5.3.9) we therefore conclude that the u-length u1 of the part of F1 which λ1 µ1 embeds into P1 is equal to (5.3.10) u1 = (l − π ) + (N1 + 1)n−1 − 2 (l − 2π ). Construction of the deformation α1 . The deformation of the fibres n−1 (k1 π ) × C1 of F1 to fibres n−1 (k2 π ) × C2 is based on the following lemma. Lemma 5.3.4. There exists a symplectic embedding α : ]0, k1 π [ × 0, N1k+2 + e1 → R2 1 which restricts to the identity on { (x, y) | y ≤ (N1 + 1)/k1 }, restricts to the affine map (5.3.11) (x, y) → kk21 x, kk21 y + 1 − k12 (N1 + 2 + k1 e1 ) on { (x, y) | y ≥ (N1 + 1)/k1 + e1 }, and is such that x (α(x, y)) ≤ x and y (α(x, y)) < 1

(5.3.12)

for all (x, y) ∈ ]0, k1 π [ × ]0, (N1 + 2)/k1 + e1 [, cf. Figure 5.7. y

y 1 N1 +2 k1 + e 1

1 k2

1 k1 e1

N1 +1 k1

α

d1 x

x k1 π

k2 π

Figure 5.7. The map α.

Proof. Choose a smooth function h : R → R such that (i) h(w) = 1 for w ≤ (ii) h (w) < 0 for w ∈

N1 +1 k1 ,

5

N1 +1 N1 +1 k 1 , k1

4 + e1 ,

k1 π

98


(iii) h(w) =

k2 k1

for w ≥

N1 +1 k1

+ e1 .

In view of the definition (5.3.7) of e1 and the inequality k2 < k1 we have e1 < d1 < e1 kk21 . We may therefore further require that / (iv)

N1 +1 k1 +e1 N1 +1 k1

1 dw = d1 . h(w)

Then the map

α : ]0, k1 π[ × 0,

N1 +2 k1

+ e1 → R , (x, y) → 2

/

y

h(y)x, 0

1 dw h(w)

is a symplectic embedding which is as required.

2

Denote by α1 the restriction to λ1 (µ1 (F1 )) of the symplectic embedding ]0, ∞[×]0, 1[ × 2n−1 (k1 π ) × 2n−1 N1k+2 + e1 → R2 × Rn−1 (x) × Rn−1 (y), 1

(u, v, x2 , . . . , xn , y2 , . . . , yn ) → (u, v, x2 , . . . , xn , y2 , . . . , yn )

where (xj , yj ) = α(xj , yj ), j = 2, . . . , n. In view of the inequalities (5.3.12), α1 maps the set λ1 (µ1 (F1 )) ∩ P1 into P1 , and in view of (5.3.11), α1 maps the set F1 = λ1 (µ1 (F1 )) \ P1 symplectically onto the set F2 := { u > l, 0 < v < 1 } × n−1 (k2 π ) × C2 , cf. Figure 5.5. Step 4. Construction of ψl . The symplectic embedding ψl : S∞ → R2n is the composition of symplectic embeddings λl−1 µl−1 (αl−2 λl−2 µl−2 ) · · · (α2 λ2 µ2 ) (α1 λ1 µ1 ) σ. Here, σ is the map defined in Step 1, µ1 is the map constructed in Step 2, λ1 and α1 are the maps constructed in Step 3, and the maps µi , λi , αi , i ≥ 2, are constructed in a similar way as µ1 , λ1 , α1 . In order to describe the maps µi , λi , αi , i ≥ 2, in more detail, we assume by induction that we have already constructed embeddings µj , λj , αj , j = 1, . . . , i − 1, where i ≤ l − 2, and that the set { (u, v, x, y) ∈ (αi−1 λi−1 µi−1 · · · α1 λ1 µ1 )(F1 ) | u > (i − 1)l } is the set Fi := { u > (i − 1)l, 0 < v < 1 } × n−1 (ki π ) × Ci

99


where

Ci :=

(y2 , . . . , yn ) | 0 < yj < (y2 , . . . , yn ) | 1 −

1 ki

1 ki ,

j = 2, . . . , n

< yj < 1, j = 2, . . . , n

if i is odd, if i is even.

The multiple folding map µi embeds a part of Fi into Pi . We fold Ni times alternatingly at u = il − π on the right and at u = (i − 1)l + π on the left into the y2 -direction, and so on. Since Ni is even, the last floor of the image of µi is Fi := ](i − 1)l + π, ∞[ × ]0, 1[ × n−1 (ki π ) × Ci where Ci

=

(y2 , . . . , yn ) |

Ni ki

< yj
(i − 1)l + π + δi } to the set Fi := { u > il, 0 < v < 1 } × n−1 (ki π ) × Ci where

 Ni +2 + e < y < + e , j = 2, . . . , n (y2 , . . . , yn ) | Nik+1  i j i k  i i   if i is odd,

Ci := N +2 N +1 i i  (y2 , . . . , yn ) | 1 − ki − ei < yj < 1 − ki − ei , j = 2, . . . , n    if i is even.

100


The u-length of the part of Fi which λi embeds into Pi is δi . In view of the identity (5.3.13) we therefore conclude that the u-length ui of the part of Fi which λi µi embeds into Pi is equal to ui = (l − π ) + (Ni + 1)n−1 − 2 (l − 2π ). (5.3.14) The symplectic embedding αi : Im λi → R2n maps the set Im λi ∩ Pi into Pi and maps Fi = Im λi \ Pi onto the set Fi+1 := { u > il, 0 < v < 1 } × n−1 (ki+1 π ) × Ci+1 . The deformation αi is constructed the same way as α1 if i is odd, and in a similar way if i is even. Next, the multiple folding map µl−1 embeds a part of Fl−1 into Pl−1 . We fold Nl−1 times alternatingly at u = (l − 1)l − π on the right and at u = (l − 2)l + π on the left into the y2 -direction, and so on. Since Nl−1 is even and l is odd, the last floor of the image of µl−1 is Fl−1 := ](l − 2)l + π, ∞[ × ]0, 1[ × n−1 (kl−1 π ) × Cl−1

where Cl−1 = (y2 , . . . , yn ) | 1 −

Nl−1 +1 kl−1

< yj < 1 −

Nl−1 kl−1 ,

j = 2, . . . , n .

We define δl−1 := 1/kl−1 and choose the δ-terms in the actual construction of µl−1 in such a way that the u-length ul−1 of the part of Fl−1 mapped to µl−1 (Fl−1 ) \ Fl−1 is equal to ul−1 = (l − π ) + (Nl−1 + 1)n−1 − 2 (l − 2π ) − δl−1 . (5.3.15) Finally, define sl−1 := kl−1 π − (Nl−1 + 1)π + π . In view of the definition (5.3.5) of Nl−1 we have +1 2 1 − Nl−1 kl−1 ≤ kl−1 . This and l > 3nπ imply that (l − 2)l + π + (n − 1)sl−1 < (l − 1)l − π. Proceeding as in the construction of λi , i even, we therefore find a symplectic embed and translates ding λl−1 : Im µl−1 → R2n which is the identity on Im µl−1 \ Fl−1 { (u, v, x, y) ∈ Fl−1 | u > (l − 2)l + π + δl−1 } to the set Fl := { u > (l − 1)l − π, 0 < v < 1 } × n−1 (kl−1 π ) × Cl where

Cl := (y2 , . . . , yn ) | 0 < yj
al − π.

Using the definition (5.3.8) of δ1 and (5.3.7), (5.3.6) and (5.3.3) we find that δ1 = d1 − e1 < d1 ≤

2 k1

< 2π.

Since l > 3nπ, we therefore find that π < l − π − δ1 . This and the definition (5.3.3) of k1 and kl−1 imply that π ψl (u, v, x, y) = u, v, (l−1)l π x, (l−1)l y ψl (u, v, x, y) = u − al + (l − 1)l, v, πl x, πl y

if u < π,

(5.3.18)

if u > al − π.

(5.3.19)

Before defining the embeddings ϕa , we further investigate the sequence (al ). Lemma 5.3.5. (i) al < al+2 for every l ∈ 2N + 1 with l > 3nπ . (ii) al → ∞ as l → ∞.

102


Proof. (i) Fix l ∈ 2N + 1 with l > 3nπ . As in Step 1 we abbreviate ki = ki (l), Ni = Ni (l),

i = 1, . . . , l − 1.

Moreover, we set l = l + 2 and abbreviate ki = ki (l ), Ni = Ni (l ), By computation, l−i l−i−1

Using the definition (5.3.3) of ki −

ki ki+1

=

1 π (l

i = 1, . . . , l − 1.

l −i i = 1, . . . , l − 2. l −i−1 , ki and ki , we therefore find

>

− i)l −

l−i l−i−1

a−π }, cf. Figure 3.6 for


103

the case a < al(a) . We finally define the symplectic embedding ϕa : Sa → T n (l(a)2 ) by ϕa = ψl(a) (βa × id2n−2 ). In view of the formulae (5.3.18) and (5.3.19) and in view of its definition, ϕa meets assertion (i) in Proposition 5.3.3. Step 6. Verification of assertion (ii) in Proposition 5.3.3. Recall that assertion (ii) in Proposition 5.3.3 claims that |ϕ (S )| a a →1 T n (l(a)2 )

as a → ∞.

Lemma 5.3.6. Assertion (5.3.23) is a consequence of n 2 T (l ) \ ψl (Sa ) l → 0 as l → ∞. T n (l 2 ) Proof. Since ψl (Sal ) = Sal and ψl (Sal ) ⊂ T n (l 2 ), we have n 2 n 2 Sa T (l ) − ψl (Sa ) T (l ) \ ψl (Sa ) l l l =1− =1− . T n (l 2 ) T n (l 2 ) T n (l 2 )

(5.3.23)

(5.3.24)

(5.3.25)

Fix ∈ ]0, 1[. The assumption (5.3.24) and (5.3.25) imply that there exists l0 ∈ 2N+1 such that Sa √ l > 1 − for all l ∈ 2N + 1 with l ≥ l0 . (5.3.26) T n (l 2 ) Choosing l0 larger if necessary, we may assume that √ l 4n > 1 − for all l ∈ 2N + 1 with l ≥ l0 . (5.3.27) l+2 Assume now that a > al0 . In view of Lemma 5.3.5 we have a ∈ ]al , al+2 ] for some l ∈ 2N + 1 with l ≥ l0 . The definition (5.3.22) of l(a) implies l(a) = l + 2. Using the estimates (5.3.26) and (5.3.27), we can therefore estimate Sa |Sa | l > T n (l(a)2 ) T n ((l + 2)2 ) n 2 T (l ) Sa l = n T ((l + 2)2 ) T n (l 2 ) Sa l 2n l = (l + 2)2n T n (l 2 ) √ √ > 1− 1− = 1 − .

104


Since ϕa (Sa ) ⊂ T n (l(a)2 ) and |ϕa (Sa )| = |Sa |, we therefore find |ϕa (Sa )| |Sa | = > 1 − . 1 ≥ n T (l(a)2 ) T n (l(a)2 ) Since ∈ ]0, 1[ was arbitrary, Lemma 5.3.6 follows.

2

In order to prove assertion (5.3.24), we fix l ∈ 2N + 1 with l > 3nπ and introduce several subsets of T n (l 2 ). We define the subsets Qi (l) of Pi (l) by Qi (l) := { (u, v, x, y) ∈ Pi (l) | x ∈ n−1 ((l − 1)l) },

i = 1, . . . , l,

and we define subsets Xi (l), Yi (l) and Zi (l) of Pi (l) by Xi (l) := Pi (l) \ Qi (l), i = 1, . . . , l,

Yi (l) := (u, v, x, y) ∈ Qi (l) | u ∈ / ](i − 1)l + π, il − π [ , i = 1, . . . , l − 1,

Zi (l) := (u, v, x, y) ∈ Qi (l) | y ∈ / 2n−1 Nik+1 , i = 1, . . . , l − 1. i We also set X(l) =

l 2

Xi (l),

Y (l) =

i=1

l−1 2

Yi (l),

Z(l) =

i=1

l−1 2

Zi (l).

i=1

The sets X(l) and Y (l) are illustrated in Figure 5.8, and for Z1 (l) we refer to Figure 5.6. l2

X1 (l)

(l − 1)l

Y1 (l) Yl−1 (l) Xl (l) u π l−π Figure 5.8. The subsets X(l) =

(l − 1)l 1l

i=1 Xi (l) and Y (l) =

l2

1l−1

n 2 i=1 Yi (l) of T (l ).

105


We recall from the construction of the embedding ψl that the unfilled space Xi (l) in Pi (l) \ ψl (Sal ) is caused by the fact that the size of the fibres of Fi is constant, that the space Y (l) was needed for folding and contains all stairs, and that the space Zi (l) is the union of the space needed to deform the fibres of Fi and the space caused by 2n the fact that the Ni have to be even integers. Denote the closure of ψl (Sal ) in R by ψl (Sal ). By construction of ψl we have |ψl (Sal )| = ψl (Sal ) and T n (l 2 ) \ ψl (Sal ) ⊂ X(l) ∪ Y (l) ∪ Z(l). We conclude that n 2 T (l ) \ ψl (Sa ) = T n (l 2 ) \ ψl (Sa ) ≤ |X(l)| + |Y (l)| + |Z(l)| . l l

(5.3.28)

Lemma 5.3.7. n 2 T (l ).

(i) |X(l)|
1 − (n − 1) kl−1 = 1 − (n − 1) 2π l .

(5.3.30)

The definitions of Zi (l) and Qi (l) and the estimates (5.3.29) and (5.3.30) now yield n−1 2 (1) \ 2n−1 Ni +1

n−1 |Zi (l)| ki Ni +1 2πn = 1 − < (n − 1) 2π = ki l < l , 2n−1 (1) |Qi (l)| i = 1, . . . , l − 1, and so |Z(l)| =

l−1 .

|Zi (l)|
π having the following property. For every a ≥ a0 there exist a number λ(a) > 0 and a symplectic embedding a : λ(a)R n (a, π) → M such that µ M \ a λ(a)R n (a, π ) < . Proof. We shall proceed along the following lines. We shall first fill almost all of M with finitely many symplectically embedded cubes whose closures are disjoint, and

108


connect these cubes by neighbourhoods of lines. In view of Proposition 5.2.3 we can then almost fill the cubes with symplectically embedded thin polydiscs, and we shall use the neighbourhoods of the lines to pass from one cube to another, cf. Figure 6.1.

M

Figure 6.1. Filling M with a thin polydisc.

Step 1. Filling M by cubes. We denote by C(s) the 2n-dimensional open cube.

C(s) = (x1 , y1 , . . . , xn , yn ) ∈ R2n | 0 < xi < s, 0 < yi < s, i = 1, . . . , n . Lemma 6.1.2. For every > 0 there exist s ∈ ]0, 1[, an integer k and a symplectic embedding k 2 γ: Ci (s) → M i=1

of a disjoint union of k translates Ci (s) of C(s) in R2n such that 2

µ M \γ Ci (s) < . Proof. We choose for each point p ∈ M a Darboux chart χp : Up → Vp ⊂ M. We can assume that the sets Up are bounded. As every manifold, M satisfies the second axiom of countability, and so M is Lindelöf, i.e., every open covering of M has a countable subcovering. We therefore find a countable subcovering {Vpi } of the open covering {Vp } of M. Since the sets Upi ⊂ R2n are bounded, we find points wi ∈ R2n such that the translates Ui := τwi (Upi ), i ≥ 1, are disjoint. We abbreviate χi = χpi τ−wi and Vi = Vpi . We have constructed countably many disjoint Darboux charts χi : Ui → Vi which cover M. We define subsets Vi of M by V1 = V1

and

Vi = Vi \

i−1 0 j =1

Vj , i ≥ 2.

6.1 Proof of lima→∞ paP (M, ω) = 1

2

Then

Vi =

i≥1

0

109

Vi = M.

i≥1

Since the open sets Vi are µ-measurable, the sets Vi are also µ-measurable. It follows that 2 . µ(Vi ) = µ Vi = µ(M). i≥1

i≥1

Since µ(M) < ∞ we therefore find m ∈ M such that m . i=1

µ(Vi ) > µ(M) − . 2

(6.1.1)

Set Ui = χi−1 (Vi ) ⊂ Ui , i = 1, . . . , m. Since Vi is µ-measurable and χi−1 is smooth, Ui is Lebesgue-measurable, i = 1, . . . , m. We therefore find s ∈ ]0, 1[ and finitely many disjoint translates Ci,j (s) ⊂ Ui ,

j = 1, . . . , ji ,

of the cube C(s) such that ji . Ci,j (s) > U − , i 2m

i = 1, . . . , m.

(6.1.2)

j =1

+ We set k = m i=1 ji . Since the sets Ui are disjoint, the k cubes Ci,j (s) are disjoint. Moreover, the embeddings χi : Ui → M are symplectic, and so the embedding γ defined by 2 2 χi C (s) : Ci,j (s) → M γ = i,j

i,j

i,j

is symplectic, and

µ(χi (Ci,j (s))) = Ci,j (s)

and

µ(Vi ) = Ui .

In view of the estimates (6.1.1) and (6.1.2) we therefore find

2 . µ(χi (Ci,j (s))) µ M \γ Ci,j (s) = µ(M) − i,j

. Ci,j (s) = µ(M) − i,j

= µ(M) −

m . i=1

0 be as in Theorem 6.1.1 and set = /3. In view of Lemma 6.1.2 we find s ∈ ]0, 1[ and a symplectic embedding γˆ =

k 2

γî :

i=1

k 2

Cˆ i (s ) → M

i=1

of a disjoint union of k translates Cˆ i (s ) of C(s ) such that 2

µ M \ γˆ Cˆ i (s ) < .

(6.1.3)

We choose s ∈ ]0, s [ so large that k (s )2n − s 2n < .

(6.1.4)

We abbreviate d := (s − s)/2. For each δ ∈ [0, d] we define Ci (δ) = { z + ((i − 1)s − δ, −δ, . . . , −δ) | z ∈ C(s + 2δ) }, and we abbreviate Ci = Ci (0) and Ci = Ci (d), i = 1, . . . , k, cf. Figure 6.2. After choosing s ∈ ]0, s [ larger if necessary we can assume that 2d ≤ s so that the cubes Ci are disjoint. We define wi ∈ R2n through the identity τwi Cˆ i (s ) = Ci , and we define the symplectic embedding γi : Ci → M by γi = γî τ−wi : Ci → M. We denote the restriction of γi to Ci by γi , and we write γ =

k 2

γi :

i=1

k 2 i=1

Ci → M

and

γ =

k 2 i=1

γi :

k 2

Ci → M.

i=1

1 + Ci = |Ci | = ks 2n and Since γ and γ are symplectic, we have µ γ 1 1 + µ γ Ci \ γ Ci = Ci \ Ci = k (s )2n − s 2n . In view of the inequalities (6.1.3) and (6.1.4) we can therefore estimate 1 1 1 1 µ(M) = µ γ Ci + µ γ Ci \ γ Ci + µ M \ γ Ci 1 = ks 2n + k (s )2n − s 2n + µ M \ γˆ Ci (s ) < ks 2n + + = ks 2n + 2 .

(6.1.5)


111

x, y C1

C2

s C1

C3

L2

C2

C3

L1 −d

u 2s

s

3s

4s

5s

Figure 6.2. The cubes Ci and Ci , i = 1, 2, 3, and the lines Li , i = 1, 2.

1 Step 2. Connecting the cubes. In this step we extend the embedding γ : Ci → M to a symplectic embedding of a connected domain. For i = 1, . . . , k −1 we abbreviate Ii = [ (2i − 1)s, 2is ] , and we define straight lines Li (t) : Ii → R2 × Rn−1 (x) × Rn−1 (y) by (t, 0, 0, 0) if i is odd, Li (t) = (t, 0, 0, s1y ) if i is even, cf. Figure 6.2. Then Li (t) ∈ Ci

if t ∈ [ (2i − 1)s, (2i − 1)s + d [,

Li (t) ∈ Ci+1

if t ∈ ] 2is − d, 2is ].

(6.1.6)

For δ > 0 we define the “δ-neighbourhood” Ni (δ) of Li in R2n by Ii × ] − δ, δ [ × (] − δ, δ [ × ] − δ, δ [)n−1 if i is odd, Ni (δ) = Ii × ] − δ, δ [ × (] − δ, δ [ × ] s − δ, s + δ [)n−1 if i is even. Proposition 6.1.3. There exist δ ∈ ]0, d/8[ and a symplectic embedding ρ:

k 2

Ci (d/3) ∪

i=1

k−1 2

Ni (δ) → M.

i=1

1 Ci is connected. Using this Proof. By construction of the map γ , the set M \ γ and the inclusions (6.1.6) we find a smooth embedding k−1 2 i=1

λi :

k−1 2 i=1

Li → M \ γ

2

Ci

112


such that λi (Li (t)) = γi (Li (t))

if t ∈ [ (2i − 1)s, (2i − 1)s + d/2 ],

(6.1.7)

(Li (t)) λi (Li (t)) = γi+1

if t ∈ [ 2is − d/2, 2is ],

(6.1.8)

and such that λi (Li (t)) ∈ M \ γ

1

Ci (d/2)

if t ∈ [ (2i − 1)s + d/2, 2is − d/2 ]. (6.1.9)

We are now going to construct a symplectic extension of λ1 to a neighbourhood of L1 . A symplectic extension of λi to a neighbourhood of Li for i ≥ 2 can be constructed the same way. We denote by

{ e1 (t), e2 (t), . . . , e2n−1 (t), e2n (t) } = ∂x∂ 1 , ∂y∂ 1 , . . . , ∂x∂ n , ∂y∂ n the standard symplectic frame of the tangent space TL1 (t) R2n . Lemma 6.1.4. There exists a smooth 1-parameter family of symplectic frames {fj (t)}, j = 1, . . . , 2n, along the curve λ1 (t) = λ1 (L1 (t)) such that f1 (t) =

d dt λ1 (t)

for all t ∈ [s, 2s]

and such that for all j = 1, . . . , 2n, fj (t) = TL1 (t) γ1 (ej (t)) if t ∈ [ s, s + d/2 ], fj (t) = TL1 (t) γ2 (ej (t)) if t ∈ [ 2s − d/2, 2s ].

(6.1.10) (6.1.11)

d λ1 (t). In view of the identities (6.1.7) Proof. For t ∈ [s, 2s] we define f1 (t) = dt and (6.1.8) for i = 1 the assertions (6.1.10) and (6.1.11) for j = 1 are met. Using (6.1.10) for j = 1 and that γ1 is symplectic we find a smooth vector field f2 (t) along λ1 (t) such that f2 (t) = TL1 (t) γ1 (e2 (t)) if t ∈ [s, s + d/2] (6.1.12) and such that ω f1 (t), f2 (t) = 1 for all t ∈ [s, 2s]. Choose a smooth function c(t) : [s, 2s] → [0, 1] such that 0 if t ≤ 2s − d/2, c(t) = 1 if t ≥ 2s − d/3,

and define a second smooth vector field f2 (t) along λ1 (t) by f2 (t)

=

0

c(t) TL1 (t) γ2 (e2 (t))

if t ≤ 2s − d/2, if t > 2s − d/2.


113

We define the smooth vector field f2 (t) along λ1 (t) by f2 (t) = c(t)f2 (t) + (1 − c(t))f2 (t). In view of the formula (6.1.12) and the definition of f2 (t) the assertions (6.1.10) and (6.1.11) for j = 2 are met, and since γ2 is symplectic, we find ω (f1 (t), f2 (t)) = 1 for all t ∈ [s, 2s]. The linear subspace Vt of Tλ1 (t) M spanned by f1 (t) and f2 (t) is therefore a symplectic subspace. Denote the symplectic complement of Vt in Tλ1 (t) M by Vt⊥ . Since the linear symplectic group Sp(n − 1; R) is path-connected, we find a smooth 1-parameter family of symplectic frames {f3 (t), . . . , f2n (t)} of Vt⊥ , t ∈ [s, 2s], such that the assertions (6.1.10) and (6.1.11) are met for all j = 3, . . . , 2n. The family of symplectic frames {f1 (t), f2 (t), f3 (t), . . . , f2n (t)} of Vt ⊕ Vt⊥ = Tλ1 (t) M is as desired. , Lemma 6.1.5. There exist κ1 > 0 and a smooth embedding σ1 : C1 (d/3) ∪ N1 (κ1 ) ∪ C2 (d/3) → M such that

σ1 C

1 (d/3)

= γ1 ,

σ1 L = λ1 , 1

σ1 C

2 (d/3)

= γ2 ,

and such that TL1 (t) σ1 (ej (t)) = fj (t) for all t ∈ [s, 2s] and j = 1, . . . , 2n. Proof. Define the smooth embedding σ : C1 (d/2) ∪ L1 ∪ C2 (d/2) → M by σ C (d/2) = γ1 , σ L = λ1 , σ C (d/2) = γ2 . 1

1

2

Applying the Whitney extension theorem to the restriction of σ to the closed set C1 (d/3) ∪ L1 ∪ C2 (d/3) we find κˆ > 0 and a smooth map ˆ ∪ C2 (d/3) → M σˆ : C1 (d/3) ∪ N1 (κ) which extends the restriction of σ to C1 (d/3) ∪ L1 ∪ C2 (d/3). We read off from the identities (6.1.7) and (6.1.8) and the inclusions (6.1.9) for i = 1 that the restriction of σ to the compact set C1 (d/3) ∪ L1 ∪ C2 (d/3) is an embedding. After choosing κˆ > 0 smaller if necessary, we can therefore assume that σˆ is an embedding. For each t ∈ [s, 2s] we define the frame {eˆj (t)} of TL1 (t) R2n by eˆj (t) = σˆ ∗ fj (t), j = 1, . . . , 2n. d Since f1 (t) = dt λ1 (t) and σˆ L = λ1 we can compute

(6.1.13)

1

eˆ1 (t) = σˆ ∗ f1 (t) d = σˆ ∗ dt λ1 (t) = σˆ ∗ TL1 (t) λ1 (e1 (t)) = σˆ ∗ TL1 (t) σˆ (e1 (t)) = e1 (t)

(6.1.14)

114


for all t ∈ [s, 2s]. Similarly, the identities (6.1.10), (6.1.11) and σˆ C (d/3) = γ1 , 1 σˆ C (d/3) = γ2 imply that for all j = 1, . . . , 2n, 2

eˆj (t) = ej (t)

if t ∈ [s, s + d/3] ∪ [2s − d/3, 2s ].

(6.1.15)

Define the smooth map α : [s, 2s] × R2n−1 → R2n by

2n .

α L1 (t) +

bi ei (t) = L1 (t) +

j =2

2n .

bi eî (t).

(6.1.16)

j =2

(6.1.14) and (6.1.15)the map α restricts to the identity on In view of the identities ˆ ∪ L1 ∪ C2 (d/3) ∩ N1 (κ) ˆ . We choose κ1 > 0 so small that the C1 (d/3) ∩ N1 (κ) ˆ We restriction of α to N1 (κ1 ) is an embedding whose image is contained in N1 (κ). can now define a smooth embedding σ1 : C1 (d/3) ∪ N1 (κ1 ) ∪ C2 (d/3) → M by

σ1 C

1 (d/3)

= γ1 ,

σ1 N

1 (κ1 )

= σˆ α,

σ1 C

2 (d/3)

= γ2 .

For z ∈ L1 we then have σ1 (z) = σˆ (α(z)) = σˆ (z) = λ1 (z). we read off Moreover, from the identity (6.1.14) and the definition (6.1.16) of α that TL1 (t) α (ej (t)) = eˆj (t). Together with the definitions (6.1.13) we therefore conclude that TL1 (t) σ1 (ej (t)) = TL1 (t) σˆ TL1 (t) α (ej (t)) = TL1 (t) σˆ (eˆj (t)) = fj (t) for all t ∈ [s, 2s] and j = 1, . . . , 2n. The proof of Lemma 6.1.5 is complete.

2

Since the maps γ1 and γ2 are symplectic and since the frames {fj (t)} along λ1 (t) are symplectic, the map σ1 guaranteed by Lemma 6.1.5 is symplectic on the set C1 (d/3) ∪ L1 ∪ C2 (d/3). Applying the proof of Lemma 3.14 in [62] to L1 ⊂ N1 (κ1 ) and to the symplectic forms ω0 and σ1∗ ω, we therefore find δ1 > 0 and a smooth embedding ψ1 : N1 (δ1 ) → N1 (κ1 ) such that ψ1 (N (δ) ∩ C (d/3)) ∪ L ∪ (N (δ) ∩ C (d/3)) = id and ψ1∗ σ1∗ ω = ω0 . (6.1.17) 1

1

1

1

2

Define the embedding ρˆ1 : N1 (δ1 ) → M by ρˆ1 = σ1 ψ1. In view of the first statement in (6.1.17) and the identities σ1 C (d/3) = γ1 and σ1 C (d/3) = γ2 we have 1

ρˆ1 N

1 (δ1 ) ∩ C1 (d/3)

= γ1

and

2

ρˆ1 N

1 (δ1 ) ∩ C2 (d/3)

and in view of the second statement in (6.1.17) we have ρˆ1∗ ω = (σ1 ψ1 )∗ ω = ψ1∗ (σ1∗ ω) = ω0 ,

= γ2 ,


115

x, y d

d/3

C1 (d/3)

C2 (d/3) u

−d/3

2s

s

−d

N1 (δ1 )

Figure 6.3. The domain C1 (d/3) ∪ N1 (δ1 ) ∪ C2 (d/3) of ρ1 .

i.e., ρˆ1 is symplectic. We can therefore define a smooth symplectic embedding ρ1 : C1 (d/3) ∪ N1 (δ1 ) ∪ C2 (d/3) → M by

ρ1 C

1 (d/3)

= γ1 ,

ρ1 N

1 (δ1 )

= ρˆ1 ,

ρ1 C

2 (d/3)

= γ2 .

(6.1.18)

Proceeding as in the construction of ρ1 we find δi > 0 and symplectic embeddings ρi : Ci (d/3) ∪ Ni (δi ) ∪ Ci+1 (d/3) → M such that ρi C (d/3) = γi , i

ρi N (δ ) = ρî , i

i

ρi C

i+1 (d/3)

= γi+1

(6.1.19)

where ρî : Ni (δi ) → M is a symplectic extension of λi , i = 1, . . . , k − 1. In view of the identities (6.1.18) and (6.1.19) the map ρ:

k 2

Ci (d/3) ∪

i=1

k−1 2

Ni (δi ) → M

i=1

defined by ρ C (d/3)∪N (δ ) = ρi C (d/3)∪N (δ ) , i = 1, . . . , k − 1, i

i

i

i

i

i

ρ C

k (d/3)

= ρk−1 C

k (d/3)

is smooth and symplectic. In view of the inclusions (6.1.9) 1 we finally find 1 some δ ∈ ] 0, min {δ1 , . . . , δk , d/8} [ such that the restriction of ρ to Ci (d/3) ∪ Ni (δ) is an embedding. The proof of Proposition 6.1.3 is thus complete. 2

116


Step 3. Replacing Ci ∪ Ni (δ) by a more set. In view of the 1 1 convenient previous two steps we are left with filling the set Ci ∪ Ni (δ) with thin polydiscs. If we would embed a part of a polydisc into the cube C1 by using the multiple folding technique described in Section 5.2, the x-width of the last floor of the embedded part of the polydisc would be s, while the x-width of N1 (δ) is δ < s. In order to pass to C2 we would therefore have to deform the fibres of the last floor. This can be done in a similar way as in Step 3 of the proof of Proposition 5.3.3. reasons we 1 1 For technical shall take a different route, however, and replace the set Ci ∪ Ni (δ) by a set all of whose fibres have x-width δ. We abbreviate ν=

s2 δ .

We define the subset K of R2n by

K = (x1 , y1 , . . . , xn , yn ) ∈ R2n | 0 < xi < δ, 0 < yi < ν, i = 1, . . . , n , and for i = 1, . . . , k we define subsets Ki of R2n by Ki = { z + ((i − 1)(ν + s), 0, . . . , 0) | z ∈ K } ,

i = 1, . . . , k.

Notice that the sets Ki are symplectomorphic to the cubes Ci . We abbreviate Iis = [ (i − 1)(ν + s) + ν, i(ν + s) ] ,

i = 1, . . . , k − 1,

and define the “δ-halfneighbourhood” Hi in R2n by Iis × ] 0, δ [ × (] 0, δ [ × ] 0, δ [)n−1 Hi = Iis × ] 0, δ [ × (] 0, δ [ × ] ν − δ, ν [)n−1

if i is odd, if i is even,

cf. Figure 6.7. Proposition 6.1.6. There exists a symplectic embedding ξ:

k 2 i=1

Ki ∪

k−1 2 i=1

Hi →

k 2

Ci (d/3) ∪

i=1

k−1 2

Ni (δ).

i=1

Proof. We start with Lemma 6.1.7. There exists a symplectic embedding α : ]0, δ[ × ]0, ν[ → ]0, s + d/3[ × ]0, s[ which restricts to the identity on { (x, y) | y < δ } and to the translation (x, y) → (x, y − ν + s) on { (x, y) | y > ν − δ }, cf. Figure 6.4.


117

y ν ν−δ

y

α

δ

s

x

x

δ

δ

s + d/3

Figure 6.4. The embedding α.

Proof. Choose a smooth function h : ]0, ν[ → [1, (s + d/3)/δ] such that (i) h(w) = 1 if w ∈ ]0, δ[ ∪ ]ν − δ/3, ν[. Since δ < d/8 and d < s we find by computation that 2δ +

ν−2δ (s+d/3)/δ

< s < ν.

We may therefore further require that / ν 1 dw = s. (ii) 0 h(w) Then the map α : ]0, δ[ × ]0, ν[ → R , 2

(x, y) →

/

y

h(y)x, 0

1 dw h(w)

is the desired symplectic embedding.

2

The (n − 1)-fold product α × · · · × α embeds the fibres (]0, δ[ × ]0, ν[)n−1 of Ki into the fibres of Ci (d/3) in such a way that 1 the fibres 1 of Hi are embedded 1 into the (δ). We next embed the base of K ∪ H into the base of C fibres of N i i i 1 1i (d/3) ∪ 1 (δ). We denote by K and C the projections of the sets K ∪ Hi and N i i 1 1 Ci (d/3) ∪ Ni (δ) onto the (u, v)-plane, cf. Figure 6.5. Lemma 6.1.8. There exists a symplectic embedding α : K → C, cf. Figure 6.5.

118


Proof. We denote by ζ the reflection (u, v) → (v, u), and we define translations τi+ and τi− by τi+ (u, v) = (u + (i − 1)2s, v),

τi− (u, v) = (u − (i − 1)(ν + s), v),

i = 1, . . . , k. Moreover, we abbreviate Iiν = ] (i − 1)(ν + s), (i − 1)(ν + s) + ν [, i = 1, . . . , k, i = 1, . . . , k − 1. Iis = [ (i − 1)(ν + s) + ν, i(ν + s) ], In view of the properties of the symplectic embedding α guaranteed by Lemma 6.1.7 the map α : K → R2 defined by + τi ζ α ζ τi− (u, v) if u ∈ Iiν , α(u, v) = + − (u, v) if u ∈ Iis , τi+1 τi+1 2

is a smooth symplectic embedding as desired, cf. Figure 6.5. v δ

u ν+s

ν

2ν + s

α v

s

δ s

2s

u

3s

Figure 6.5. The embedding α¯ : K → C.

We finally define ξ to be the restriction to

1

Ki ∪

1

Hi of the symplectic embedding

α × α × · · · × α : K × (]0, δ[ × ]0, ν[)n−1 → C × (]0, s + d/3[ × ]0, s[)n−1 . 1 1 1 1 Ci (d/3) ∪ Ni (δ), and so the proof of By construction, ξ Ki ∪ Hi ⊂ Proposition 6.1.6 is complete. 2


119

Step 4. Filling Ki ∪ Hi with thin polydiscs,. Let k, s, d and δ be the numbers found in the previous three steps, and recall that = /3 and ν = s 2 /δ. For each aˆ > 3π we let N (a) ˆ and ˆ π ) → R n ((N (a) ˆ + 1)π ) ϕaˆ : R n (a, be the natural number and the symplectic embedding found in Proposition 5.2.3. By Proposition 5.2.3 (ii) there exists aˆ 0 > 3π such that for all aˆ ≥ aˆ 0 , n ϕaˆ R (a, ˆ π) > 1 − 2n . (6.1.20) R n N (a) ks ˆ +1 π In view of the definition (5.2.8) we have N (a) ˆ ≥ N(aˆ 0 ) whenever aˆ ≥ aˆ 0 , and in view of (5.2.11) we have N (a) ˆ → ∞ as aˆ → ∞. Choosing aˆ 0 larger if necessary, we can therefore assume that ν ≤ δ for all aˆ ≥ aˆ 0 . (6.1.21) N (a) ˆ +1 We set a0 = k aˆ 0 , and we define the function λ : [a0 , ∞[ → R by λ(a) = 9

N

s a k

(6.1.22)

. +1 π

Proposition 6.1.9. For each a ≥ a0 there exists a symplectic embedding a : λ(a)R n (a, π) →

k 2

Ki ∪

i=1

k−1 2

Hi .

i=1

Proof. Fix a ≥ a0 . We set aˆ = a/k, and we abbreviate N = N(a) ˆ and λ = λ(a). In abuse of notation we denote the dilatation z → λz, z ∈ R2n , also by λ. Moreover, we define the linear symplectomorphism σ of R2n by σ (u, v, x, y) = λδ u, λδ v, λδ x, λδ y . ˆ π) → R2n defined by Then σ (λR n ((N + 1)π)) = K, and so the map ψaˆ : λR n (a, ψaˆ = σ λ ϕaˆ λ−1 ˆ π ) into K. Using Proposition 5.2.3 (i), the definitions symplectically embeds λR n (a, 2 of ψaˆ , σ and λ, and ν = s /δ, we find that ψaˆ (u, v, x, y) = σ (u, v, x, y)

ψaˆ (u, v, x, y) = σ (u, v, x, y) + ν −

if u < λπ,

ν aˆ N+1 π , 0, 0,

ν−

ν N +1

if u > λ(aˆ − π ),

1y

(6.1.23) (6.1.24)

120


cf. the left picture in Figure 6.6. Denote the reflection (u, v, x, y) → (u, v, x, −y) by ζy . Since ψaˆ symplectically embeds λR n (a, ˆ π ) into K, the map ψaˆ defined by ψaˆ = τν1y ζy ψaˆ τλπ1y ζy symplectically embeds λR n (a, ˆ π ) into K as well, cf. Figure 6.6. We read off from the identities (6.1.23) and (6.1.24) that (6.1.25) ψaˆ (u, v, x, y) = σ (u, v, x, y) + 0, 0, 0, ν − N ν+1 1y if u < λπ, ν aˆ ψaˆ (u, v, x, y) = σ (u, v, x, y) + ν − N+1 if u > λ(aˆ − π ), (6.1.26) π , 0, 0, 0 cf. the right picture in Figure 6.6. y

y

ν

ν

ν N+1

u ν N+1

u

ν

ν

Figure 6.6. The embeddings ψaˆ and ψaˆ : λR n (a, ˆ π) → K. n ˆ π) into K to We are now going to use the embeddings ψaˆ and 1 ψaˆ of λR 1 (a, n construct a symplectic embedding a of λR (a, π ) into Ki ∪ Hi , cf. Figure 6.7. As in Step 1 of Section 3.2 we find a symplectic embedding

β : ]0, ν[×]0, δ[ → ]0, ν + s[×]0, δ[

which restricts to the identity on u ≤ ν − N ν+1 and to the translation (u, v) →

ν (u + s, v) on u ≥ ν − 21 N+1 . For i = 1, . . . , k we define

Ri := (u, v, x, y) ∈ λR n (a, π) | (i − 1)λaˆ < u ≤ iλaˆ . Then λR n (a, π) =

k 2 i=1

Ri .


121

We denote the translation (u, v, x, y) → (u + (i − 1)(ν + s), v, x, y) of R2n by τi . For each odd i ∈ {1, . . . , k − 1} we define the map ψi : Ri → R2n by   τi ψaˆ τi−1 (u, v, x, y) if u < λ(i aˆ − π ), ψi (u, v, x, y) = −1  τ (β × id (u, v, x, y) if u ≥ λ(i aˆ − π ). i 2n−2 ) ψaˆ τi The estimate (6.1.21) and formula (6.1.26) imply that ψi is a smooth symplectic embedding of Ri into Ki ∪ Hi for which ψi (u, v, x, y) = σ (u, v, x, y) + (ui , 0, 0, 0) if u ≥ λ i aˆ − π2 (6.1.27) where ui is such that the right end of Ri is mapped to the right end of Hi , cf. Figure 6.7. For each even i ∈ {2, . . . , k − 1} we define the map ψi : Ri → R2n by   τi ψaˆ τi−1 (u, v, x, y) if u < λ(i aˆ − π ), ψi (u, v, x, y) = −1  τ (β × id (u, v, x, y) if u ≥ λ(i aˆ − π ). i 2n−2 ) ψaˆ τi The estimate (6.1.21) and formula (6.1.24) imply that ψi is a smooth symplectic embedding of Ri into Ki ∪ Hi for which ψi (u, v, x, y) = σ (u, v, x, y)+ ui , 0, 0, ν− N ν+1 1y if u ≥ λ i aˆ − π2 (6.1.28) where ui is such that the right end of Ri is mapped to the right end of Hi , cf. Figure 6.7. We finally define the symplectic embedding ψk : Rk → Kk by   τi ψaˆ τi−1 (u, v, x, y) if k is odd, ψk (u, v, x, y) =  τ ψ τ −1 (u, v, x, y) if k is even. i aˆ i y ν

K1

K2

K3 H2

H1 δ

u ν

ν+s

2ν + s

2ν + 2s

Figure 6.7. The embedding a : λR n (a, π) →

3ν + 2s

1k

i=1 Ki ∪

1k−1

i=1 Hi .

122


In view of the identities (6.1.23), (6.1.25), (6.1.27) and (6.1.28) the embedding a : λR n (a, π) →

k 2 i=1

Ki ∪

k−1 2

Hi

i=1

defined by a |Ri = ψi ,

i = 1, . . . , k,

is a smooth symplectic embedding. The proof of Proposition 6.1.9 is complete.

2

Step 5. End of the proof of Theorem 6.1.1. We let > 0 be as in Theorem 6.1.1, set = /3 and let k and s be as introduced after the proof of Lemma 6.1.2. We choose a0 = a0 () as before Proposition 6.1.9, fix a ≥ a0 and define λ(a) as in (6.1.22). Lemma 6.1.10. We have

λ(a)R n (a, π) > ks 2n − .

(6.1.29)

Proof. We set again aˆ = a/k, N = N (a) ˆ and λ = λ(a). Since the embedding ϕaˆ : R n (a, ˆ π) → R n ((N + 1)π) is volume preserving, we have n λR (a, π) = λ2n k R n (a, ˆ π ) = kλ2n ϕaˆ R n (a, ˆ π) , (6.1.30) and multiplying the inequality (6.1.20) by kλ2n R n ((N + 1)π ) = kλ2n ((N + 1)π )n = ks 2n we find that

ˆ π )) > ks 2n − . kλ2n ϕaˆ (R n (a,

(6.1.31)

Lemma 6.1.10 now follows from combining the identity (6.1.30) with the estimate (6.1.31). 2 Composing the symplectic embeddings a , ξ and ρ guaranteed by Proposition 6.1.9, Proposition 6.1.6 and Proposition 6.1.3 we obtain the symplectic embedding a := ρ ξ a : λ(a)R n (a, π ) → M. Using the estimates (6.1.5) and (6.1.29) we find µ M \ a λ(a)R n (a, π) = µ(M) − µ a λ(a)R n (a, π ) = µ(M) − λ(a)R n (a, π ) < ks 2n + 2 − ks 2n − = 3 = . This is the required estimate in Theorem 6.1.1 and so the proof of Theorem 6.1.1 is complete. 2

6.2 Proof of lima→∞ paE (M, ω) = 1

6.2

123

Proof of lim paE (M, ω) = 1 a→∞

We recall from the introduction that for every a ≥ π the real number paE (M, ω) is defined by |λE(π, . . . , π, a)| paE (M, ω) = sup Vol(M, ω) λ where the supremum is taken over all those λ for which λE 2n (π, . . . , π, a) symplectically embeds into (M, ω). Corollary 5.3.2 (i) implies that the first statement in Theorem 3 in Section 1.3.2 is a consequence of Theorem 6.2.1. For every > 0 there exists a number a0 = a0 () > π having the following property. For every a ≥ a0 there exist a number λ(a) > 0 and a symplectic embedding a : λ(a) T n (a, π) → M such that µ M \ a λ(a) T n (a, π ) < . Proof. We shall proceed along the same lines as in the proof of Theorem 6.1.1. We shall first fill almost all of M with finitely many symplectically embedded balls whose closures are disjoint, and connect these balls by neighbourhoods of lines. Using Corollary 5.3.2 (ii) and Proposition 5.3.3 we can then almost fill the balls with symplectically embedded parts of λ(a) T n (a, π), and we shall use the neighbourhoods of the lines to pass from one ball to another. The proof of Theorem 6.2.1 is substantially more difficult than the proof of Theorem 6.1.1, however. The first reason is that for n ≥ 3 there is no elementary method of symplectically filling M with balls. We shall overcome this difficulty by using a result of Mc Duff and Polterovich in [61]. The second reason is that symplectically filling a ball with a part of λ(a) T n (a, π ) is more difficult than symplectically filling a cube with a polydisc. We have overcome this difficulty in Corollary 5.3.2 (ii) and Proposition 5.3.3. The third reason is that the fibres of λ(a) T n (a, π ) are not constant. This will merely cause technical complications. Step 1. Filling M by balls. We denote by B(r) the 2n-dimensional open ball B(r) = { z ∈ R2n | |z| < r }. Lemma 6.2.2. For every > 0 there exists an integer k0 with the following property. For each integer k > k0 there exist r = r(k) > 0 and a symplectic embedding ηk :

k 2

Bi (r) → M

i=1

of a disjoint union of k translates Bi (r) of B(r) in R2n such that

2 µ M \ ηk Bi (r) < .

(6.2.1)

124


Proof. We follow [61, Remark 1.5.G]. Fix1 > 0. In view of Lemma 6.1.2 we find s > 0 and a symplectic embedding γ : Ci (s) → M of a disjoint union of m translates of C(s) such that

2 (6.2.2) µ M \γ Ci (s) < . 3 Choose l0 ∈ N so large that for all l ≥ l0 , (l − 1)n >1− . ln 3ms 2n

(6.2.3)

We define the integer k0 by k0 = m n! l0n . We fix k > k0 and define the integer l > l0 through the inequalities m n! (l − 1)n < k ≤ m n! l n .

(6.2.4)

The crucial ingredient of the proof of Lemma 6.2.2 is√the following result which is proved in [61, Corollary 1.5.F]. We abbreviate ς = s/ π . Lemma 6.2.3 (Mc Duff–Polterovich). There exist rl > 0 and a symplectic embedding βl :

n! ln 2

Bˆj (rl ) → C 2n (π) = D(π ) × · · · × D(π )

j =1

of a disjoint union of n! l n translates Bˆj (rl ) of B(rl ) such that

2 2n . Bˆj (rl ) < C (π) \ βl 3mς 2n Remark 6.2.4. If n = 2, Lemma 6.2.3 follows from Lemma 5.3.1 (i) and Lemma 3.1.5, see also 9.3.2 in Chapter 9 and [81]. For general n, however, the only known proof in [61] uses non-elementary, algebro-geometric methods. 3 Continuing with the proof of Lemma 6.2.2 we define r = ςrl and Bˆj (r) = ς Bˆj (rl ). For each i = 1, . . . , m we choose wi ∈ R2n such that the m n! l n balls Bi,j (r) = τ−wi Bˆj (r) are disjoint. In view √ of Lemma 3.1.5 the disc D(π ) is symplectomorphic √ to the square ] 0, π [ × ] 0, π [. We therefore find a symplectomorphism √ σ : C 2n (π) → C( π . We finally denote the dilatation z → ς z of R2n also by ς and the translation C(s) → Ci (s) by τi . We can now define a symplectic embedding 2 2 Bi,j (r) → Ci (s) β: i,j

i


by

β B

i,j (r)

125

= τi ς σ βl ς −1 τwi B

i,j (r)

.

In view of the second inequality in (6.2.4) we can choose k members B1 (r), . . . , Bk (r) of the family {Bi,j (r)}. We define the symplectic embedding ηk :

k 2

Bi (r) → M

i=1

by ηk = γ β. In order to verify the estimate (6.2.1) we first use Lemma 6.2.3 to estimate m 2 2

. 2 Ci (s) \ β Bi,j (r) = (ς σ βl ) Bˆj (rl ) (ς σ ) C 2n (π ) \ i

i,j

=

i=1 m .

j

2 ς 2n C 2n (π ) \ βl Bˆj (rl )

i=1

< mς =

j 2n

3mς 2n

. 3

Moreover, since β is volume preserving, the inclusion β implies that m n! l n |B(r)| ≤ ms 2n .

(6.2.5)

1

1 Ci (s) Bi,j (r) ⊂ (6.2.6)

The left inequality in (6.2.4) and the estimates (6.2.6) and (6.2.3) now yield 1 1 Bi,j (r) \ Bi (r) = (m n! l n − k) |B(r)| < m n! (l n − (l − 1)n ) |B(r)|

n (6.2.7) ≤ ms 2n 1 − (l−1) n l < . 3 Using the fact that β and γ are both volume preserving and using the estimates (6.2.2), (6.2.5) and (6.2.7) we finally find 1 Bi (r) µ M \ ηk 1 1 1 1 1 =µ M \γ Ci (s) + Ci (s) \ β Bi,j (r) + Bi,j (r) \ Bi (r)
0 be as in Theorem 6.2.1 and set = /5. In view of Lemma 6.2.2 we find an integer k0 such that for each integer k > k0 there exists rk > 0 and a symplectic embedding k 2 ηk : Bi (rk ) → M i=1

of a disjoint union of k translates Bi (rk ) of B(rk ) such that

2 Bi (rk ) < . µ M \ ηk

(6.2.8)

Fix k > k0 . In order to bring Proposition 5.3.3 √ intoplay we shall next replace the balls Bi (rk ) by trapezoids. We choose s ∈ 0, π rk so large that k

1 n!

π n rk2n − (s )2n < ,

and we choose s ∈ ]0, s [ so large that 2n 1 (s ) − s 2n < . k n!

(6.2.9)

(6.2.10)

We abbreviate d := (s − s)/(2n). For each δ ∈ [0, d] we define the trapezoid T (δ) by T (δ) = n (s + 2nδ) × 2n (s + 2nδ) ⊂ Rn (x) × Rn (y). Moreover, we define the translates Ti (δ) by Ti (δ) = { z + ((i − 1)s − δ, −δ, . . . , −δ) | z ∈ T (δ) }, and we abbreviate Ti = Ti (0) and Ti = Ti (d), i = 1, . . . , k, cf. Figure 6.8 and Figure 6.9. Notice that Ti (δ) ⊂ Ti (δ ) whenever δ ≤ δ . After choosing s ∈ ]0, s [ larger if necessary we can assume that 2nd ≤ s so that the trapezoids Ti are disjoint. Composing the linear symplectomorphism T (d) = n (s ) × 2n (s ) → n (s )2 × 2n (1) = T n (s )2 , (x, y) → s x, s1 y with the symplectic embedding T n ((s )2 ) → B 2n π rk2 = B(rk ) guaranteed by Corollary 5.3.2 (ii), we obtain a symplectic embedding σ : T (d) → B(rk ). We define the points vi and wi in R2n through the identities τvi Ti = T (d) and τwi B(rk ) = Bi (rk ). We can now define the symplectic embedding ϑi : Ti → M by ϑi = ηk τwi σ τvi : Ti → M.

6.2 Proof of lima→∞ paE (M, ω) = 1 x2

127

y2

s − d

s − d

s

s × u

−d

s

v

s − d

−d

s

s − d

Figure 6.8. The trapezoids T1 and T1 for n = 2.

We denote the restriction of ϑi to Ti by ϑi , and we write ϑ=

k 2 i=1

ϑi :

k 2

Ti → M

i=1

and ϑ =

k 2 i=1

ϑi :

k 2

Ti → M.

i=1

1 + 1 2n Ti = |Ti | = k n! s , Since ϑ, ϑ and ηk are symplectic, we have µ ϑ 1 + 1 1 Ti \ ϑ Ti = Ti \ Ti = k n! (s )2n − s 2n µ ϑ and 1 1 µ ηk Bi (rk ) \ ϑ Ti = k (|B(rk )| − |T (d)|) = k

1 n!

π n rk2n − (s )2n .

In view of the inequalities (6.2.8), (6.2.9) (6.2.10) we can therefore estimate 1 1 1 µ(M) = µ ϑ Ti + µ ϑ Ti \ ϑ Ti 1 1 1 + µ ηk Bi (rk ) \ ϑ Ti + µ M \ ηk Bi (rk ) 2n 1 2n 1 = k n! s + k n! (s ) − s 2n (6.2.11) n 2n 1 2n 1 Bi (rk ) + k n! π rk − (s ) + µ M \ ηk 1 2n s + + + < k n! 1 2n s + 3 . = k n!

1 Step 2. Connecting the trapezoids. We next extend the embedding ϑ : Ti → M to a symplectic embedding of a connected domain. For i = 1, . . . , k − 1 we define straight lines Li (t) : [ (2i − 1)s, 2is ] → R2 × Rn−1 (x) × Rn−1 (y)

128


by Li (t) = (t, 0, 0, 0), cf. Figure 6.9. Then Li (t) ∈ Ti Li (t) ∈ Ti+1

if t ∈ [ (2i − 1)s, (2i − 1)s + (2n − 2)d [, if t ∈ ] 2is − d, 2is ].

(6.2.12)

x, y

s

T1

T2

T1 −d

T2

L1 s

u 2s

3s

Figure 6.9. The trapezoids Ti and Ti , i = 1, 2, and the line L1 .

For δ > 0 we define the “δ-neighbourhood” Ni (δ) of Li in R2n by Ni (δ) = [ (2i − 1)s, 2is ] × ] − δ, δ [2n−1 . For later use we shall verify that the left end of Ni (δ) and the right end of Ti (δ) fit together nicely. In the sequel we denote by pu,v , pu,x and pv,y the projections of R2n onto R2 (u, v), Rn (u, x) and Rn (v, y). Lemma 6.2.5. For each δ ∈ ]0, d ] and each i ∈ {1, . . . , k − 1} we have { (u, v, x, y) ∈ Ni (δ) | u = (2i − 1)s + δ } ⊂ Ti (δ). Proof. We may assume that i = 1. We compute that { (u, x2 , . . . , xn ) | u = s + δ, −δ < x2 , . . . , xn < δ } ⊂ { (u, x2 , . . . , xn ) | u = s + δ, −δ < x2 , . . . , xn and x2 + · · · + xn < (n − 1)δ } = { (u − δ, x2 − δ, . . . , xn − δ) | u − δ = s + δ, 0 < x2 , . . . , xn and u + x2 + · · · + xn < s + 2nδ } ⊂ { (u − δ, x2 − δ, . . . , xn − δ) | 0 < u, x2 , . . . , xn and u + x2 + · · · + xn < s + 2nδ }, i.e.,

pu,x { (u, v, x, y) ∈ N1 (δ) | u = s + δ } ⊂ pu,x (T1 (δ)) .


129

Moreover, ] − δ, δ [n ⊂ ] − δ, s + (2n − 1)δ [n , i.e., pv,y { (u, v, x, y) ∈ N1 (δ) | u = s + δ } ⊂ pv,y (T1 (δ)). 2

Lemma 6.2.5 thus follows. The following proposition parallels Proposition 6.1.3. Proposition 6.2.6. There exist δ ∈ ]0, d [ and a symplectic embedding ρ:

k 2

Ti (δ) ∪

i=1

k−1 2

Ni (δ) → M.

i=1

1 Proof. By construction of the map ϑ the set M \ ϑ ( Ti ) is connected. Using this and the inclusions (6.2.12) we find a smooth embedding k−1 2

λi :

k−1 2

i=1

Li → M \ ϑ

2

Ti

i=1

such that λi (Li (t)) = ϑi (Li (t)) λi (Li (t)) =

if t ∈ [ (2i − 1)s, (2i − 1)s + (2n − 1)d/2 ], if t ∈ [ 2is − d/2, 2is ],

ϑi+1 (Li (t))

and such that λi (Li (t)) ∈ M \ ϑ

1

Ti (d/2)

if t ∈ [ (2i − 1)s + (2n − 1)d/2, 2is − d/2 ].

For i = 1, . . . , k − 1 and δ ∈ [0, d ] we define the truncated trapezoid T˘i (δ) = { z ∈ Ti (δ) | u < (2i − 1)s + δ }, cf. Figure 6.10. Replacing the maps γ1 , . . . , γk and the sets C1 (d/3), C1 (d/2), . . . , Ck−1 (d/3), Ck−1 (d/2), Ck (d/3), Ck (d/2) in the proof of Proposition 6.1.3 by the maps ϑ1 , . . . , ϑk and the sets T˘1 (d/3), T˘1 (d/2), . . . , T˘k−1 (d/3), T˘k−1 (d/2), Tk (d/3), Tk (d/2) we find δˆ > 0 and a symplectic embedding ρ:

T˘i (d/3) ∪ Ni δˆ ∪ Tk (d/3) → M,

k−1 2 i=1

130

6 Proof of Theorem 3 x, y

T˘1 (d/3)

u

−d/3

s

−d

N1 (δ1 )

Figure 6.10. The set T˘1 (d/3) ∪ N1 (δ1 ).

cf. Figure 6.10. We define δ by

d/3 ˆ . δ = min δ, 2n − 1

Then δ < d and Ti (δ) ⊂ T˘i (d/3) for all i = 1, . . . , k − 1. The restriction of ρ to the domain k k−1 2 2 Ti (δ) ∪ Ni (δ) i=1

is as desired.

i=1

2

Step 3. The choice of k and of a0 . We recall that = /5, and we choose the integer k0 as after the proof of Lemma 6.2.2. We now choose the integer k such that k ≥ 2 and k > k0 and 1 n 1 1+ √ −1≤ (6.2.13) . n µ(M) k The inequality (6.2.13) will be crucial for the proof of Lemma 6.2.9 below. Let s and d be the numbers associated with k after the proof of Lemma 6.2.2, and let δ be as in Proposition 6.2.6. For each aˆ > 3π we let l(a) ˆ and ˆ 2 ϕaˆ : Saˆ → T n l(a) be the natural number and the symplectic embedding found in Proposition 5.3.3. We abbreviate n! . (6.2.14) q =1− k s 2n


131

By Proposition 5.3.3 (ii) there exists aˆ 0 > 3π such that for all aˆ ≥ aˆ 0 , |ϕ (S )| aˆ aˆ > q. T n l(a) ˆ 2

(6.2.15)

In view of Lemma 5.3.5 and the definition (5.3.22) we have that l(a) ˆ ≥ l(aˆ 0 ) whenever aˆ ≥ aˆ 0 and that l(a) ˆ → ∞ as aˆ → ∞. Choosing aˆ 0 larger if necessary, we can therefore assume that s for all aˆ ≥ aˆ 0 . (6.2.16) l(a) ˆ ≥ π δ a . We set a0 = kn aˆ 0 . In the sequel we fix a ≥ a0 . We set aˆ = kn Step 4. The set ki=1 Si and the choice of λ(a). With each k-tuple (u1 , . . . , uk ) ∈ Rk we associate the k-tuple (v1 , . . . , vk ) defined by vi = u1 + · · · + ui . We say that a k-tuple (u1 , . . . , uk ) is admissible if ui > 0 for all i and if vk = a. Lemma 6.2.7. There exists a unique admissible k-tuple (u1 , . . . , uk ) such that n−1 ui 1 − vi−1 = u1 , i = 2, . . . , k. (6.2.17) a Proof. Fix u1 ∈ ]0, a[. We inductively associate with u1 numbers u2 , . . . , uk as follows. Assume that i ∈ {2, . . . , k} and that we have already constructed u2 , . . . , ui−1 . We set vj = u1 + · · · + uj , j = 1, . . . , i − 1. We define ui by ui 1 −

vj0 n−1 a

= u1

where j0 = max{ j | vj < a }. The function f : ]0, a[ → R defined by f (u1 ) = vk is then continuous and strictly increasing. Since f (u1 ) → 0 as u1 → 0 and f (u1 ) → ∞ as u1 → a we conclude that there exists a unique u1 ∈ ]0, a[ with f (u1 ) = a. By construction, the k-tuple (u1 , . . . , uk ) associated with this u1 is admissible and meets the identities (6.2.17). The proof of Lemma 6.2.7 is complete. 2 Let (u1 , . . . , uk ) be the k-tuple guaranteed by Lemma 6.2.7. As before we abbreviate vi = u1 + · · · + ui , and we set v0 = 0. We consider the subsets S1 = ]0, v1 [ × ]0, 1[ × n−1 (π) × 2n−1 (1), Si = [vi−1 , vi [ × ]0, 1[ × n−1 π − πa vi−1 × 2n−1 (1), 2 ≤ i ≤ k, 1 of R2 × Rn−1 (x) × Rn−1 (y). Then T n (a, π) ⊂ Si , cf. Figure 6.11. Notice that |Si | = ui

π n−1 (n−1)!

1−

vi−1 n−1 , a

1 ≤ i ≤ k.

132


The identities (6.2.17) therefore imply that |S1 | = · · · = |Sk | .

(6.2.18)

x π S1

S2

S3

S4

v2

v1

v3

Figure 6.11. The set

u a

1k

i=1 Si for k = 4.

Lemma 6.2.8. We have aˆ 0 ≤

a kn

≤ u1 < u2 < · · · < uk ≤

a √ n . k

(6.2.19)

Proof. The first inequality 1 in (6.2.19) is equivalent to our assumption a0 ≤ a. The inclusion T n (a, π) ⊂ Si and the identities (6.2.18) yield 1 u1 π n−1 aπ n−1 = T n (a, π) ≤ Si = k |S1 | = k n! (n − 1)! a and so kn ≤ u1 . The inequalities u1 < u2 < · · · < uk1follow from the identities (6.2.17). Finally, the identities (6.2.18) and the inclusion Si ⊂ Sa yield

π kuk and so uk ≤

n−1

1 π n−1 = k |Sk | = Si ≤ |Sa | = a (n − 1)! (n − 1)! a uk

2

a √ n . k

We now define the real number λ = λ(a) by |λS1 | = q. |T1 | k 2 Si \ λ T n (a, π) < . Lemma 6.2.9. λ i=1

(6.2.20)


133

Proof. In view of Lemma 6.2.8 we have u1 < u2 < · · · < uk , and so k 2

Si ⊂ T n a + uk , π + πa uk ,

(6.2.21)

i=1

cf. Figure 6.12. x π + πa uk π

a Figure 6.12. T n (a, π ) ⊂

Since the embedding ϑ :

a + uk

u

n a + u ,π + π u . k i=1 Si ⊂ T a k

1k

1

Ti → M is symplectic, we have 1 + 1 2n µ(M) ≥ µ ϑ Ti = |Ti | = k n! s .

(6.2.22)

In view of the inclusion (6.2.21), the last inequality in (6.2.19) and the estimates (6.2.13) and (6.2.22) we can now estimate 1 Si \ T n (a, π) ≤ T n a + uk , π + π uk \ T n (a, π ) a n−1 n−1 −a = π n! (a + uk ) 1 + uak n n−1 1 + uak − 1 a = π n! n−1 1 n ≤ π n! 1 + √ −1 a n k

1 a ≤ π n−1 n! µ(M)

≤ π n−1 ks12n a. Using the definitions (6.2.20) and (6.2.14) and the second inequality in (6.2.19) we find that s 2n |T1 | |T1 | s 2n ks 2n n! 2n λ =q = ≤ . < = π n−1 |S1 | |S1 | nu1 π n−1 aπ n−1 u1 (n−1)! We conclude that 1 λ Si \ λ T n (a, π) = λ2n 1 Si \ T n (a, π ) < and so the proof of Lemma 6.2.9 is complete.

2

134


Step 5. Embedding λ

Si into

Ti (δ) ∪

Ni (δ)

Proposition 6.2.10. There exists a symplectic embedding a : λ

k 2 i=1

Si →

k 2

Ti (δ) ∪

i=1

k−1 2

Ni (δ).

i=1

Proof. We start with introducing several geometric quantities and with replacing the sets λSi by more convenient sets. 5.A. Preliminaries. For i = 1, . . . , k we set hi = 1 −

vi−1 a .

Notice that hi π is the x-width of Si and that 1 = h1 > h2 > · · · > hk . We define ai by

ai =

ui hi .

Then a1 = u1 , and in view of Lemma 6.2.8 we have ai ≥ ui ≥ u1 > aˆ 0 . We denote the natural number l(ai ) associated with ai in Proposition 5.3.3 by li = l(ai ). Since ai > aˆ 0 the estimate (6.2.16) shows that

We finally define λi by

li ≥ π δs .

(6.2.23)

λi = lsi .

(6.2.24)

√ √ hi λ and δ > hi πλ. l 2n 2n i we have Proof. Since |Ti | = sn! and T n li2 = n! n 2 |Ti | = λ2n li , i T

Lemma 6.2.11.

λi >

and since ai ≥ aˆ 0 , the estimate (6.2.15) shows that |S | ai 2 > q. T n l i

Together with the definition (6.2.20) of λ and the identities (6.2.18) we can now estimate h−n |λS1 | |Sai | λ2n |Si | i |Si | = > q = = , T n l 2 |T1 | |Ti | |Ti | λ−2n i i


135

and so the first statement in Lemma 6.2.11 follows. The second statement follows 2 from the first one, from the definition of λi and from the estimate (6.2.23). We define the linear symplectomorphism β of R2 by β(u, v) = λs u, λs v . Moreover, we set u˜ i =

λ2 s ui

(6.2.25)

and v˜i = u˜ 1 + · · · + u˜ i , and we abbreviate

3 Si := (β × id2n−2 ) (λSi ),

1 ≤ i ≤ k.

Then 3 S1 = ]0, v˜1 [ × ]0, s[ × n−1 (πλ) × 2n−1 (λ), 3 Si = [v˜i−1 , v˜i [ × ]0, s[ × n−1 (hi π λ) × 2n−1 (λ),

2 ≤ i ≤ k.

We shall next embed 3 S1 into T1 and shall then successively embed 3 Si into Ni−1 (δ) ∪ Ti (δ), i = 2, . . . , k. 5.B. The embedding ψ1 : S1 → T1 . We recall that a1 = u1 and s = λ1 l1 , and we read off from the first statement in Lemma 6.2.11 that λ1 > λ. Therefore, t1 :=

λ1 l1 a1

− u˜ 1 > 0,

and so the affine symplectomorphism β1 of R2 defined by β1 (u, v) = l1 (u + t1 ), l11 v

S1 into ]0, λ1 a1 [ × ]0, λ1 [. Using once more that λ1 > λ we conclude embeds pu,v 3 that −1 S1 ⊂ S a 1 . λ1 (β1 × id2n−2 ) 3 We define the linear symplectomorphism σ1 of R2n by σ1 (u, v, x, y) = l11 u, l1 v, l11 x, l1 y . 3 Composing the affine embedding λ−1 1 (β1 × id2n−2 ) : S1 → Sa1 with the symplectic embedding ϕa1 : Sa1 → T n (l12 ) guaranteed by Proposition 5.3.3 and with the linear diffeomorphism σ1 λ1 : T n (l12 ) → T1 we obtain the symplectic embedding 3 ψ1 := σ1 λ1 ϕa1 λ−1 1 (β1 × id2n−2 ) : S1 → T1 .

136


We abbreviate s1 = s − λ1 . Using Proposition 5.3.3 (i) and the definitions of β1 , σ1 and λ1 we find that ψ1 (u, v, x, y) = u − v˜1 + s1 , v, π1 x, πy if u > v˜1 − λl11 π. 5.C. The embedding ψj :

j

j

→

i=1 Si

i=1 Ti (δ) ∪

j −1 i=1

Ni (δ), j = 2, . . . , k.

For i = 1, . . . , k we abbreviate si = (2i − 1)s − λi ,

di = min

6

u˜ i 2 ,δ

7

εi = min u˜ i − di , λlii π .

,

(6.2.26)

Proceeding by induction we fix j ∈ {1, . . . , k − 1} and assume that we have already constructed a symplectic embedding ψj :

j 2

3 Si →

i=1

j 2

Ti (δ) ∪

i=1

j2 −1

Ni (δ)

(6.2.27)

i=1

which is such that Im ψj ⊂ (u, v, x, y) | u < sj and ψj (u, v, x, y) = u − v˜j + sj , v, √1

1 x, hj π

8 hj πy

if u > v˜j − εj ,

(6.2.28)

cf. Figure 6.13. We are going to construct a symplectic extension ψj +1 :

j2 +1

3 Si →

i=1

j2 +1

Ti (δ) ∪

i=1

j 2

Ni (δ)

(6.2.29)

i=1

of ψj which is such that Im ψj +1 ⊂ (u, v, x, y) | u < sj +1 and ψj +1 (u, v, x, y) = u − v˜j +1 + sj +1 , v, √ 1

1 x, hj +1 π

8 hj +1 πy

if u > v˜j +1 − εj +1 . We extend ψj by formula (6.2.28) to the smooth symplectic embedding 3j : ψ

j2 +1

3 Si → R2n ,

i=1

and we denote by Fj the end

3j 3 3j | sj ≤ u < sj + u˜j +1 , Fj = ψ Sj +1 = (u, v, x, y) ∈ Im ψ cf. Figure 6.13.

(6.2.30)


Tj (δ)

137

Tj +1 (δ)

u sj (2j − 1)s

2j s Nj (δ) Figure 6.13. The end Fj .

5.C.1. The map βj . We recall that uj +1 = hj +1 aj +1 and s = λj +1 lj +1 , and we read 8 off from the first statement in Lemma 6.2.11 that λj +1 > hj +1 λ. Therefore, tj +1 :=

λj +1 lj +1 aj +1

− u˜j +1 > 0.

(6.2.31)

As in Step 1 of the folding construction described in Section 3.2 we find a symplectic embedding λ +1 βj : sj , sj + u˜j +1 × ]0, s[ → sj , 2j s + ljj+1 aj +1 × ]0, s[ 6 which restricts to the identity on (u, v) | sj ≤ u ≤ sj + lation (u, v) → (u + 2j s − sj + tj +1 , v)

dj +1 2

7

, restricts to the trans-

on { (u, v) | u ≥ sj + dj +1 },

(6.2.32)

and is such that Im βj ∩ {(2j − 1)s + δ ≤ u ≤ 2j s − δ} ⊂ {0 < v < δ},

δ ≤ u ≤ 2j s − 2δ ⊂ {s − δ < v < s}, Im βj ∩ 2j s − δ + 2n

δ ≤ u ≤ 2j s ⊂ {0 < v < δ}, Im βj ∩ 2j s − 2δ + 2n

(6.2.33) (6.2.34) (6.2.35)

cf. Figure 6.14. Lemma 6.2.12. We have

βj × id2n−2 Fj ∩ (u, v, x, y) | sj ≤ u ≤ 2j s − δ ⊂ Tj (δ) ∪ Nj (δ). Proof. In view of the inclusion (6.2.33) we have

pu,v βj × id2n−2 Fj ∩ sj ≤ u ≤ 2j s − δ ⊂ pu,v Tj (δ) ∪ Nj (δ) .

138


v dj +1

s

u

sj βj

v dj +1 2

s

δ 2n

δ 2n

dj +1 + tj +1

δ

u (2j − 1)s + δ

sj

2j s − 2δ

2j s − δ

2j s

Figure 6.14. The map βj .

Moreover, the formula (6.2.28) implies that the fibres of Fj are equal to

hj +1 √1

1 λn−1 (π) hj π

×

8

hj π λ2n−1 (1) = n−1 (wj ) × 2n−1

8 hj π λ

where we abbreviated hj +1 wj = 8 λ. hj

(6.2.36)

In view of the inequality hj +1 < hj and the second estimate in Lemma 6.2.11 these fibres are contained in the fibres of Nj (δ). Lemma 6.2.12 thus follows in view of Lemma 6.2.5. 2 5.C.2. Rescaling the fibres. We define νj by : νj = (lj +1 − 1)

hj hj +1 .

(6.2.37)

In view of the inequalities (6.2.23) we have lj +1 > 3, and so νj > 2. We abbreviate ej =

1 νj

8 hj π λ

(6.2.38)


139

and we define 8

Cj = (y2 , . . . , yn ) | 0 < yi < hj π λ, i = 2, . . . , n , 8

Cj = (y2 , . . . , yn ) | − hj π λ − 2ej < yi < −2ej , i = 2, . . . , n ,

Cj = (y2 , . . . , yn ) | −2ej < yi < −ej , i = 2, . . . , n ,

Cj +1 = (y2 , . . . , yn ) | 0 < yi < ej , i = 2, . . . , n , cf. Figure 6.15. y3 8

hj π λ Cj ej Cj +1 ej

8

y2 hj πλ

Cj

Cj

Figure 6.15. The cubes Cj , Cj , Cj and Cj +1 for n = 3.

Our next goal is to rescale the fibres n−1 (wj ) × Cj of (βj × id2n−2 ) Fj over {u > 2j s − δ} to the fibres n−1 (νj wj ) × Cj +1 . We shall use the same method as in Step 3 of the proof of Proposition 5.3.3 and first lower the fibres n−1 (wj ) × Cj at δ u = 2j s − δ + 2n to the fibres n−1 (wj ) × Cj , then deform these fibres to the fibres δ n−1 (νj wj ) × Cj , and finally lift these fibres at u = 2j s − 2δ + 2n to the fibres n−1 (νj wj ) × Cj +1 . The lowering map ϕj− . Using the definition of ej and the inequality νj > 2 we estimate 8 (6.2.39) 2ej + hj π λ ≤ 2ej νj ,

140


and using the definitions of ej and wj , the second statement in Lemma 6.2.11 and s we estimate δ < d ≤ 2n 2ej νj wj = 2hj +1 π λ2
−δ 8 for all (x, y) ∈]0, wj [ × −2ej − hj π λ, ∞ , cf. Figure 6.16.

(6.2.42)


141

y

y

wj −2ej

8

νj wj

x

α

−ej

x

ej

hj π λ

Figure 6.16. The map α.

Proof. Choose a smooth function f : R → [1, νj ] such that (i) f (w) = νj for w ≤ −2ej , (ii) f (w) = 1 for w ≥ 0. Since νj > 2 we have 2ej ν1j < ej < 2ej . We may therefore further require that / 0 1 dw = ej . (iii) f (w) −2ej Using (i) and (iii), the definition (6.2.38) of ej , the inequality νj > 2 and the second inequality in Lemma 6.2.11 we can estimate /

0 −2ej −

√

hj π λ

18 1 dw = hj π λ + ej = f (w) νj

2 νj

8 hj π λ < δ.

We conclude that the map 8 α : ]0, wj [× − 2ej − hj π λ, ∞ → R2 ,

/ (x, y) → f (y)x, 0

y

1 dw f (w)

is a symplectic embedding which is as required. In view of (6.2.41) the symplectic embedding αj : ϕj− βj × id2n−2 Fj → R2n defined by αj (u, v, x2 , y2 , . . . , xn , yn ) = (u, v, α(x2 , y2 ), . . . , α(xn , yn ))

2

142


maps the fibres n−1 (wj ) × Cj over the base (u, v) | u > 2j s − 2δ to the fibres n−1 (νj wj ) × Cj . The lifting ϕj+ . In view of the estimate (6.2.40) we find for i = 2, . . . , n a cut off 4 5 δ δ , 2j s − 2δ + i 2n function ci+ : R → 0, νj swj with support 2j s − 2δ + (i − 1) 2n and such that / ∞ 0

ci+ (t) dt = 2ej .

The symplectic embedding ϕj+ : αj ϕj− βj × id2n−2 Fj → R2n ,

(u, v, x, y) → (u , v , x , y )

defined by

u = u, v = v +

n .

ci+ (u)xi ,

xi

= xi ,

yi

/ = yi +

i=2

u

0

ci+ (t) dt, i = 2, . . . , n,

maps the fibres n−1 (νj wj ) × Cj over the base {(u, v) | u > 2j s} to the fibres n−1 (νj wj ) × Cj +1 . We abbreviate the composition φj = ϕj+ αj ϕj− βj × id2n−2 : Fj → R2n . Lemma 6.2.14. We have

φj Fj ∩ (u, v, x, y) | sj ≤ u ≤ 2j s ⊂ Tj (δ) ∪ Nj (δ) ∪ Tj +1 (δ). Proof. We fix (u, v, x, y) ∈ βj × id2n−2 Fj ∩ {sj ≤ u ≤ 2j s}, and we set (u , v , x , y ) = ϕj+ αj ϕj− (u, v, x, y). Then u = u. Assume first that u ≤ 2j s − δ. Then (u , v , x , y ) = (u, v, x, y), and so Lemma 6.2.12 shows that (u , v , x , y ) ∈ Tj (δ) ∪ Nj (δ). Assume now that 2j s − δ < u ≤ 2j s. We read off from the inclusions (6.2.34) and (6.2.35) and from the definitions of ϕj− , αj and ϕj+ that v ∈ ] − δ, s + δ[, cf. Figure 6.17.


143

v s+δ s

−δ

u 2j s − δ

2j s

Figure 6.17. The set pu,v φj Fj ∩ { 2j s − δ < u ≤ 2j s } for n = 3.

Moreover, we have x ∈ n−1 (wj ), and so the first inequality in (6.2.42) implies that x ∈ n−1 (νj wj ). Using the definitions (6.2.37) and (6.2.36) of νj and wj , the first estimate in Lemma 6.2.11 and the definition (6.2.24) of λj +1 we compute : 8 hj hj +1 √ λ = (lj +1 − 1) hj +1 λ < lj +1 λj +1 = s, νj wj = (lj +1 − 1) hj +1 hj

and so

x ∈ n−1 (s). 8 Finally, we have y ∈ Cj = 2n−1 hj π λ , and so the definitions of ϕj− and ϕj+ and 8 the second inequality in (6.2.42) imply that y ∈ ] − δ, hj π λ[n−1 . Using the second 8 estimate in Lemma 6.2.11 we find hj π λ < δ < s, and so y ∈ ] − δ, s [n−1 . We conclude that (u , v , x , y ) ∈ ]2j s − δ, 2j s] × ] − δ, s + δ [ × n−1 (s) × ] − δ, s [n−1 ⊂ Tj +1 (δ). The proof of Lemma 6.2.14 is thus complete.

2

d Recall that βj is the identity on (u, v) | sj ≤ u ≤ sj + j2+1 . We can therefore 3j 3 Sj +1 → R2n by the identity to the extend the symplectic embedding φj : Fj = ψ symplectic embedding  j 2

  +1 id on ψ j2 3 3j S , i 3 3j : ψ 3j 3j = φ Si → R2n , φ i=1   i=1  3j 3 Sj +1 . φj on ψ

144


In view of the formulae (6.2.28) and (6.2.32) and the definition (6.2.37) of νj we find 3j (u, v, x, y) 3j ψ φ 8 (6.2.43) = u − v˜j + 2j s + tj +1 , v, (lj +1 − 1) √ 1 π1 x, lj +11−1 hj +1 πy hj +1

if u ≥ v˜j + dj +1 . 5.C.3. End of the construction of ψj +1 . We denote by Fj +1 the set 5 4 λ +1 aj +1 × ]0, s[ × n−1 (νj wj ) × Cj +1 . Fj +1 = 2j s, 2j s + ljj+1 In view of the construction of φj we have that φj Fj ∩ {(u, v, x, y) | u > 2j s } ⊂ Fj +1 and that the right face of φj Fj is equal to the right face of Fj +1 . We are going to embed Fj +1 into Tj +1 . We denote by τj the translation (u, v, x, y) → (u + 2j s, v, x, y). Moreover, we define the linear symplectomorphisms σj +1 and ξj +1 of R2n by

σj +1 (u, v, x, y) = lj 1+1 u, lj +1 v, lj 1+1 x, lj +1 y and

(lj +1 −1)lj +1 π x, y . ξj +1 (u, v, x, y) = u, v, (lj +1 −1)l π j +1

Using the definitions (6.2.37), (6.2.36), (6.2.38) and (6.2.24) of νj , wj , ej and λj +1 and the first inequality in Lemma 6.2.11 we find that −1 −1 (Fj +1 ) ξj +1 λj−1 +1 σj +1 τj √ √ hj +1 λ hj +1 λ n−1 n−1 π λj +1 ×2 = ]0, aj +1 [ × ]0, 1[ × λj +1 ⊂ ]0, aj +1 [ × ]0, 1[ × n−1 (π ) × 2n−1 (1) = Saj +1 . −1 −1 with the symplectic Composing the affine diffeomorphism ξj +1 λj−1 +1 σj +1 τj embedding ϕaj +1 : Saj +1 → T n (lj2+1 )

guaranteed by Proposition 5.3.3 and with the affine diffeomorphism τj σj +1 λj +1 : T n (lj2+1 ) → Tj +1 we obtain the symplectic embedding −1 −1 ρj := τj σj +1 λj +1 ϕaj +1 ξj +1 λj−1 +1 σj +1 τj : Fj +1 → Tj +1 .


145

Using Proposition 5.3.3 (i) we find ρj (u, v, x, y) = (u, v, x, y) and

ρj (u, v, x, y) = u + s − λj +1 −

if u < 2j s +

λj +1 lj +1 π

λj +1 1 lj +1 aj +1 , v, lj +1 −1 x, (lj +1

if u > 2j s +

− 1)y

λj +1 lj +1 (aj +1

(6.2.44)

− π ).

(6.2.45)

In view of the identity (6.2.44) we can extend the restriction of ρj to the subset φj Fj ∩ { (u, v, x, y) | u > 2j s } by the identity to the symplectic embedding  j 2

  j +1 

3 3 3 2 id on φ ψ S , j j i 3 3j ψ 3j ρ 3j = Si → R2n , ρ 3j : φ i=1   i=1  3j ψ 3j 3 Sj +1 . ρj on φ We finally define the symplectic embedding ψj +1 by ψj +1

3j ψ 3j : := ρ 3j φ

j2 +1

3 Si → R2n .

i=1

In view of the induction hypothesis (6.2.27) and Lemma 6.2.14 and the inclusion ρj (Fj +1 ) ⊂ Tj +1 we have ψj +1 :

j2 +1

3 Si →

i=1

j2 +1

Ti (δ) ∪

i=1

j 2

Ni (δ),

i=1

i.e., (6.2.29) holds true. Moreover, we deduce from formulae (6.2.43) and (6.2.45), from v˜j +1 = v˜j + u˜j +1 and from the definitions of sj +1 , tj +1 and εj +1 given in (6.2.26) and (6.2.31) that 8 if u > v˜j +1 − εj +1 , ψj +1 (u, v, x, y) = u − v˜j +1 + sj +1 , v, √ 1 π1 x, hj +1 πy hj +1

i.e., formula (6.2.30) holds true. This completes the inductive construction of the symplectic embedding ψj +1 . Composing the linear symplectomorphism β × id2n−2 : λ

k 2

Si →

i=1

k 2

3 Si

i=1

given by formula (6.2.25) with the inductively constructed symplectic embedding ψk :

k 2 i=1

3 Si →

k 2 i=1

Ti (δ) ∪

k−1 2 i=1

Ni (δ)

146


we finally obtain the symplectic embedding a := ψk (β × id2n−2 ) : λ

k 2

Si →

i=1

k 2

Ti (δ) ∪

i=1

k−1 2

Ni (δ).

i=1

2

The proof of Proposition 6.2.10 is complete.

Step 6. End of the proof of Theorem 6.2.1. We let > 0 be as in Theorem 6.2.1, set = /5, choose k and a0 = a0 () as in Step 3 and let s be the number associated with k and after the proof of Lemma 6.2.2. We fix a ≥ a0 and define λ = λ(a) as in (6.2.20). Using Lemma 6.2.9, the identities (6.2.18) and the definitions (6.2.20) and (6.2.14) we can estimate 1 |λ T n (a, π)| > λ Si − = k |λS1 | − (6.2.46) = kq |T1 | − 1 2n = k n! s − 2 .

Composing the inclusion ιa : λ T n (a, π) → λ

k 2

Si

i=1

with the symplectic embeddings a and ρ guaranteed by Proposition 6.2.10 and Proposition 6.2.6 we obtain the symplectic embedding a := ρ a ιa : λ T n (a, π ) → M. In view of the estimates (6.2.11) and (6.2.46) we find µ M \ a λ T n (a, π) = µ(M) − µ a λ T n (a, π ) = µ(M) − |λ T n (a, π )| 1 2n 1 2n < k n! s + 3 − k n! s − 2 = 5

= . This is the required estimate in Theorem 6.2.1 an so the proof of Theorem 6.2.1 is complete. 2

6.3 Asymptotic embedding invariants

147

6.3 Asymptotic embedding invariants Consider again a connected 2n-dimensional symplectic manifold (M, ω) of finite volume Vol(M, ω). In view of Theorem 3 the asymptotic symplectic invariants lim paE (M, ω) = 1

a→∞

and

lim paP (M, ω) = 1

a→∞

(6.3.1)

are uninteresting. In order to recapture some information on the geometry of (M, ω) one can try to study the convergence speeds in (6.3.1). We define the symplectic invariants σE (M, ω) and σP (M, ω) in [0, ∞] by there exists a constant C < ∞ , σE (M, ω) = sup s such that 1 − paE (M, ω) ≤ Ca −s for all a > π there exists a constant C < ∞ σP (M, ω) = sup s . such that 1 − paP (M, ω) ≤ Ca −s for all a > π In this section we notice that σE (M, ω) ≥ n1 or σP (M, ω) ≥ n1 for large classes of 2n-dimensional symplectic manifolds and thereby improve Theorem 3 for many symplectic manifolds. Consider a domain U in R2n . The distance d(p, ∂U ) between a point p ∈ U and the boundary ∂U is d (p, ∂U ) = inf {d(p, q) | q ∈ ∂U } . We say that U is very connected if there exists > 0 such that the set U \ {p ∈ U | d (p, ∂U ) < } is connected. Theorem 6.3.1. Assume that U is a very connected bounded domain in R2n with piecewise smooth boundary and that (M, ω) is a compact connected 2n-dimensional symplectic manifold. (i)E σE (U ) ≥

1 n

(ii)E σE (M, ω) ≥

if n ≤ 3 or if U is a ball. 1 n

if n ≤ 3.

(i)P σP (U ) ≥ n1 . (ii)P σP (M, ω) ≥ n1 . Rough outline of the proof of Theorem 6.3.1. Assertion (i)E for a 2n-ball follows from Corollary 7.1.6 (i) below which is proved by symplectic wrapping. The other assertions can be proved by the symplectic folding methods previously described. Assume first that U is a 2n-cube. If n = 2, assertion (i)E follows from Proposition 4.4.2, and if n = 3, assertion (i)E can be proved by using that in dimension 2

148


a cube can be filled with small simplices. Assertion (i)P for a cube follows from Proposition 5.2.1. Assume next that U is a very connected bounded domain in R2n with piecewise smooth boundary. Then assertions (i)E and (i)P can be proved by first exhausting U with an increasing sequence U1 ⊂ U2 ⊂ · · · of connected unions of equal cubes and then extending the embedding techniques used to fill a cube to the sets Ui . Assume finally that (M, ω) is a compact connected 2n-dimensional symplectic manifold. Then assertions (ii)E and (ii)P can be proved by first choosing finitely many Darboux charts ϕi : Ui → Vi ⊂ M such that the Ui are very connected bounded domains with piecewise smooth boundary and such that the Vi are disjoint and Vi = M, then connecting the Vi by lines, and finally applying the technique used to prove (i)E and (i)P to the sets Vi = ϕi (Ui ) and using thinner and thinner neigh2 bourhoods of the lines to pass from one Vi to another.

Chapter 7

Symplectic wrapping

In this chapter we first describe the symplectic wrapping construction invented by Traynor in [81], and then compare the embedding results obtained by symplectic folding and symplectic wrapping.

7.1 The wrapping construction Symplectic wrapping views the whole ellipsoid or the whole polydisc as a Lagrangian product of a cube and a simplex or of a cube and a cuboid, and then wraps the cube around the base of the cotangent bundle of the torus via a linear map. Symplectic wrapping has therefore a more algebraic flavour than symplectic folding. For the sake of brevity we shall only study symplectic wrapping embeddings of skinny ellipsoids into balls and of skinny polydiscs into cubes. Theorem 7.1.1. Assume that a > 0 and that k1 < · · · < kn−1 are relatively prime numbers. (i) The ellipsoid E 2n (π, . . . , π, a) symplectically embeds into the ball a 2n max (kn−1 + 1) π, + B k1 · · · kn−1 for any > 0. (ii) The polydisc P 2n (π, . . . , π, a) symplectically embeds into the cube a 2n max kn−1 π, (n − 1)π + . C k1 · · · kn−1 Proof. Theorem 7.1.1 (i) for n = 2 is Theorem 6.4 in [81]. We shall closely follow [81]. We consider again the Lagrangian splitting Rn (x) × Rn (y) of R2n , set 2(a1 , . . . , an ) = {0 < xi < ai , 1 ≤ i ≤ n} ⊂ Rn (x), 6 7 .n yi < 1 ⊂ Rn (y), (b1 , . . . , bn ) = 0 < y1 , . . . , yn i=1 bi

150

7 Symplectic wrapping

and abbreviate 2n (a) = 2(a, . . . , a) and n (b) = (b, . . . , b). We also set T n = Rn (x)/π Zn and abbreviate κ = k1 · · · kn−1 and

AE = max (kn−1 + 1) π, κa ,

AP = max kn−1 π, (n − 1)π + κa . Choose > 0. We set = /AE . The embeddings provided by symplectic wrapping are compositions of symplectic embeddings αE

E(π, . . . , π, a) −→ 2n (1) × (1 + )(π, . . . , π, a) β a − → 2 kπ1 , . . . , knπ−1 , κπ × (1 + ) k1 , . . . , kn−1 , κπ γ − → T n × n AEπ+ δE

−→ B 2n (AE + ) respectively αP

P (π, . . . , π, a) −→ 2n (1) × 2(π, . . . , π, a) β π a − → 2 kπ1 , . . . , kn−1 , κπ × 2 k1 , . . . , kn−1 , κπ γ − → T n × 2n AπP δP

−→ C 2n (AP ) . 1. The maps αE and αP . Recall from the proof of Lemma 5.3.1 (i) that there exists a symplectomorphism α1 × · · · × αn : P (π, . . . , π, a) → 2(π, . . . , π, a) × 2n (1) which embeds the subset E(π, . . . , π, a) into (1 + )(π, . . . , π, a) × 2n (1). We denote the reflection (x, y) → (x, −y) of Rn (x) × Rn (y) by ρ and the permutation (x, y) → (y, x) by σ . The composition αP := σ (α1 × · · · × αn ) ρ symplectomorphically maps P (π, . . . , π, a) to 2n (1)×2(π, . . . , π, a), and its restriction to E(π, . . . , π, a), which we denote by αE , symplectically embeds E(π, . . . , π, a) into 2n (1) × (1 + )(π, . . . , π, a).

151

7.1 The wrapping construction

2. The map β. The map β is the linear symplectomorphism of Rn (x) × Rn (y) given by the diagonal matrix π kn−1 1 π k1 diag , . ,..., , κπ, , . . . , k1 kn−1 π π κπ 3. The map γ . In order to describe the wrapping map γ we need an elementary lemma. Lemma 7.1.2. Let M : Rn (x) → Rn (x) be the linear map given by the matrix 

− k11

1

       

1 .. 0

0

− k12 .. .

1

1 − kn−1

.

        

1 n n Then the composition and let p : Rn (x) → Rn (x)/π Z π = T πbe the projection. n n p M : R (x) → T embeds 2 k1 , . . . , kn−1 , κπ into T n . π , κπ for which (p M)(x) = Proof. Let x and x be points in 2 kπ1 , . . . , kn−1 (p M)(x ). Then x − xi < π , i = 1, . . . , n − 1, (7.1.1) i ki

and

x − xn < κ π, n

(7.1.2)

and there exist integers l1 , . . . , ln−1 such that xi − and an integer ln such that

xn ki

= xi −

xn ki

+ li π

xn = xn + ln π.

(7.1.3) (7.1.4)

Inserting the identity (7.1.4) into the estimate (7.1.2) we obtain |ln | < κ

(7.1.5)

and inserting (7.1.4) into the identities (7.1.3) we obtain xi − xi = klni + li π, i = 1, . . . , n − 1.

(7.1.6)

The identities (7.1.6) and the estimates (7.1.1) yield l n + li < 1 , i = 1, . . . , n − 1. ki ki

(7.1.7)

152


Assume that ln = 0. Then the estimates (7.1.7) imply that ln + ki li = 0 for i = 1, . . . , n − 1. Since the numbers k1 , . . . , kn−1 are relatively prime, we conclude that ln = mk1 · · · kn−1 for some integer m = 0. In particular, |ln | ≥ k1 · · · kn−1 = κ in contradiction to (7.1.5). Therefore ln = 0. The estimates (7.1.7) now imply that also l1 = · · · = ln−1 = 0, and of the identities (7.1.4) and (7.1.3). πso x = xπ in view The restriction of p M to 2 k1 , . . . , kn−1 , κπ is therefore injective, as claimed. 2 We denote by M ∗ the transpose of the inverse of M. Then the map M × M ∗ : Rn (x) × Rn (y) → Rn (x) × Rn (y) is a linear symplectomorphism. In view of Lemma 7.1.2 the map π

γ:2

π k1 , . . . , kn−1 , κπ

× Rn (y) → T n × Rn (y)

defined by γ = (p M) × M ∗ is a symplectic embedding. Lemma 7.1.3. We have π a ⊂ T n × n AEπ+ , κπ × (1 + ) k1 , . . . , kn−1 , κπ γ 2 kπ1 , . . . , kn−1 and

π a γ 2 kπ1 , . . . , kn−1 ⊂ T n × 2n AπP . , κπ × 2 k1 , . . . , kn−1 , κπ

Proof. In view of the definition of γ and since (1 + )AE = AE + it is enough to show that a ⊂ n AπE M ∗ k1 , . . . , kn−1 , κπ and that

a ⊂ 2n AπP . M ∗ 2 k1 , . . . , kn−1 , κπ

We compute that M ∗ is given by the matrix 



1

      

1 ..

.

0 1 k1

1 k2

   .   

0

1 ...

1 kn−1

1


153

a and set y = M ∗ y. Using the definiWe assume first that y ∈ k1 , . . . , kn−1 , κπ a tions of k1 , . . . , kn−1 , κπ and AE we then find y1 yn−1 a yn + · · · + (kn−1 + 1) + a k1 kn−1 κπ κπ 6 a 7 < max kn−1 + 1, κπ AE . = π

y1 + · · · + yn = (k1 + 1)

Therefore y ∈ n

AE

as claimed. a and set y = M ∗ y. Using the We assume next that y ∈ 2 k1 , . . . , kn−1 , κπ a definitions of 2 k1 , . . . , kn−1 , κπ and AP we then find a ⊂ 2n AπP y ∈ 2 k1 , . . . , kn−1 , n − 1 + κπ π

2

as claimed. 4. The maps δE and δP . We define symplectic embeddings δ˜E : 2n (π) × n AEπ+ → B 2n (AE + ) and

δ˜P : 2n (π) × 2n

by (x1 , . . . , xn , y1 , . . . , yn ) →

AP π

→ C 2n (AP )

√ √ y1 cos 2x1 , . . . , yn cos 2xn , √ √ − y1 sin 2x1 , . . . , − yn sin 2xn .

The embedding δ˜E extends to a symplectic embedding δE : T n × n AEπ+ → B 2n (AE + ) and the embedding δ˜P extends to a symplectic embedding δP : T n × 2n AπP → C 2n (AP ). This completes the construction of the symplectic embeddings involved in symplectic wrapping, and Theorem 7.1.1 is proved. 2 Assume that a > 0 and that k1 < · · · < kn−1 are relatively prime numbers. As in the proof of Theorem 7.1.1 we abbreviate 6 7 AE (a; k1 , . . . , kn−1 ) = max (kn−1 + 1) π, k1 ···ka n−1 , 6 7 AP (a; k1 , . . . , kn−1 ) = max kn−1 π, (n − 1)π + k1 ···ka n−1 .

154


2n : ]2π, ∞[ → R and In view of Theorem 7.1.1 we are interested in the functions wEB 2n : ]2π, ∞[ → R defined by wPC 2n (a) := inf {AE (a; k1 , . . . , kn−1 ) | k1 < · · · < kn−1 are relatively prime} , wEB 2n wPC (a) := inf {AP (a; k1 , . . . , kn−1 ) | k1 < · · · < kn−1 are relatively prime} . 2n and w 2n can be replaced Notice that the infimum in the definitions of the functions wEB PC 2n and w 2n for by the minimum. We shall next explicitly compute the functions wEB PC n = 2 and n = 3.

Corollary 7.1.4. Assume that a > 2π.

4 (i) The ellipsoid E(π, a) symplectically embeds into the ball B 4 wEB (a) + for any > 0 where (k + 1)π if (k − 1)(k + 1) < πa ≤ k(k + 1), 4 wEB (a) = a if k(k + 1) < πa ≤ k(k + 2). k

4 (ii) The polydisc P (π, a) symplectically embeds into the cube C 4 wPC (a) where kπ if (k − 1)2 < πa ≤ (k − 1)k, 4 wPC (a) = a a 2 k + π if (k − 1)k < π ≤ k . Proof. (i) follows from Theorem 7.1.1 (i), from the identity

4 (a) = min max (k + 1)π, ak wEB k∈N

and from a straightforward computation. (ii) follows from Theorem 7.1.1 (ii), from the identity

4 wPC (a) = min max kπ, ak + π k∈N

and from a straightforward computation.

Corollary 7.1.5. Assume that a > 2π .

2

6 (i) The ellipsoid E(π, π, a) symplectically embeds into the ball B 6 wEB (a) + for any > 0 where  (k + 1)π if (k − 2)(k − 1)(k + 1) < a ≤ (k − 1)k(k + 1), π 6 wEB (a) = a  a if (k − 1)k(k + 1) < π ≤ (k − 1)k(k + 2). (k−1)k

155


6 (a) where (ii) The polydisc P (π, π, a) symplectically embeds into the cube C 6 wPC  (k + 1)π if (k − 1)2 k < πa ≤ (k − 1)k(k + 1), 6 wPC (a) =  a + 2π if (k − 1)k(k + 1) < a ≤ k 2 (k + 1). k(k+1) π Proof. (i) Fix a > 2π and assume that k1 < k2 are relatively prime numbers. Then k2 − 1 and k2 are also relatively prime, and AE (a; k2 − 1, k2 ) ≤ AE (a; k1 , k2 ). 6 7 6 a (a) = min max (k + 1)π, (k−1)k . wEB

It follows that

k∈N

(7.1.8)

Assertion (i) follows from Theorem 7.1.1 (i), from the identity (7.1.8) and from a straightforward computation. (ii) The same argument as above implies that for each a > 2π , 6 7 6 a (a) = min max kπ, 2π + (k−1)k wPC k∈N 6 7 a = min max (k + 1)π, k(k+1) + 2π . (7.1.9) k∈N

Assertion (ii) follows from Theorem 7.1.1 (ii), from the identity (7.1.9) and from a straightforward computation. 2 2n (a) and w 2n (a) is more involved, and we do not For n ≥ 4 the computation of wEB PC 2n and w 2n . The next corollary describes know explicit formulae for the functions wEB PC the asymptotic behaviour of these functions for all n ≥ 2.

Corollary 7.1.6. (i) There exists a constant CE depending only on n such that |E(π, . . . , π, a)| ≥ 1 − CE a −1/n for all a > 2π. B 2n w2n (a) EB

(ii) There exists a constant CP depending only on n such that |P (π, . . . , π, a)| ≥ 1 − CP a −1/n for all a > 2π. C 2n w2n (a) PC

Proof. (i) We choose n − 1 prime numbers p1 < p2 < · · · < pn−1 and define l to be the least common multiple of the differences pj − pi , Fix a > 2π. We set

1 ≤ i < j ≤ n − 1.

: 1 n a + pn−1 m := l π

(7.1.10)

156


where $r% denotes the minimal integer which is greater or equal to r, and ki := ml − pn−i ,

i = 1, . . . , n − 1.

(7.1.11)

We claim that the numbers k1 < k2 < · · · < kn−1 are relatively prime. Indeed, assume that (7.1.12) d | ml − pi and d | ml − pj for some i = j . Then d divides the difference (ml − pi ) − (ml − pj ) = pj − pi and so d divides the least common multiple l. But then (7.1.12) implies that d divides both pi and pj , and so d = 1 as claimed. Using the definitions (7.1.11) and (7.1.10) we find : a k1 = ml − pn−1 ≥ n . π Therefore, (kn−1 + 1) kn−1 . . . k1 ≥ k1n ≥ We conclude that (kn−1 + 1) π ≥

a . π

a k1 · · · kn−1

and so 2n (a). (kn−1 + 1) π = AE (a; k1 , . . . , kn−1 ) ≥ wEB

By definition (7.1.10) we have : : a 1 n a + pn−1 + 1 = n + pn−1 + l. ml ≤ l l π π Using definition (7.1.11) and the estimate (7.1.14) we find : a kn−1 + 1 = ml − p1 + 1 ≤ n + pn−1 + l. π

(7.1.13)

(7.1.14)

(7.1.15)

The estimates (7.1.13) and (7.1.15) now yield 2n 2n √ √ n n B w (a) a + (pn−1 + l) n π (kn−1 + 1)n π EB . (7.1.16) ≤ ≤ |E(π, . . . , π, a)| a a √ Recall that the number (pn−1 + l) n π only depends on n. In view of the estimate (7.1.16) we therefore find a constant CE depending only on n such that 2n 2n B w (a) EB ≤ 1 + CE a −1/n for all a > 2π |E(π, . . . , π, a)|

7.2 Folding versus wrapping

and so

|E(π, . . . , π, a)| ≥ 1 − CE a −1/n B 2n w2n (a)

157

for all a > 2π.

EB

Assertion (i) is proved. (ii) can be proved in the same way as (i).

7.2

2

Folding versus wrapping

Recall from Chapters 4, 5 and 6 that symplectic folding yields good symplectic embedding results of 2n-dimensional ellipsoids and polydiscs into any connected 2n-dimensional symplectic manifold of finite volume. The reason is that symplectic folding is local in nature in the sense that each fold is achieved on a set of small volume. Symplectic wrapping described in Section 7.1, however, is global in nature and thus yields good symplectic embedding results of ellipsoids and polydiscs into special symplectic manifolds only. E.g., the best 2n-dimensional symplectic wrapping embedding of an ellipsoid into a cube respectively of a polydisc into a ball fill less 1 respectively nn!n of the volume. than n! In this section we shall compare the embedding results for embeddings of skinny ellipsoids into balls and skinny polydiscs into cubes yielded by the two methods. 7.2.1

Embeddings E 2n (π, . . . , π, a) → B 2n (A)

1. The case n = 2. Recall from Theorem 1 that for a ∈ [π, 2π] the ellipsoid E(π, a) does not symplectically embed into the ball B 4 (A) if A < 2π . Also recall 4 : ]2π, ∞[ → R constructed that the functions fEB : ]2π, ∞[ → R and wEB ≡ wEB in 4.3.1 and defined in Corollary 7.1.4 (i) describe our results for the embedding problem E(π, a) → B 4 (A) obtained by multiple symplectic folding and by symplectic wrapping. √ According to Proposition 4.3.7 the difference fEB (a) − π a between fEB and the volume condition is bounded by 2π . Computer calculations suggest that this difference is √ monotone increasing and converging to π as a → ∞. The difference wEB (a) − πa between wEB and the volume condition √ attains its local minima at k(k + 1)π, where it is equal to mk = (k + 1)π − k(k + 1) π√ , and it attains its local maxima at k(k + 2)π , where it is equal to Mk = (k + 2)π − k(k + 2) π . The sequence (mk ) strictly decreases to π2 , and (Mk ) strictly decreases to π . We conclude that the difference |fEB (a) − wEB (a)| is bounded by 2π . For a > 2π small we have fEB (a) < wEB (a). E.g., fEB (3π ) = 2.3801 . . . π and fEB (4π) = 2.6916 . . . π, and so Fact 7.2.1. The ellipsoid E(π, 3π ) symplectically embeds into B 4 (2.381 π ), and E(π, 4π) symplectically embeds into B 4 (2.692 π ).

158


The inequality wEB (a) < fEB (a) happens first at a = 5.1622 . . . π. In general, the computer calculations for fEB suggest that the functions wEB and fEB yield alternately better estimates: For all k ∈ N we seem to have wEB (a) < fEB (a) on an interval around k(k + 1)π and fEB (a) < wEB (a) on an interval around k(k + 2)π; moreover, we seem to have lim (fEB (k(k + 2)π) − wEB (k(k + 2)π )) = 0.

k→∞

We checked the above statements for k ≤ 5 000. We extend both functions fEB (a) and wEB (a) to functions on [π, ∞[ by setting fEB (a) = a and wEB (a) = a for a ∈ [π, 2π]. The characteristic function for the embedding problem E(π, a) → B 4 (A) is the function χEB on [π, ∞[ defined by

χEB (a) = inf A | E(π, a) symplectically embeds into B 4 (A) . The following proposition summarizes what we know about this function. Proposition 7.2.2. For a ∈ [π, 2π] we have χEB (a) = a, and on ]2π, ∞[ the function χEB is bounded from below and above by √ max(2π, π a ) ≤ χEB (a) ≤ min wEB (a), fEB (a) , see Figure 7.1. In particular, lim sup →0+

3 χEB (2π + ) − 2π ≤ . 7

(7.2.1)

The function χEB is monotone increasing and hence almost everywhere differentiable. Moreover, χEB is Lipschitz continuous with Lipschitz constant at most 1; more precisely, (a), f (a) min w EB EB χEB (a ) − χEB (a) ≤ (a − a) for all a ≥ a ≥ π. a Proof. The estimates of χEB (a) from below are provided by the second Ekeland– Hofer capacity, which yields χEB (a) ≥ a for a ∈ [π, 2π ] andχEB (a) ≥ 2π for a > 2π, and√by the volume condition |E(π, a)| ≤ B 4 χEB (a) , which translates to χEB (a) ≥ πa. The estimate (7.2.1) follows from χEB ≤ fEB and from Proposition 4.3.5. The remaining claims follow as in the proof of Proposition 4.3.11. 2 2. The case n ≥ 3. Recall from Theorem 1 that for a ∈ [π, 2π ] the ellipsoid E 2n (π, . . . , π, a) does not symplectically embed into the ball B 2n (A) if A < a. For a > 2π, the n’th Ekeland–Hofer capacity still implies that E 2n (π, . . . , π, a) does not symplectically embed into B 2n (A) if A 2π , Theorem 3.1.1, which we proved by symplectic folding, shows that E 2n (π, . . . , π, a) symplectically embeds into B 2n a2 + π + for every > 0. For a large, the results obtained by symplectic folding, which are Theorem 6.3.1 (i)E for n = 3 and the first statement in Theorem 3, are weaker than the results obtained 2n : ]2π, ∞[ → R by symplectic wrapping, which are described by the function wEB √ 6 (a) − 3 π 2 a between the defined before Corollary 7.1.4. For n = 3 the difference wEB 6 computed in Corollary 7.1.5 (i) and the volume condition is bounded by function wEB √ 2n (a)− n π n−1 a 2π. For n ≥ 4 it follows from Corollary 7.1.6 (i) that the difference wEB 2n and the volume condition is bounded. between wEB 7.2.2

Embeddings P 2n (π, . . . , π, a) → C 2n (A)

1. The case n = 2. Recall that the functions fPC : ]2π, ∞[ → R and wPC ≡ 4 : ]2π, ∞[ → R defined in Proposition 4.4.4 and Corollary 7.1.4 (ii) describe our wPC results for the embedding problem P (π, a) → C 4 (A) obtained by multiple symplectic folding and by symplectic wrapping. Comparing Proposition 4.4.4 with Corollary 7.1.4 (ii) we see that wPC (a) ≤ fPC (a) for all a > 2π , cf. Figure 7.2. Equality only holds √ for a ∈ ]2π, 4π [ and 2 2 a ∈ [k π, (k + 1)π ], k ≥ 2. The difference wPC (a) − π a between wPC and the volume condition attains its local minima at k(k − 1)π , k ≥ 3, where it is equal to

160


√ mk = kπ − k(k − 1)π , and it attains its local maxima at k 2 π , where it is equal to π . The sequence (mk ) strictly decreases to π2 . Extend the function wPC (a) to a function on [π, ∞[ by setting wPC (a) = a for a ∈ [π, 2π ]. The characteristic function for the embedding problem P (π, a) → C 4 (A) is the function χPC on [π, ∞[ defined by

χPC (a) = inf A | P (π, a) symplectically embeds into C 4 (A) . The following proposition summarizes what we know about this function. Proposition 7.2.3. The function χPC : [π, ∞[ → R is bounded from below and above by √ π a ≤ χPC (a) ≤ wPC (a), see Figure 7.2. It is monotone increasing and hence almost everywhere differentiable. Moreover, χPC is Lipschitz continuous with Lipschitz constant at most 1; more precisely, χPC (a ) − χPC (a) ≤

wPC (a) (a − a) for all a ≥ a ≥ π. a

√ Proof. The estimate π a ≤ χPC (a) from below is provided by the volume condition |P (π, a)| ≤ |C 4 (χPC (a)) |. The remaining claims follow as in the proof of Proposition 4.3.11. 2 A A=a

5π

wPC (a)

4π fPC (a) 3π

A=

2π

√ πa

a 2π

4π

6π

9π

Figure 7.2. What is known about χPC (a). 2n : ]2π, ∞[ → R and 2. The case n ≥ 3. We recall that the two functions fPC 2n : ]2π, ∞[ → R defined in Proposition 5.2.1 and before Corollary 7.1.4 describe wPC

7.2 Folding versus wrapping

161

our results for the embedding problem P 2n (π, . . . , π, a) → C 2n (A) obtained by multiple symplectic folding and by symplectic wrapping. Comparing the case n = 3 of Proposition 5.2.1 with Corollary 7.1.5 (ii) we see 6 (a) ≤ w 6 (a) for all a > 2π . Equality that in contrast to thecase n = 2 we have fPC PC 2 only holds for a ∈ (k − 1)k + 2 π, (k − 1)k(k + 1)π , k ≥ 2. The difference √ 6 (a) − 3 π 2 a between f 6 and the volume condition is bounded by fPC PC √ 3 d3 = 13 − 1586 π ≈ 1.338π. 2n and w 2n is more involved since we For n ≥ 4 the comparison of the functions fPC PC √ 2n . The difference f 2n (a) − n π n−1 a between do not know an explicit formula for wPC PC 2n and the volume condition is bounded by d where fPC n

√ 4 d4 = 5 − 194 π,

√ 5 d5 = 4 − 164 π,

8 n dn = 3 − 2n−1 + 2 π,

n ≥ 6.

The sequence (dn ), n ≥ 3, strictly √ decreases to π . It follows from Corollary 7.1.6 (ii) 2n (a) − n π n−1 a is also bounded. Therefore, the difference that the difference w PC 2n f − w 2n is bounded. PC PC We conclude this section by motivating the conjecture alluded to at the end of Section 2.3. We say that a polydisc P 2n (π, . . . , π, a), a > π , is reducible if it symplectically embeds into a cube C 2n (A) for some A < a. It follows from Theorem 1.3 (ii) in [50] that a polydisc cannot be reduced by a local squeezing method. A symplectic embedding which reduces P 2n (π, . . . , π, a) must therefore be of global nature. In view of Proposition 5.2.1 and Corollary 7.1.4 (ii) symplectic folding and symplectic wrapping both show that P 2n (π, . . . , π, a) is reducible if a > 2π . However, none of the two methods can reduce P 2n (π, . . . , π, a) if a ≤ 2π . Since we believe that only some kind of folding or wrapping can reduce a polydisc, we conjecture Conjecture 7.2.4. The polydisc-analogue of Theorem 1 holds true. In particular, the polydiscs P 2n (π, . . . , π, a) symplectically embeds into the cube C 2n (A) for some A < a if and only if a > 2π.

Chapter 8

Proof of Theorem 4

In this chapter we prove Theorem 4 stated in Section 1.3.3 in two different ways, first by symplectic folding, and then by a symplectic lifting construction. We first state a generalization of the second statement ζˆ (a) = 0

for all a ∈]0, π [

(8.0.1)

in Theorem 4, and prove a generalization of ζˆ (π ) = π . In Section 8.2 we give a further motivation for Problem ζ coming from convex geometry and from the special behaviour of symplectic capacities on bounded convex subsets of (R2n , ω0 ). In Section 8.3 we prove the generalization of (8.0.1) by symplectic folding, and in Section 8.4 we use symplectic lifting to prove a yet more general result which also proves the first statement in Theorem 4.

8.1 A more general statement We consider arbitrary subsets S of R2n which symplectically embed into the cylinder Z 2n (π ), and we measure the intersections ϕ(S) ∩ Dx by arbitrary extrinsic symplectic capacities c on R2 as defined in Definition C.5. We refer to Appendix C.2 for a thorough study of such capacities and only mention that any (normalized) intrinsic symplectic capacity on R2 as defined in Definition C.1 is a (normalized) extrinsic symplectic capacity on R2 . A symplectic embedding of a subset S of R2n into another subset S of R2n is by definition a symplectic embedding of an open neighbourhood of S into R2n which maps S into S . As before we denote by Z 2n (a) the open standard symplectic cylinder D(a) × R2n−2 , and we denote by Ez ⊂ R2n the affine plane Ez := R2 × {z},

z ∈ R2n−2 .

Given any subset T of Z 2n (π) we abbreviate c (T ∩ Ez ) := c { (u, v) ∈ R2 | (u, v, z) ∈ T ∩ Ez } . Assume now that the subset S of R2n symplectically embeds into Z 2n (π ). We define the symplectic invariant ξc (S) ∈ [0, π ] by ξc (S) := inf sup c (ϕ (S) ∩ Ez ) ϕ

z

(8.1.1)

8.1 A more general statement

163

where ϕ varies over all symplectic embeddings of S into Z 2n (π ). The main result of this chapter is Theorem 8.1.1. Assume that the subset S of R2n symplectically embeds into Z 2n (a) for some a < π. Then ξc (S) = 0 for any extrinsic symplectic capacity c on R2 . Assume now that S is a subset of R2n which embeds into Z 2n (π ) by a symplectomorphism of R2n . We define the ambient symplectic invariant ζc (S) ∈ [0, π ] by ζc (S) := inf sup c (ϕ (S) ∩ Ez ) ϕ

z

where now ϕ varies over all symplectomorphisms of R2n which embed S into Z 2n (π ). Then ξc (S) ≤ ζc (S). Corollary 8.1.2. Assume that S is a relatively compact subset of R2n whose closure embeds into Z 2n (π) by a symplectomorphism of R2n . Then ζc (S) = 0 for any extrinsic symplectic capacity c on R2 . The second statement (8.0.1) in Theorem 4 in Section 1.3.3 is a special case of Corollary 8.1.2. By Theorem 8.4.2 below, Corollary 8.1.2 can be extended to certain unbounded subsets of R2n . The following result shows that some condition on S, however, must be imposed. Proposition 8.1.3. For the unit circle S 1 ⊂ R2 we have ζc B 2n (π ) ∈ [c(S 1 ), π ] for any normalized extrinsic symplectic capacity c on R2 . According to Corollary C.10 (i) we have c(S 1 ) = 0 for any intrinsic symplectic capacity c on R2 , and so the statement in Proposition is empty for these capacities. 2n 8.1.3 In fact, Theorem 8.4.2 below shows that ζc B (π ) = 0 for intrinsic symplectic capacities. On the other hand, Proposition C.12 says that for any a ∈ [0, π ] there exists a normalized extrinsic symplectic capacity c on R2 such that c(S 1 ) = a. Examples of normalized extrinsic symplectic capacities on R2 with c(S 1 ) = π are the first Ekeland– Hofer capacity, [20, Theorem 1], the displacement energy, [38, Theorem 1.9], and the outer cylindrical capacity zˆ , see Theorem C.8 (iii). Proof of Corollary 8.1.2. Fix > 0. We denote the closure of S by S. By assumption there exists a symplectomorphism ϕ of R2n such that ϕ(S) ⊂ Z 2n (π ). Since S is compact, ϕ(S) is compact in R2n . We therefore find a ∈ ]0, π [ and A > 0 such that ϕ(S) ⊂ D(a) × B 2n−2 (A).

(8.1.2)

D(a) × B 2n−2 (A) ⊂ Z 2n (a ).

(8.1.3)

Choose a ∈ ]a, π [. Then

164


In view of Theorem 8.1.1 there exists a symplectic embedding

such that

ψ : Z 2n (a ) → Z 2n (π )

(8.1.4)

sup c ψ Z 2n (a ) ∩ Ez < .

(8.1.5)

z

Applying Proposition A.1 to the bounded starshaped domain D(a) × B 2n−2 (A) we find a symplectomorphism of R2n such that |D(a)×B 2n−2 (A) = ψ|D(a)×B 2n−2 (A) .

(8.1.6)

The inclusion (8.1.2), the identity (8.1.6) and the inclusion (8.1.3) show that ( ϕ)(S) ⊂ ψ Z 2n (a ) .

(8.1.7)

In view of the inclusions (8.1.7) and (8.1.4) the symplectomorphism ϕ of R2n embeds S into Z 2n (π). Moreover, the inclusion (8.1.7), the monotonicity of the symplectic capacity c and the estimate (8.1.5) yield sup c ϕ (S) ∩ Ez ≤ sup c ψ Z 2n (a ) ∩ Ez < . z

z

We conclude that ζc (S) ≤ ζc (S) < . Since > 0 was arbitrary, Corollary 8.1.2 follows. 2 Proof of Proposition 8.1.3. We abbreviate

Sz1 := (u, v, z) ∈ Ez | u2 + v 2 = 1 ,

z ∈ R2n−2 .

Let ϕ be a symplectomorphism of R2n which embeds B 2n (π ) into Z 2n (π ). According to Lemma 1.2 in [50] there exists z0 ∈ R2n−2 such that Sz10 ⊂ ∂ ϕ B 2n (π) ∩ Ez0 . Since ϕ is a diffeomorphism of R2n and ϕ B 2n (π ) ⊂ Z 2n (π ), the boundary 2n 2n ∂ ϕ B (π ) of ϕ B (π) is tangent to the boundary S 1 × R2n−2 of Z 2n (π ) at each point of Sz10 . This, the inclusion ϕ B 2n (π ) ⊂ Z 2n (π ) and the compactness of Sz10 imply that there exists an > 0 such that the annulus

A := (u, v, z0 ) ∈ Ez0 | 1 − < |(u, v)| < 1 is contained in ϕ B 2n (π) ∩ Ez0 . For each r ∈ ]1 − , 1[ we then have rSz10 ⊂ A ⊂ ϕ B 2n (π ) ∩ Ez .

8.2 A further motivation for Problem ζ

165

In view of the conformality and the monotonicity of the symplectic capacity c we obtain that r 2 c(S 1 ) = c(rS 1 ) = c rSz10 ≤ c(A ) ≤ c ϕ B 2n (π ) ∩ Ez for each r ∈ ]1 − , 1[, and so c ϕ B 2n (π) ∩ Ez ≥ c(S 1 ). Since this estimate holds for any symplectomorphism ϕ of R2n which embeds B 2n (π ) into Z 2n (π ) we conclude that ζc B 2n (π) ≥ c(S 1 ). On the other hand, the definition of ζc and the normalization of c yield ζc B 2n (π) ≤ sup c B 2n (π) ∩ Ez = c D(π ) = π. z

The proof of Proposition 8.1.3 is complete.

2

Lemma 1.2 in [50] and its proof suggest that the answer to the following question is affirmative. It would be interesting to have a proof of this. Question 8.1.4. Is it true that ξc B 2n (π) ≥ c(S 1 ) for every extrinsic symplectic capacity c on R2 ? In particular, ξµˆ B 2n (π) = π ?

8.2 A further motivation for Problem ζ Recall from Section 1.3.3 that Problem ζ was stimulated by the search for symplectic rigidity phenomena beyond the Nonsqueezing Theorem. In this section we give yet another motivation for this problem coming from convex geometry and from the special behaviour of symplectic capacities on bounded convex domains in (R2n , ω0 ). We denote by K n the set of bounded convex domains in Rn . Notice that K 2n is not invariant under the group D(n) of symplectomorphisms of (R2n , ω0 ). This can be seen by symplectic folding or, even easier, by symplectic lifting as described in Section 8.4 below, cf. Figure 8.9. We thus consider

Ks2n = ϕ(K) | K ∈ K 2n , ϕ ∈ D(n) . It seems that the symplectic geometry of sets in Ks2n is special. One instance for this is the following observation due to Viterbo, [85]. Proposition 8.2.1. For any two normalized intrinsic symplectic capacities c and c on (R2n , ω0 ) and any K ∈ Ks2n we have c (K) ≤ n2 c(K).

166


We refer to Definition C.1 for the definition of an intrinsic symplectic capacity on (R2n , ω0 ). The proof of Proposition 8.2.1 is similar to the proof of Proposition 8.2.2 below and is given in Appendix C.1. Notice that Proposition 8.2.1 does not extend to bounded starshaped domains in view of the Symplectic Hedgehog Theorem stated in Section 1.2.2. Looking for other peculiarities of the geometry of convex domains, one can proceed as follows: Start from an inequality of Euclidean invariants on K 2n , symplectify these invariants to symplectic invariants on Ks2n , and check whether the inequality survives. We refer to Section C.4 and to [76] for examples of this procedure. Here, we start from the intersection invariant σ and the projection invariant π . Fix a convex domain K ∈ K n . Given an element H of the Grassmannian H of hyperplanes in Rn through the origin, let pH : Rn → H be the orthogonal projection onto H and let |pH (K)| be the (n − 1)-dimensional Lebesgue measure of pH (K) ⊂ H . The projection invariant π(K) = min |pH (K)| H ∈H

is the volume of the “smallest shadow” of K. Similarly, let |K ∩ (H + x)| of K ∈ be the (n − 1)-dimensional Lebesgue measure of the intersection of K with a translate H + x of H . The intersection invariant Kn

σ (K) = min maxn |K ∩ (H + x)| H ∈H x∈R

of K is the “smallest largest intersection” of K. The following proposition was pointed out to me by Daniel Hug. Proposition 8.2.2. For any convex domain K ∈ K n it holds that σ (K) ≤ π(K) ≤ nn−1 σ (K). Proof. According to a result of John, there exists a unique open ellipsoid E of minimal volume containing K, and this ellipsoid, called John’s ellipsoid, satisfies 1 E ⊂ K ⊂ E, n

(8.2.1)

see [28], [85] and the references therein. Since π and σ are monotone with respect to inclusion and since π(E) = σ (E), we have π(K) ≤ π(E) = nn−1 π n1 E = nn−1 σ n1 E ≤ nn−1 σ (K), 2

as claimed.

Remarks 8.2.3. 1. If K is centrally symmetric, then John’s ellipsoid satisfies K ⊂ E, so that the constant nn−1 in Proposition 8.2.2 can be replaced by n

√1 E n

n−1 2

.

⊂

8.2 A further motivation for Problem ζ

167

2. In dimension 2, σ (K) = π(K) for all K ∈ K 2 , see [28, Theorem 8.3.5]. This is not so in dimension n ≥ 3: The set of K with σ (K) < π(K) is open and dense in 3 K n in the Hausdorff topology, see [53]. Let H1 = &x1 , . . . , xn−1 ' ∈ H . Since the orthogonal group O(n) acts transitively on H, π(K) = min |pH1 (A(K))|, A∈O(n)

σ (K) = min maxn |A(K) ∩ (H1 + x)|. A∈O(n) x∈R

Symplectifying these Euclidean invariants on K 2n , we obtain the symplectic invariants πs and σs on Ks2n defined by πs (K) = σs (K) =

inf |p1 (ϕ(K))| ,

ϕ∈D(n)

inf

sup |ϕ(K) ∩ Ez | ,

ϕ∈D(n) z∈R2n

where p1 : R2n → E1 := R2 (x1 , y1 ) is the projection, where again Ez = E1 × {z} for z ∈ R2n−2 , and where |U | is the area of a domain U ⊂ R2 . The cylindrical capacity of K is zˆ (K) := inf{a | there exists ϕ ∈ D(n) such that ϕ (K) ⊂ Z 2n (a)}. Remarks C.17, Theorem C.18 and the Extension after Restriction Principle Proposition A.1 show that πs (K) = zˆ (K) for all K ∈ Ks2n . (8.2.2) 2n 2n In particular, πs B (a) = a. Notice that σs B (a) ≤ ζ (a) for the function ζ considered in the introduction. If Proposition 8.2.2 would survive to πs (K) ≤ c σs (K) for some c > 0, we would thus find the lower bound ζ (a) ≥ 1c a. However, Proposition 8.2.4. σs (K) = 0 for any K ∈ Ks2n and ζ (a)/a → 0 as a → 0. Proof. The following simple argument due to Polterovich is taken from [59]. (i) Using the monotonicity and conformality of σs and arguing as in the proof of Proposition 8.3.1 below we see that it suffices to prove σs (K) = 0 for B = B 4 (π ). Choose a linear symplectomorphism M of R4 mapping the symplectic plane E1 to a symplectic plane E so close to a Lagrangian plane that the ω0 -area of B ∩ E is less than π. Then the ω0 -area and hence the area of (M )−1 (B) ∩ Ez is less than π for every z ∈ R2 . To be more explicit, fix ∈ (0, 1). The linear symplectomorphism  1  0 0 1  0 0 0   M =   0 1 1 0  0 0 0

168


maps E1 to E := { (x1 , x2 , x2 , 0) | x1 , x2 ∈ R } ⊂ R4 = { (x1 , y1 , x2 , y2 ) }. Since the ω0 -area of B ∩ (E + z) is maximal for z = 0, σs (B) is at most the area of

2 (M )−1 (B) ∩ E1 = (x1 , y1 ) | 1 x1 + ( y1 )2 + y12 ≤ 1 . This set is an ellipsoid of area

√ π < π. 1+ 2 set M B 4 (a) lies

in Z 4 (π ) for a small enough, so (ii) For fixed > 0, the that ζ (a) ≤ a for these a. In particular, ζ (a)/a → 0 as a → 0. More explicitly, 2 shows let 02 = (0.786 . . . )2 be the positive zero of4 x + x − 1. A computation 4 that for √ ≤ 0 the set M B (a) lies in Z (π ) provided that a ≤ π 2 , so that 2 ζ (a) < a/π a. The symplectomorphisms M used in the proof of Proposition 8.2.4, which made our first naive attempt to symplectify Proposition 8.2.2 fail, drastically increased the E1 -shadow of, say, a ball. In a second attempt we thus exclude such maps and introduce constrained symplectic projection and intersection invariants: For > 0 the set D(K; ) of symplectomorphisms of R2n mapping K into Z 2n (ˆz(K) + ) is non-empty, and we set πsz (K) = lim

inf

σsz (K) = lim

inf

→0 ϕ∈D(K;)

|p1 (ϕ(K))| , sup |ϕ(K) ∩ Ez | .

→0 ϕ∈D(K;) z∈R2n

Clearly, πs (K) ≤ πsz (K) ≤ zˆ (K), so that the identity (8.2.2) implies πsz (K) = πs (K) = zˆ (K) for all K ∈ Ks2n . In particular, πsz B 2n (a) = a. Notice that σsz B 2n (π ) = lima π ζ (a). An inequal ity of the form πsz B 2n (π) ≤ c σsz B 2n (π) for some c > 0 would thus be equivalent to the asymptotics lima π ζ (a) ≥ 1c π . Corollary 8.1.2 shows that there are no such inequalities, so that our second attempt of symplectifying Proposition 8.2.2 also fails.

8.3

Proof by symplectic folding

In this section we prove Theorem 8.1.1 by symplectic folding. We fix an extrinsic symplectic capacity c on R2 . We assume without loss of generality that c is normalized. Step 1. Reduction to a 4-dimensional problem Proposition 8.3.1. Assume that ξc Z 4 (a) = 0 for all a ∈ ]0, π [. Then Theorem 8.1.1 holds true.

169

8.3 Proof by symplectic folding

Proof. Let S be a subset of R2n for which there exist a < π and a symplectic embedding ϕ : S → Z 2n (a). We fix > 0. By assumption we find a symplectic embedding ψ : Z 4 (a) → Z 4 (π) such that (8.3.1) sup c ψ Z 4 (a) ∩ Ez < . z∈R2

The composition ψ×id2n−4

ϕ

→ Z 2n (a) = Z 4 (a) × R2n−4 −−−−−−→ Z 4 (π ) × R2n−4 = Z 2n (π ) ρ: S − symplectically embeds S into Z 2n (π). Moreover, the monotonicity of the symplectic capacity c and the estimate (8.3.1) yield sup c ρ(S) ∩ Ez = sup c (ψ × id2n−4 ) (ϕ(S)) ∩ Ez z∈R2n−2

z∈R2n−2

≤ sup c (ψ × id2n−4 ) Z 2n (a) ∩ Ez z∈R2n−2

= sup c ψ Z 4 (a) ∩ Ez z∈R2

< . Since > 0 was arbitrary, we conclude that ξc (S) = 0, as claimed.

2

Step 2. Reformulation of the 4-dimensional problem. We shall use coordinates u, v, x, y on (R4 , du ∧ dv + dx ∧ dy). Fix a < π and > 0. We may assume a > π/2 and < (π − a)/9. Denote by R ⊂ { (u, v) | 0 < u < π, 0 < v < 1 } the convex hull of the image of the map γ drawn in Figure 8.4 below. The domain R is a rectangle with smooth corners. We set A := { (u, v, x, y) | < u, 0 < v < 1 − , 0 < x < 1, 0 < y < a }, Z := R × R2 . Proposition 8.3.2. Assume there exists a symplectic embedding : A → Z such that (8.3.2) sup c ((A) ∩ Ez ) ≤ 2. z∈R2

Then there exists a symplectic embedding : Z 4 (a) → Z 4 (π ) such that sup c Z 4 (a) ∩ Ez ≤ 2. z∈R2

(8.3.3)

170


Proof. We start with Lemma 8.3.3. (i) There exists a symplectomorphism α : R2 → { (u, v) ∈ R2 | < u, 0 < v < 1 − }. (ii) There exists a symplectomorphism σ : D(a) → { (x, y) ∈ R2 | 0 < x < 1, 0 < y < a }. (iii) There exists a symplectomorphism ω of R2 such that ω(R) ⊂ D(π ). Proof. (i) follows from the proof of Lemma 3.1.5 or from Proposition B.6 (ii). Here we give an explicit construction. Choose orientation preserving diffeomorphisms g : R → ]0, ∞[ and h : R → ]0, 1 − [, and denote by g and h their derivatives. Then the maps v α1 : R2 → ]0, ∞[ × R, (u, v) → g(u), , g (u) u + , h(v) α2 : ]0, ∞[ × R → ], ∞[ × ]0, 1 − [ , (u, v) → h (v) are symplectomorphisms. The map α := α2 α1 is therefore as desired. (ii) follows from Lemma 3.1.5 or from Proposition B.6 (i). (iii) Choose r ∈ R so large that R ⊂ D(πr 2 ), and choose a diffeomorphism χ of D(2πr 2 ) which is a translation near the origin, maps the boundary of D(area(R)) to the boundary of R, and is the identity near the boundary of D(2π r 2 ). By Lemma 3.1.5 and its proof, we may assume that χ is a symplectomorphism. Extend χ to the symplectomorphism of R2 which is the identity outside D(2π r 2 ), and let ω be the inverse of this extension. Then ω(R) = D(area(R)) ⊂ D(π ), as desired. 2 Let : A → Z be a symplectic embedding as assumed in the proposition, let α, σ and ω be symplectomorphisms as guaranteed by Lemma 8.3.3, and denote the linear symplectomorphism (u, v, x, y) → (x, y, u, v) of R4 by τ . Then the composition (α×σ )τ

ω×id

: Z 4 (a) −−−−−→ A − → Z −−−→ Z 4 (π ) is a symplectic embedding. Moreover, the identity (α × σ ) τ Z 4 (a) = A, the monotonicity of the symplectic capacity c on R2 and the assumed estimate (8.3.2) imply that sup c Z 4 (a) ∩ Ez = sup c ((ω × id) ((A)) ∩ Ez ) z∈R2

z∈R2

= sup c ((A) ∩ Ez ) z∈R2

≤ 2.

171

8.3 Proof by symplectic folding

2

This completes the proof of Proposition 8.3.2.

Step 3. Construction of the embedding . We are going to construct a symplectic embedding : A → Z satisfying the estimate (8.3.2) by a variant of multiple symplectic folding. Instead of folding alternatingly on the right and on the left, we will always fold on the right. If we neglect all quantities which can be chosen arbitrarily small, the image (A) ⊂ Z will look as in Figure 8.1. y

2a

a

u a + 2 π

2

Figure 8.1. The embedding : A → Z for a = 3π 5 .

The symplectic embedding will be the composition of the three maps β × id, ϕ and γ × id. Here, the smooth area preserving embedding β : { (u, v) | < u, 0 < v < 1 − } → R2 . The is similar to the map constructed in Step 1 of Section 4.2. Indeed, set δ = 2a map β restricts to the identity on { < u ≤ 2}, and it maps {2 < u ≤ 3} to the black region in {2 < u ≤ a + 9} drawn in Figure 8.2. Moreover,

β(u, v) = β(u − i2, v) + (i(a + 8), 0) for u ∈ ] + i2, + (i + 1)2] and i = 1, 2, 3, . . . , see Figure 8.2. The “lifting” map ϕ is similar to the map constructed in Step 3 of Section 4.2. Indeed, choose a cut off function f0 : R → [0, 1 − − δ] with support [3, a + 8] such that / f0 (s) ds > a. aˆ := R

172

8 Proof of Theorem 4 v

1−

u 2 3 β v 1 1−

δ

u 2

a + 9 π

Figure 8.2. The left part of the image of β.

+ Set fi (s) = f0 (s − i(a + 8)), i = 1, 2, 3, . . . , and f (s) = i≥0 fi (s). The 4 4 symplectomorphism ϕ : R → R is defined by / u f (s) ds . (8.3.4) ϕ(u, v, x, y) = u, v + f (u)x, x, y + 0

The image ϕ ((β × id)(A)) ⊂ R4 is illustrated in Figure 8.3. Denote the projection (u, v, x, y) → (u, v) by p. The left part of the set p (ϕ ((β × id)(A))) is equal to the upper domain in Figure 8.4. We finally spiral the set ϕ ((β × id)(A)) into Z. The smooth area preserving local embedding γ : p (ϕ ((β × id)(A))) → R, which is explained by Figure 8.4, restricts to the identity on { < u ≤ a + 8 }, and it maps the black region in { a + 8 < u ≤ a + 9 } to the black region in its image. Moreover, γ (u, v) = γ (u − i(a + 8), v) for u ∈ ] + i(a + 8), + (i + 1)(a + 8)] and i = 1, 2, 3, . . . , see Figure 8.4. The map γ is constructed in the same way as the map in Step 4 of Section 3.2. Since γ is an embedding on each part {(u, v) ∈ p (ϕ ((β × id)(A))) | + i(a + 8) a, the map γ × id : ϕ ((β × id)(A)) → R4 is a symplectic embedding, cf. Figure 8.3 and Figure 8.5. We conclude that the composition := (γ × id) ϕ (β × id) is a symplectic embedding of A into Z. Step 4. Verification of the estimate (8.3.2) in Proposition 8.3.2. We have to show that for any point (x0 , y0 ) ∈ R2 the estimate (8.3.5) c (A) ∩ E(x0 ,y0 ) ≤ 2 holds true. To this end, we may assume that x0 ∈ ]0, 1[ since otherwise the intersection (A) ∩ E(x0 ,y0 ) is empty. Moreover, by construction of the embedding we have (u + 2, v, x, y) = (u, v, x, y) + (0, 0, 0, a) ˆ for all (u, v, x, y) ∈ A. It follows that for each y0 ∈ R there exists i ∈ Z such that ˆ 2a] ˆ and y0 + i aˆ ∈ ]a, p (A) ∩ E(x0 ,y0 ) ⊂ p (A) ∩ E(x0 ,y0 +i a) ˆ , ˆ 2a]. ˆ In order to cf. Figure 8.5. We may therefore assume that (x0 , y0 ) ∈ ]0, 1[ × ]a, verify the estimate (8.3.5) we distinguish two cases.

174


1 1−

δ u 2 3

a + 8

π

a + 8

π

γ v

1 1−

δ u

Figure 8.4. The folding map γ .

Case A. Assume first y0 ∈ ]a, ˆ aˆ + a[. Then E(x0 ,y0 ) intersects the “floor” F2 := { (u, v, x, y) | 3 < u ≤ 4 } = { (u, v, x, y) | < u ≤ 2, 0 < v < 1 − , 0 < x < 1, aˆ < y < 2aˆ } and the two “stairs” Si := { (u, v, x, y) | 2i < u ≤ (2i + 1) },

i = 1, 2,

only, cf. Figure 8.5. The set F := F2 ∩ E(x0 ,y0 ) has area (1 − ). The set S1 ∩ E(x0 ,y0 ) consists of the black thickened “arc” in Figure 8.6 and of the branch B1 = 1 A drawn a B1 (x0 , y0 ), which for (x0 , y0 ) = 2 , aˆ + 2 looks like in Figure 8.6. Indeed, since the preimage of S1 under is contained in {0 < y < a}, we read off from definition

175

8.3 Proof by symplectic folding y

S2

2aˆ aˆ + a y0

F2

S1

aˆ a

u a + 9 π

2

Figure 8.5. The floor F2 and the stairs S1 and S2 .

(8.3.4) that B1 (x0 , y0 ) = { (u, v + f (u)x0 , x0 , y0 ) | u1 < u < a + 8, 0 < v < δ } where u1 = u1 (y0 ) is defined through the equation / u1 f (s) ds = y0 − a.

(8.3.6)

(8.3.7)

3

Similarly, the branch B2 = B2 (x0 , y0 ) = S2 ∩E(x0 ,y0 ) looks for (x0 , y0 ) = 21 , aˆ + a2 as in Figure 8.6. Indeed, since the preimage of S2 under is contained in the set {aˆ < y < aˆ + a}, we read off from (8.3.4) that B2 (x0 , y0 ) ∩ {u > 3} = { (u, v + f (u)x0 , x0 , y0 ) | 3 < u < u2 , 0 < v < δ } (8.3.8) where u2 = u2 (y0 ) is defined through the equation / u2 f (s) ds = y0 − a. ˆ (8.3.9) 3

Subtracting (8.3.9) from (8.3.7) and using that f (s) < 1 for all s ∈ R we obtain / u1 0 < aˆ − a = f (s) ds < u1 − u2 . (8.3.10) u2

176


1 1− A F B2

B1

δ u

2

3

u2 u1

a + 8

π

Figure 8.6. The set (A) ∩ E(x0 ,y0 ) for (x0 , y0 ) = 21 , aˆ + a2 .

Let µ be the area of the set (A) ∩ E(x0 ,y0 ) . In order to estimate the capacity of (A) ∩ E(x0 ,y0 ) we next embed (A) ∩ E(x0 ,y0 ) into a set whose capacity is known. Lemma 8.3.4. There exists a symplectomorphism φ of R2 such that φ p (A) ∩ E(x0 ,y0 ) ⊂ D µ + 2 . Proof. The set p (A) ∩ E(x0 ,y0 ) ⊂ R2 is diffeomorphic to an open disc, and in view of the estimate (8.3.10) its boundary is a piecewise smooth embedded closed curve in R2 , cf. Figure 8.6. We therefore find a simply connected domain U ⊂ R2 such that closed curve in R2 , and such the boundary of U is a smoothly embedded 2 that p (A) ∩ E(x0 ,y0 ) ⊂ U and area(U ) = µ + . Applying the argument given in the proof of Lemma 8.3.3 (iii) to U , we find a symplectomorphism φ of R2 such 2 that φ(U ) = D(µ + ). Then φ p (A) ∩ E(x0 ,y0 ) ⊂ φ(U ) = D(µ + 2 ), as desired. 2 In view of the monotonicity and the conformality of the normalized symplectic capacity c on R2 , Lemma 8.3.4 implies c (A) ∩ E(x0 ,y0 ) = c φ p (A) ∩ E(x0 ,y0 ) ≤ c D(µ + 2 ) (8.3.11) 2 = µ+ . It remains to estimate the number µ = area (A) ∩ E(x0 ,y0 ) . We abbreviate Q = { 2 < u ≤ 3, 0 < v < 1 − }. Using the definitions of β and γ and the identities

8.4 Proof by symplectic lifting

177

(8.3.6) and (8.3.8) we can estimate area(B2 ∪ B1 ∪ A) = area p(B2 ) ∪ p(B1 ) ∪ γ −1 (p(A)) < area { (u, v + f (u)x0 ) | (u, v) ∈ β(Q) } = area { (u, v) | (u, v) ∈ β(Q) } = area β(Q) = area(Q) = (1 − ). Therefore, µ = area (A) ∩ E(x0 ,y0 ) = area (F ) + area (B2 ∪ B1 ∪ A) < (1 − ) + (1 − ). In view of (8.3.11) we conclude that c (A) ∩ E(x0 ,y0 ) < 2. Case B. Assume now y0 ∈ [aˆ + a, 2a]. ˆ Then E(x0 ,y0 ) intersects the stairs S2 only, and (A) ∩ E(x0 ,y0 ) = { (u, v + f (u)x0 , x0 , y0 ) | ul < u < ur , 0 < v < δ } where ul = ul (y0 ) and ur = ur (y0 ) lie in [3, a + 8] and are defined through the equations / ur / ul f (s) ds = y0 − (aˆ + a) and f (s) ds = y0 − a. ˆ 3

3

The set p (A) ∩ E(x0 ,y0 ) ⊂ R2 is diffeomorphic to an open disc, and its boundary is piecewise smooth. Moreover, area (A) ∩ E(x0 ,y0 ) = (ur − ul ) δ < (a + 5) < . 2a Arguing as above, we find c (A) ∩ E(x0 ,y0 ) < 2. This finishes the verification of the estimate (8.3.5) and hence of the estimate (8.3.2) in Proposition 8.3.2. Step 5. End of the proof of Theorem 8.1.1. We have constructed a symplectic embedding : A → Z satisfying the estimate (8.3.2). In view of Proposition 8.3.2 there exists a symplectic embedding : Z 4 (a) → Z 4 (π ) satisfying the estimate (8.3.3). Since a < π and > 0 were arbitrary, we conclude that ξc (Z 4 (a)) = 0 for all a ∈ ]0, π [. Proposition 8.3.1 now implies that Theorem 8.1.1 holds true. 2

8.4

Proof by symplectic lifting

In this section we give another proof of Theorem 8.1.1 which relies on a symplectic lifting construction. In order to explain the idea of the construction, we assume that S = B 4 (a) ⊂ Z 4 (π). We slice B 4 (a) by planes u = const and then lift a large part

178


of the interior of the i’th slice by i into the y-direction, cf. Figures 8.8 and 8.9 below. Symplectic lifting is even more elementary than symplectic folding and in the problem at hand leads to stronger results. We consider again a (normalized) extrinsic symplectic capacity c on R2 . If c satisfies the stronger monotonicity axiom A1. Monotonicity: c(S) ≤ c(T ) if S symplectically embeds into T , then c is called a (normalized) intrinsic symplectic capacity on R2 . Examples of normalized intrinsic symplectic capacities on R2 are the outer Lebesgue measure µ, the Gromov width and the Hofer–Zehnder capacity, see Appendix C.1. For examples of extrinsic symplectic capacities on R2 which are not intrinsic we refer to Proposition C.7. Definition 8.4.1. A subset S of Z 2n (π) is partially bounded if at least one of the coordinate functions x2 , . . . , xn , y2 , . . . , yn is bounded on S. The following theorem was proved in [74]. Theorem 8.4.2. Consider a subset S of Z 2n (π) and an extrinsic symplectic capacity c on R2 . (i) ξc (S) = 0 if c is intrinsic. (ii) ξc (S) = 0 if S ⊂ Z 2n (a) for some a < π. If S is partially bounded, then (i) and (ii) hold with ξc (S) replaced by ζc (S). Notice that Theorem 8.4.2 (ii) is Theorem 8.1.1, and that Theorem 4 in Section 1.3.3 is a special case of the last assertion in Theorem 8.4.2. We do not know ζc (Z 2n (a)) for any symplectic capacity c on R2 and any a ∈ ]0, π [. We also do not know anything about ζc (Z 2n (π )) if c(S 1 ) = 0. Proof of Theorem 8.4.2. Consider a bounded subset T of R2 . The simply connected hull Tˆ of T is the union of its closure T and the bounded components of R2 \ T . We denote by µ the Lebesgue measure on R2 , and we abbreviate µ(T ˆ ) = µ(Tˆ ). Notice that the outer Lebesgue measure µ is a normalized intrinsic symplectic capacity on R2 and that µˆ is a normalized extrinsic symplectic capacity on R2 which is not intrinsic. Since T is bounded, c(T ) ≤ µ(T ) for every normalized intrinsic symplectic capacity ˆ ) for every normalized extrinsic symplectic capacity c on c on R2 and c(T ) ≤ µ(T R2 , see Theorem C.8. We can thus assume that c = µ in (i) and c = µˆ in (ii). As in the previous proof of Theorem 8.1.1, the main ingredient in the proof of Theorem 8.4.2 is a special embedding result in dimension 4. We shall again use coordinates z = (u, v, x, y) on (R4 , du ∧ dv + dx ∧ dy), denote by E(x,y) ⊂ R4 the affine plane E(x,y) = R2 × {(x, y)}, and for S ⊂ R4 abbreviate µ S ∩ E(x,y) = µ p S ∩ E(x,y) , µˆ S ∩ E(x,y) = µˆ p S ∩ E(x,y) .


179

Fix an integer k ≥ 2. We set =

π , k

δ=

, 4k

and we define closed rectangles P , P and Q in R2 (u, v) by P = [0, π ] × [0, 1], P = [δ, π − δ] × [δ, 1 − δ], Q = [3δ, π − 3δ] × [3δ, 1 − 3δ]. The support of a map ϕ : R4 → R4 is defined by

supp ϕ = z ∈ R4 | ϕ(z) = z . Proposition 8.4.3. There exists a symplectomorphism ϕ of R4 with supp ϕ ⊂ P ×R2 and such that for each (x, y) ∈ R2 , µ ϕ P × R × [0, 1] ∩ E(x,y) ≤ 2, (8.4.1) (8.4.2) µˆ ϕ (Q × R × [0, 1]) ∩ E(x,y) ≤ 2. Proof. We define closed rectangles R, R and R in R2 (u, v) by R = [0, ] × [0, 1], R = [δ, − δ] × [δ, 1 − δ], R = [2δ, − 2δ] × [2δ, 1 − 2δ], and we define closed rectangular annuli A and A in R2 (u, v) by A = R \ R,

A = R \ R .

Then R = A ∪ A ∪ R , cf. Figure 8.7. We choose smooth cut off functions f1 , f2 : R → [0, 1] such that 0, t ∈ / [δ, − δ], f1 (t) = 1, t ∈ [2δ, − 2δ], 0, t ∈ / [δ, 1 − δ], f2 (t) = 1, t ∈ [2δ, 1 − 2δ],

and we define the smooth function H : R4 → R by H (u, v, x, y) = −f1 (u)f2 (v)(1 + )x.

180


v 1

δ

u δ

Figure 8.7. The decomposition R = A ∪ A ∪ R .

The Hamiltonian vector field XH of H defined by ω0 (XH (z), ·) = dH (z), is given by

   XH (u, v, x, y) = (1 + )  

z ∈ R2n ,

−f1 (u)f2 (v)x f1 (u)f2 (v)x 0 f1 (u)f2 (v)

(8.4.3)    . 

(8.4.4)

The time-1-map φH is a lifting map with the following properties. (P1) supp φH ⊂ R × R2 , (P2) φH fixes A × R2 , (P3) φH embeds A × R2 into A × R2 , (P4) φH translates R × R2 by (1 + )1y , where we abbreviated 1y = (0, 0, 0, 1). For each subset T of R2 (u, v) and each i ∈ {1, . . . , k} we define the translate Ti of T by Ti = {(u + (i − 1), v) | (u, v) ∈ T } . With this notation we have P =

k 0 i=1

cf. Figure 8.8.

Ri =

k 0 i=1

Ai ∪ Ai ∪ Ri ,

181


v

1

δ

u δ

π

Figure 8.8. The decomposition P =

-k

i=1 Ri =

-k

i=1 Ai ∪ Ai ∪ Ri for k = 4.

Abbreviate Hi (u, v, x, y) = iH (u − (i − 1), v, x, y). We define the smooth 3 : R4 → R by function H k . 3 H (z) = Hi (z) i=1

and we define the symplectomorphism ϕ of R4 by ϕ = φH3. In view of the identity (8.4.4) we see that ϕ is of the form ϕ(u, v, x, y) = (u , v , x, y ),

(8.4.5)

and in view of the Properties (P1)–(P4) we find 3 supp ϕ ⊂ P × R2 , P1 3 ϕ fixes k Ai × R2 , P2 i=1 3 ϕ embeds A × R2 into A × R2 , i = 1, . . . , k, P3 i i 3 ϕ translates R × R2 by i(1 + )1y , i = 1, . . . , k. P4 i Verification of the estimates (8.4.1) and (8.4.2). Fix (x, y) ∈ R2 . We abbreviate P = p ϕ P × R × [0, 1] ∩ E(x,y) , Q = p ϕ Q × R × [0, 1] ∩ E(x,y) . Lemma 8.4.4. We have µ(P ) ≤ 2. Proof. Using the definitions =

π k

and δ =

4k

we estimate

µ Ai ∪ Ai = − ( − 4δ)(1 − 4δ) ≤ , k

i = 1, . . . , k.

(8.4.6)

182

8 Proof of Theorem 4 y 5 + 4

4 + 3

3 + 2

2+

1+ 1

u 2δ

π

Figure 8.9. The intersection of ϕ (P × R × [0, 1]) with a plane {(u, v, x, y) | v, x constant} for v ∈ [2δ, 1 − 2δ].

3 – P4 3 we have Case A: y ∈ [i ∗ (1 + ), i ∗ (1 + ) + 1]. According to Properties P2 ∗ P ∩ Ri = ∅ if i = i , and so P ⊂ Ri ∗ ∪

k 0

Ai ∪ Ai .

i=1

Together with the estimate (8.4.6) we therefore find µ(P ) ≤ + k

= 2. k

(8.4.7)

3 – P4 3 we Case B: y ∈ / ki=1 [i(1 + ), i(1 + ) + 1]. According to Properties P2 have P ∩ Ri = ∅ for all i, and so P ⊂

k 0

Ai ∪ Ai .

i=1

Therefore,

µ(P ) ≤ .

(8.4.8)


183

The estimates (8.4.7) and (8.4.8) yield that µ(P ) ≤ 2.

2

Lemma 8.4.5. We have µ(Q) ˆ ≤ 2. Proof. In view of the special form (8.4.5) of the map ϕ we have Q = p ϕ (Q × {x} × [0, 1]) ∩ E(x,y) . For i = 1, . . . , k we abbreviate the intersections Ai = Q ∩ Ai ,

Ai = Q ∩ Ai ,

Ri = Q ∩ Ri .

(8.4.9)

Each of the sets Ai and Ai consists of one closed rectangle if i ∈ {1, k} and of two closed rectangles if i ∈ {2, . . . , k − 1}, cf. Figure 8.10. The crucial observation in the proof is that for each i the simply connected hull of the part p ϕ Ai × {x} × [0, 1] ∩ E(x,y) 3 the closed of Q is a simply connected subset P3 of Ai . Indeed, according to property and simply connected set ϕ Ai × {x} × [0, 1] is contained in Ai × {x} × R, and so the simply connected hull of ϕ Ai × {x} × [0, 1] ∩ E(x,y) is a simply connected subset of Ai × {(x, y)}. v

1

3δ u 3δ

π

Figure 8.10. The subsets Ai , Ai and Ri of Q, i = 1, . . . , 4.

ˆ the simply connected hull of Q. Denote by Q 3 – P4 3 we have Q ∩ Ai = Ai and Case A: y ∈ [0, 1]. According to Properties P2 Q ∩ Ri = ∅ for all i. In view of the above observation we conclude that ˆ ⊂ Q

k 0 i=1

Ai ∪ Ai .

184


Together with the estimate (8.4.6) we therefore find ˆ ≤k µ(Q)

= . k

(8.4.10)

3 – P4 3 we have Case B: y ∈ [i ∗ (1 + ), i ∗ (1 + ) + 1]. According to Properties P2 ∗ Q ∩ Ai = ∅ for all i and Q ∩ Ri = ∅ if i = i . In view of the above observation we conclude that k 0 ˆ ⊂ Ri ∗ ∪ Ai . Q i=1

Therefore, ˆ ≤ + = 2. µ(Q)

(8.4.11) 3 – P4 3 Case C: y ∈ / [0, 1] ∪ i=1 [i(1 + ), i(1 + ) + 1]. According to Properties P2 we have Q ∩ Ai = Q ∩ Ri = ∅ for all i. In view of the above observation we conclude that k 0 ˆ ⊂ Ai . Q -k

i=1

Therefore, ˆ ≤ . µ(Q)

(8.4.12)

ˆ ≤ 2. This The estimates (8.4.10), (8.4.11) and (8.4.12) yield that µ(Q) ˆ = µ(Q) completes the proof of Lemma 8.4.5. 2 In view of Lemmata 8.4.4 and 8.4.5 the estimates (8.4.1) and (8.4.2) hold true. The proof of Proposition 8.4.3 is thus complete. 2 End of the proof of Theorem 8.4.2 (i). Fix k ≥ 2 and set = πk . Arguing as in the proof of Lemma 8.3.3 (iii) we find a symplectomorphism α of R2 (u, v) such that 2 P ⊂ α B (π ) . Choose an orientation preserving diffeomorphism f : R → ]0, 1[ and denote by f its derivative. Then the map x 2 β : R → R × ]0, 1[, (x, y) → , f (y) (8.4.13) f (y) is a symplectomorphism. We define the symplectic embedding : R2n → R2n by = (α −1 × id) ϕ (α × β) × id2n−4 where ϕ is the map guaranteed by Proposition 8.4.3. Since supp ϕ ⊂ P × R2 ⊂ α B 2 (π ) × R2

(8.4.14)


185

we have Z 2n (π) ⊂ Z 2n (π). Recall that Dz = B 2 (π ) × {z}, z ∈ R2n−2 . For each subset S of Z 2n (π) and each point z = (x, y, z3 , . . . , zn ) ∈ R2n−2 we have (S) ∩ Dz ⊂ Z 2n (π) ∩ Dz = (α −1 × id) ϕ (α × β) Z 4 (π ) ∩ D(x,y) ⊂ (α −1 × id) ϕ α B 2 (π ) × R × [0, 1] ∩ E(x,y) . Using this, the facts that µ is monotone and α −1 preserves µ, the inclusions (8.4.14) and the estimates (8.4.1) and (8.4.6) we can estimate µ (S) ∩ Dz ≤ µ ϕ α B 2 (π) × R × [0, 1] ∩ E(x,y) = µ ϕ P × R × [0, 1] ∩ E(x,y) + µ α B 2 (π ) \ P ≤ 3. Since this holds true for all z ∈ R2n−2 and since k ≥ 2 was arbitrary, we conclude that ξµ (S) = 0. Assume now that S ⊂ Z 2n (π) is partially bounded. There exists i ∈ {2, . . . , n} and b > 0 such that xi (S) ⊂ ] − b, b[ or yi (S) ⊂ ] − b, b[. We can assume without loss of generality that i = 2. If x(S) ⊂ ] − b, b[, we define the symplectomorphism σ of R2 (x, y) by σ (x, y) = (−y, x), and we let σ be the identity mapping otherwise. Define the symplectomorphism τ of R2 (x, y) by 1 1 . τ (x, y) = 2bx, y + 2b 2 The composition id2 × (τ σ ) × id2n−4 maps S into B 2 (π ) × R×]0, 1[×R2n−4 . Replacing the symplectomorphism β in (8.4.13) by the symplectomorphism τ σ of R2 and proceeding as above, we find that ζµ (S) = 0. End of the proof of Theorem 8.4.2 (ii). Choose a < π so large that S ⊂ Z 2n (a). We choose k ≥ 2 so large that a < µ(Q). Arguing again as in the proof of Lemma 8.3.3 (iii) we then find a symplectomorphism α of R2 (u, v) such that α B 2 (a) ⊂ Q and α B 2 (π ) ⊃ P , cf. Figure 8.11. Choose a symplectomorphism β : R2 → R × ]0, 1[ as above and define the symplectic embedding : R2n → R2n by = (α −1 × id) ϕ (α × β) × id2n−4 . Since supp ϕ ⊂ P × R2 ⊂ α(B 2 (π)) × R2 we have (Z 2n (a)) ⊂ Z 2n (π ). For each z = (x, y, z3 , . . . , zn ) ∈ R2n−2 we have (S) ∩ Dz ⊂ Z 2n (a) ∩ Dz = (α −1 × id) ϕ (α × β) Z 4 (a) ∩ D(x,y) ⊂ (α −1 × id) ϕ Q × R × [0, 1] ∩ E(x,y) .

186


v Q

1

P

1 α 1

u

π

u

Figure 8.11. The symplectomorphism α.

Using this, the facts that µˆ is monotone and α −1 preserves µˆ and the estimate (8.4.2) we can estimate µˆ ((S) ∩ Dz ) ≤ µˆ ϕ (Q × R × [0, 1]) ∩ E(x,y) ≤ 2. Since this holds true for all z ∈ R2n−2 and since we can choose k as large as we like, we conclude that ξµˆ (S) = 0. If S ⊂ Z 2n (a) is partially bounded, we replace β by a symplectomorphism τ σ 2 as above and find that ζµˆ (S) = 0. The proof of Theorem 8.4.2 is complete.

Addendum: Symplectic lifting via Hamiltonian deformations Hofer-close to the identity. The symplectic embeddings in the definitions of ξc (S) and ζc (S) were not further specified. Following a suggestion of Polterovich, we next ask whether the vanishing phenomenon described by Theorem 8.4.2 persists if we restrict ourselves to symplectic embeddings which are close to the identity mapping in a symplectically relevant sense. We denote by H (2n) the set of smooth and bounded functions H : R2n → R whose support is contained in Z 2n (π ) and whose Hamiltonian vector field XH defined by (8.4.3) generates a flow on R2n . The time-1-map of this flow is then again denoted by φH . Moreover, we abbreviate (H ( = sup H (z) − inf H (z). z∈R2n

z∈R2n

(8.4.15)

For each subset S of Z 2n (π) and each extrinsic symplectic capacity c on R2 we define 6 7 ηc (S) = inf sup c (φH (S) ∩ Ez ) + (H ( H

z

where H varies over H(2n). Clearly, ξc (S) ≤ ζc (S) ≤ ηc (S). The following result improves the last statement in Theorem 8.4.2.


187

Theorem 8.4.6. Consider a partially bounded subset S of Z 2n (π ) and an extrinsic symplectic capacity c on R2 . (i) ηc (S) = 0 if c is intrinsic. (ii) ηc (S) = 0 if S ⊂ Z 2n (a) for some a < π. Of course, ηc Z 2n (π) = π for every normalized extrinsic symplectic capacity c on R2 . Question 8.4.7. Is it true that ηµ Z 2n (a) = 0 for all a ∈ ]0, π [ ? Theorem 8.4.6 can be proved by refining the lifting construction used in the proof of Theorem 8.4.2. The proof is much trickier, however, since one needs to control the behaviour of the lifting maps on those parts of their compact supports which are not translated in the y-direction. We refer to [74] for the proof. In order to see Theorem 8.4.6 in its right perspective we consider the set H Z 2n (π ) of smooth functions H = H (t, z) : [0, 1] × R2n → R whose support is contained in t 1 [0, 1]×Z 2n (π ) and whose Hamiltonian flow φH exists for all t ∈ [0, 1], set φH = φH , 2n and for H ∈ H Z (π) abbreviate / (H ( =

1

0

sup H (t, z) − inf H (t, z) dt. z∈R2n

z∈R2n

Notice that H (2n) ⊂ H Z 2n (π) and that this definition extends definition (8.4.15). Abbreviating

Ham Z 2n (π) = φH | H ∈ H Z 2n (π ) we define the energy E(φ) of φ ∈ Ham Z 2n (π ) by

E(φ) = inf (H ( | φ = φH for some H ∈ H Z 2n (π ) . In the framework of Hofer geometry the energy of a Hamiltonian diffeomorphism is its distance from the identity mapping, see [39], [50], [70]. Notice that 7 6 ηc (S) ≥ inf sup c(φ(S) ∩ Ez ) + E(φ) φ

z

where φ varies over Ham Z 2n (π) . Theorem 8.4.6 therefore says that the vanishing phenomenon described by Theorem 8.4.2 persists if we restrict ourselves to Hamiltonian diffeomorphism of Z 2n (π) whose Hofer distance to the identity mapping is arbitrarily small.

Chapter 9

Packing symplectic manifolds by hand

This last chapter is devoted to explicit symplectic packings of some symplectic manifolds by equal balls. After recalling the symplectic packing problem, we give several motivations for it, and then collect the known packing numbers of interest to us. In Section 9.3 we describe a simple construction of explicit maximal packings of the 4-ball and CP2 and of ruled symplectic 4-manifolds by few equal balls, and in Section 9.4 we briefly look at packings in higher dimensions. We consider again a connected symplectic manifold (M, ω) of , 2n-dimensional 1 n , and as before we abbreviate the Lebesgue finite volume Vol(M, ω) = n! ω M measure of an open subset U of R2n by |U |. In Chapters 2 to 6 we studied the numbers |λE(π, . . . , π, a)| paE (M, ω) = sup Vol(M, ω) λ where the supremum is taken over all those λ for which the ellipsoid λE 2n (π, . . . , π, a) symplectically embeds into (M, ω), and we proved in Chapter 6 that lim paE (M, ω) = 1.

a→∞

(9.0.1)

Notice that the invariant pπE (M, ω) measures the maximal ball which symplectically embeds into (M, ω). Instead of “stretching the ball to ellipsoids”, we shall now increase the number of balls and study for each k ∈ N the k’th symplectic packing number k B 2n (a) pk (M, ω) = sup a Vol(M, ω) 1 where the supremum is taken over all those a for which the disjoint union ki=1 B 2n (a) of k equal balls symplectically embeds into (M, ω). The problem of understanding the numbers pk (M, ω) ∈ ]0, 1] is called the symplectic packing problem, a problem much studied in recent years. If pk (M, ω) < 1, one says that there is a packing obstruction, and if pk (M, ω) = 1, one says that (M, ω) admits a full packing by k balls. The first examples of packing obstructions were found by Gromov, [31], and many further packing obstructions and also some exact values of pk were obtained by McDuff and Polterovich in [61]. Finally, Biran showed in [7], [8] that P (M, ω) := inf {k0 ∈ N | pk (M, ω) = 1

for all k ≥ k0 } < ∞

(9.0.2)

189

9.1 Motivations for the symplectic packing problem

for an interesting class of closed symplectic 4-manifolds containing sphere bundles over a surface and for all closed symplectic 4-manifolds with [ω] ∈ H 2 (M; Q). According to Lemma 6.2.2 and [61, Remark 1.5.G], lim pk (M, ω) = 1

k→∞

(9.0.3)

for every connected symplectic manifold (M, ω) of finite volume. By (9.0.1), the asymptotics of paE and pk are thus the same. The analogue of Biran’s result (9.0.2) for paE is not known for any connected symplectic manifold of finite volume. Question 9.0.8. For which connected symplectic manifolds (M, ω) of finite volume does there exist a0 such that paE (M, ω) = 1 for all a ≥ a0 ? Besides sporadic results on the first packing number p1 and on packing numbers for ellipsoids in [81], [54], all known computations of packing numbers are contained in [61],[7], [8]. We refer to Biran’s excellent survey [10] for the methods used, and only mention that in [61], [7], [8] the problem of symplectically embedding k equal balls into (M, ω) is first reformulated as the problem of deforming a symplectic form on the k-fold blow-up of (M, ω) along a certain family of cohomology classes, and that this problem is then solved using tools from classical algebraic geometry, Seiberg–Witten–Taubes theory, and Donaldson’s symplectic submanifold theorem, respectively. As a consequence, the symplectic packings found are not explicit. For some of the symplectic manifolds considered in [61], [7], [8] and some values of k, explicit maximal symplectic packings were constructed by Karshon [41], Traynor [81], Kruglikov [45], and Maley, Mastrangeli and Traynor [54]. In this final chapter we describe a very simple and explicit construction realizing the packing numbers pk (M, ω) for those symplectic 4-manifolds (M, ω) and numbers k considered in [41], [81], [45], [54] as well as for some other closed symplectic 4-manifolds and small values of k. In the range of k for which these constructions fail to give maximal packings, they give a feeling that the balls in the packings from [61], [7], [8] must be “wild”. We shall also construct an explicit full packing of B 2n (a) by l n equal balls for each l ∈ N in a most simple way. As in the previous chapters, balls and ellipsoids will always be endowed with the standard symplectic form ω0 . Since the packing numbers pk B 2n (a), ω0 do not depend on a, we shall usually pack the unit ball B 2n := B 2n (π ), ω0 .

9.1

Motivations for the symplectic packing problem

1. Higher Gromov widths. The Gromov width wG (M, ω) := sup{a | B 2n (a) symplectically embeds into (M, ω)} of a symplectic manifold (M, ω) measures the size of a largest round symplectic chart of (M, ω). It is the smallest normalized symplectic capacity as defined in [39, p. 51],

190

9 Packing symplectic manifolds by hand

and we refer to [7], [8], [12], [27], [31], [40], [42], [48], [49], [56], [60], [61], [64], [77] and to Theorem C.8 (i) for results on the Gromov width and to Section 9.3 below for explicit symplectic embeddings realizing wG (M, ω) or estimating it from below. If (M, ω) has finite volume, the first packing number p1 (M, ω) is equivalent to the Gromov width, n 1 wG (M, ω) . p1 (M, ω) Vol(M, ω) = n! Similarly, the higher packing numbers pk (M, ω), k ≥ 2, are equivalent to the higher Gromov widths 6 2k 7 k (M, ω) := sup a B 2n (a) symplectically embeds into (M, ω) , wG i=1

which form a distinguished sequence of embedding capacities as considered in [16]. 2. “Superrecurrence for symplectomorphisms” via packing obstructions? In view of the Poincaré recurrence theorem, for which we refer to [35], volume preserving mappings have strong recurrence properties. The solution of the Arnold conjecture for the torus by Conley and Zehnder, which we stated in Section 1.2.1, demonstrated that Hamiltonian diffeomorphisms have yet stronger recurrence properties. As was pointed out to me by Leonid Polterovich, the original motivation for Gromov to study the packing numbers pk was his search for recurrence properties of arbitrary symplectomorphisms which are stronger than those of volume preserving mappings. We explain the relation between “superrecurrence for symplectomorphisms” and symplectic packing obstructions by means of an example. Let B and B be the open balls in R2n centred at the origin of volumes 2n − 21 and 1, respectively. For every compactly supported volume preserving diffeomorphism ϕ of B set

R(ϕ) = min m ∈ N | ϕ m (B ) ∩ B = ∅ . Of course, R(ϕ) ≤ 2n − 2, and using Moser’s deformation argument, for which we refer to [39, p. 11], it is easy to construct a ϕ with R(ϕ) = 2n − 2. The packing 1 obstruction p2 (B) = 2n−1 proved by Gromov in [31] shows, however, that R(ϕ) = 1 if ϕ is symplectic. This motivation for symplectic packings lost some of its appeal by the work of Mc Duff–Polterovich and Biran. Indeed, in dynamics one usually asks for recurrence into small neighbourhoods of a point. To establish recurrence of small balls we would need packing obstructions for large k. In view of (9.0.3), these obstructions asymptotically always vanish, and in view of (9.0.2), they completely vanish for many symplectic 4-manifolds.


191

3. Between Euclidean and volume preserving Volume preserving packings. Consider a connected n-dimensional , manifold M endowed with a volume form such that the volume Vol(M, ) = M is finite, and denote the Lebesgue measure of an open subset U of Rn by |U |. We write B n (A) for √ the open ball of radius A/π in Rn . For k ∈ N we set k |B n (A)| vk (M, ) = sup Vol(M, ) where the supremum is taken over all A for which there exists a volume preserving 1k n embedding i=1 B (A) → (M, ). Moser’s deformation method readily implies for all k ∈ N. Proposition 2 of Section 1.3.2 shows more: For that vk (M, ) = 11 any partition M = ki=1 Mi of M into subsets Mi such that Int Mi is connected and Vol (Int Mi , ) = k1 Vol(M, ) for all i there exists a volume preserving embedding 1k 1k 1 n n i=1 B (A) → i=1 Int Mi with |B (A)| = k Vol(M, ). If the volume form comes from a symplectic form ω, the sequence (1 − pk (M, ω))k∈N is a measure for how far the symplectic geometry of (M, ω) is from the volume geometry of (M, ). Euclidean packings. Given a bounded domain U in Rn , define its k’th Euclidean packing number as k |B n (a)| k (U ) = sup |U | where the supremum is taken over all a for which k disjoint translates of B n (a) fit into U . Then k (U ) ≤ pk (U ) ≤ vk (U ) = 1 for all k ∈ N, and it is interesting to understand “on which side” pk (U ) lies. To fix the ideas, we assume that U is the unit ball B n := B n (π) in Rn . The precise values of k (B n ) are known only for small √ k: If 1 ≤ k ≤ n + 1, the smallest ball containing k balls of radius 1 has radius 1 + 2 − 2/k, and the centres of the balls are arranged as vertices of a regular (k − 1)-dimensional simplex inscribed in the ball and concentric with it. Moreover, if√n + 2 ≤ k ≤ 2n, the smallest ball B containing k balls of radius 1 has radius 1 + 2, and the packing configuration of 2n balls in B is unique up to isometry, the centres being the midpoints of the faces of an n-dimensional Euclidean √ cube whose edges have length 2 2. In particular,  n 9k  if 1 ≤ k ≤ n + 1,   n 1+ 2− 2k k B = (9.1.1) k   if n + 2 ≤ k ≤ 2n.  1+√2 n While for 1 ≤ k ≤ n + 1 these numbers were known to Rankin in 1955, for n + 2 ≤ k ≤ 2n they were obtained only recently by W. Kuperberg, [46]. An obvious upper bound for k (B n ) is k k B n ≤ n for all k ≥ 2. (9.1.2) 2

192


Given a bounded domain U in Rn , let conv(U ) be the convex hull of U . For each k ≥ 1 we set k |B n | convk (B n ) = sup (9.1.3) |conv(U )| where the supremum is taken over all configurations U of k disjoint translates of B n in Rn . Since B n is convex, k (B n ) ≤ convk (B n ) for all k ∈ N. Let Skn = conv(U ) be the sausage obtained by choosing U=

k−1 2

(B n + iu)

(9.1.4)

i=0

where u is a unit vector in Rn . With κn := |B n | we then have |Skn | = κn +2(k−1)κn−1 . The sausage conjecture of L. Fejes Tóth from 1975 states that equality in (9.1.3) is attained exactly for U as in (9.1.4), and this conjecture was proved by Betke and Henk, [5], for n ≥ 42. Therefore, : : kκn k π 1 n n if n ≥ 42. < k (B ) ≤ convk (B ) = κn + 2(k − 1)κn−1 k−1 2 n+1 (9.1.5) For arbitrary n, an older result of Gritzmann, [30], states that : : √ π 1 n n . k (B ) ≤ convk (B ) < 2 + 3 2 n In order to get an idea of the values k (B n ) for large k we notice that the limit n := lim k (B n ) k→∞

exists and is equal to the highest density of a packing of Rn , see Section 2.1 of Chapter 3.3 in [34]. The highest density of a packing of R2 is π 2 = √ = 0.9069 . . . 12 as in the familiar hexagonal lattice packing in which each disk touches 6 others (Thue, 1910). The highest density of a packing of R3 is π 3 = √ = 0.74048 . . . 18 as in the face centred cubic lattice packing which is usually found in fruit stands and in which each ball touches 12 other balls. This was conjectured by Keppler in 1611, and Gauss proved in 1831 that no lattice packing has a higher density. The Keppler conjecture was settled only recently by Hales, see [33] and the references therein. For


193

4 ≤ n ≤ 36, the currently best upper bound for n was given recently by Cohn and Elkies in [17]. E.g., π2 = 0.61685 ≤ 4 ≤ 0.647742. 16 Here, the lower bound is the density of the packing associated with the “checkerboard lattice” consisting of all vectors (a, b, c, d) ∈ Z4 with a + b + c + d ∈ 2Z, and it is known that this is the highest possible density for a 4-dimensional lattice packing. A result of Blichfeldt from 1929 states that n ≤ (n + 2)2−(n+2)/2 ,

(9.1.6)

and the best known lower and upper bounds for n of asymptotic nature are cn2−n ≤ n ≤ 2−(0.599+o(1))n

as n → ∞

for any constant c < log 2, see Section 2 of Chapter 3.3 in [34]. We refer to [19], to Sections 3.3 and 3.4 of [34], and to [88] for more information on Euclidean packings, its long history and its many relations and applications to other branches of mathematics (such as discrete geometry, group theory, number theory and crystallography) and to problems in physics, chemistry, engineering and computer science. The symplectic packing numbers pk B 4 are listed in Table 9.1 below. For n ≥ 3, the results known about pk B 2n are k pk B 2n = n for 2 ≤ k ≤ 2n , 2 pl n B 2n = 1 for all l ∈ N,

(9.1.7) (9.1.8)

see [61, Corollary 1.5.C and 1.6.B] and Section 9.4.1 below. The identities (9.1.8) yield another proof of (9.0.3) for the ball, (9.1.9) lim pk B 2n = 1 for all n.

2

k→∞

Of course, k B < pk B 2 = vk B 2 = 1 for all k ≥ 2. Comparing (9.1.1) or 4 (9.1.2) for n = 4 with the values pk B listed in Table 9.1 we see that k B 4 < pk B 4 for all k ≥ 2. Moreover, (9.1.2) and (9.1.7) show that 1 k B 2n ≤ n pk B 2n for 2 ≤ k ≤ 2n and all n ∈ N. 2 Inequality (9.1.5) and (9.1.8) yield an explicit k(2n) such that k B 2n < pk B 2n for all k ≥ k(2n) and 2n ≥ 42. It is conceivable that k B 2n < pk B 2n for all k ≥ 2 and n ∈ N, but we do not know the answer to

194


Question 9.1.1. Is it true that 28 B 6 < p28 B 6 ? Finally, comparing (9.1.9) with (9.1.6) we see that pk B 2n is much larger than 2n k B for sufficiently large k and large n. 4. Relations to algebraic geometry. A symplectic packing of (M, ω) by k equal balls corresponds to a symplectic blow-up of (M, ω) at k points with equal weights. Via this correspondence, the symplectic packing problem is intimately related to old problems in algebraic geometry: The symplectic packing problem for the complex projective plane CP2 (completely solved by Biran in [7]) is related to an old (and still open) conjecture of Nagata on the minimal degree of an irreducible algebraic curve in CP2 passing through N ≥ 9 points with given multiplicities, see [9], [10], [61], [87] for details. Moreover, the symplectic packing problem is closely related to the problem of computing Seshadri constants of ample line bundles, which are a measure of their local positivity, see [9], [10], [12], [52].

9.2 The packing numbers of the 4-ball and CP2 and of ruled symplectic 4-manifolds In this section we collect the known packing numbers of interest to us. 9.2.1 The packing numbers of the 4-ball and CP2 . Let ωSF be the unique U(3)invariant Kähler form on CP2 whose integral over CP1 equals 1. According to a result of Taubes, [80], every symplectic form on CP2 is diffeomorphic to a ωSF for some a > 0. In view of the symplectomorphism 4 5 8 4 B (π ), ω0 → CP2 \ CP1 , π ωSF , z = (z1 , z2 ) → z1 : z2 : 1 − |z|2 (9.2.1) further discussed in [62, Example 7.14] we have pk B 4 ≤ pk CP2 for all k. It is shown in [61, Remark 2.1.E] that in fact (9.2.2) pk B 4 = pk CP2 for all k. A complete list of these packing numbers was obtained in [7] (see Table 9.1). Table 9.1. pk B 4 = pk CP2

k pk

1

2

3

1

1 2

3 4

4

5

6

7

8

≥9

1

20 25

24 25

63 64

288 289

1

9.2 The packing numbers of the 4-ball and CP2 and of ruled symplectic 4-manifolds 195

Explicit maximal packings were found by Karshon [41] for k ≤ 3 and by Traynor [81] for k ≤ 6 and k = l 2 (l ∈ N). We will give even simpler maximal packings for these values of k in 9.3.1.

9.2.2 The packing numbers of ruled symplectic 4-manifolds. Denote by g the closed orientable surface of genus g. There are exactly two orientable S 2 -bundles with base g , namely the trivial bundle π : g × S 2 → g and the nontrivial bundle π : g S 2 → g , see [62, Lemma 6.9]. Such a manifold M is called a ruled surface. A symplectic form ω on a ruled surface is called compatible with the given ruling π if it restricts on each fibre to a symplectic form. Such a symplectic manifold is then called a ruled symplectic 4-manifold. It is known that every symplectic structure on a ruled surface is diffeomorphic to a form compatible with the given ruling π via a diffeomorphism which acts trivially on homology, and that two cohomologous symplectic forms compatible with the same ruling are isotopic [51]. A symplectic form ω on a ruled surface M is thus determined up to diffeomorphism by the class [ω] ∈ H 2 (M; R). In order to describe the set of cohomology classes realized by (compatible) forms on M we fix an orientation of g and an orientation of the fibres of the given ruled surface M. These orientations determine an orientation of M in a natural way, see below. We say that a compatible symplectic form ω is admissible if its restriction to each fibre induces the given orientation and if ω induces the natural orientation on M. Notice that every symplectic form on M is diffeomorphic to an admissible form for a suitable choice of orientations of g and the fibres. Consider first the trivial bundle g × S 2 , and let {B = [g × pt], F = [pt × S 2 ]} be a basis of H 2 (M; Z). Here and henceforth we identify homology and cohomology via Poincaré duality. The natural orientation of g × S 2 is such that B · F = 1. A cohomology class C = bB + aF can be represented by an admissible form if and only if C · F > 0 and C · C > 0, i.e., a>0

and

b > 0,

standard representatives being split forms. We write g (a) × S 2 (b) for this ruled symplectic 4-manifold. In case of the nontrivial bundle g S 2 a basis of H 2 (g S 2 ; Z) is given by {A, F }, where A is the class of a section with selfintersection number −1 and F is the fibre class. The homology classes of sections of g S 2 of self-intersection 2 number k are Ak = A + k+1 2 F with k odd. The natural orientation of g S is such that Ak · F = A · F = 1 for all k. Set B = A + F /2. Then {B, F } is a basis of H 2 (g S 2 ; R) with B · B = F · F = 0 and B · F = 1. As for the trivial bundle, the necessary condition for a cohomology class bB + aF to be representable by an admissible form is a > 0 and b > 0. It turns out that this condition is sufficient only if g ≥ 1: A cohomology class bB + aF can be represented by an admissible form if

196


and only if a > b/2 > 0

if g = 0,

a > 0 and b > 0

if g ≥ 1,

see [62, Theorem 6.11]. We write (g S 2 , ωab ) for this ruled symplectic 4-manifold. A “standard Kähler form” in the class [ωab ] is explicitly constructed in [57, Section 3] and [62, Exercise 6.14]. When constructing our explicit symplectic packings, it will always be clear which symplectic form in [ωab ] is chosen. We begin with the trivial sphere bundle over the sphere. Proposition 9.2.1. Assume that a ≥ b. Abbreviate pk = pk (S 2 (a) × S 2 (b)), and denote by $x% the minimal integer which is greater than or equal to x. Then k b kb if ≤ 1. pk = 2a 2 a Moreover, a+b 2 3 p3 = b, 2ab 3 a + 2b 2 4 5 b, p4 = p3 , p5 = 3 2ab 5 a + 2b 2a + 2b 2 3 b, , p6 = ab 5 7 7 a + 3b 3a + 4b 4a + 4b 2 b, , , p7 = 2ab 7 13 15

b , p1 = 2a

b p2 = , a

1 0, , 1 , 2 1 0, , 1 , 3 1 3 0, , , 1 , 3 4 1 8 7 0, , , , 1 . 4 11 8

on on on on

In particular, for k ≤ 7 we have pk (S 2 (a) × S 2 (b)) = 1 exactly for (k = 2, (k = 4, ab = 21 ), (k = 6, ab = 13 ), (k = 6, ab = 43 ) and (k = 7, ab = 78 ).

b a

= 1),

We explain our notation by an example: p3 =

3 2 2ab b

if 0
b2 > 0. Abbreviate pk = pk (S 2 S 2 , ωab ), and set &k' = k if k is odd and &k' = k + 1 if k is even. Then pk =

kb 2a

if

&k' b ≤ 1. 2 a

Moreover, 2a + b 2 1 2 b, on 0, , 2 , ab 4 3 2 2a + 3b 3 2 2 b, on 0, , 2 , p3 = p2 , p4 = 2 ab 8 5 2 2a + 3b 2a + b 5 2 6 b, , on 0, , , 2 , p5 = 2ab 8 5 5 7 2 2a + 5b 2a + 2b 2a + b 3 2 10 4 b, , , on 0, , , , 2 , p6 = ab 12 7 5 7 11 3 7 2a + 5b 6a + 9b 4a + 4b 4a + 3b 6a + 3b 2 b, , , , , p7 = 2ab 12 28 15 13 16 2 1 22 8 14 on 0, , , , , , 2 . 7 2 23 7 9 p1 =

b , 2a

p2 =

In particular, for k ≤ 7 we have pk (S 2 S 2 , ωab ) = 1 exactly for (k = 3, (k = 5, ab = 25 ), (k = 6, ab = 43 ) (k = 7, ab = 27 ) (k = 7, ab = (k = 7, ab = 14 9 ).

b 2 a = 3 ), 8 7 ) and

In 9.3.2 we will construct explicit maximal packings of (S 2 S 2 , ωab ) for all k with ≤ 1, for k ≤ 5 and 0 < b2 < a arbitrary, and for k = 6 and ab ∈ ]0, 23 ] ∪ [ 43 , 2[. 2l for some l, m ∈ N with m > l, we will construct Moreover, given ωab with ab = 2m−l 2 2 explicit full packings of (S S , ωab ) by l(2m − l) balls. &k' b 2 a

198


Corollary 9.2.4. We have   8a a b 2 2 max 2 , 8 ≤ P (S S , ωab ) ≤  8ab b

(2a−b)2

if b ≤ a if b ≥ a

2 2 except possibly for ab ∈ 27 , 87 , 14 9 , in which case the lower bound for P (S S , ωab ) is 7. For (S 2 S 2 , ω11 ) we thus have Table 9.3. pk (S 2 S 2 , ω11 )

k

1

2

3

4

5

6

7

≥8

pk

1 2

9 16

27 32

25 32

9 10

48 49

14 15

1

Proposition 9.2.5. Let g ≥ 1 and let a > 0 and b > 0. Then kb . pk (g (a) × S 2 (b)) = pk (g S 2 , ωab ) = min 1, 2a In particular, P (g (a) × S 2 (b)) = P (g S 2 , ωab ) = 2a b . In 9.3.2 we will construct explicit maximal packings of g (a) × S 2 (b) and of (g S 2 , ωab ) for all k with 2k ab ≤ 1 and explicit full packings for k = 2ml 2 if a = mb or b = ma (l, m ∈ N). For the proofs of Propositions 9.2.1, 9.2.3 and 9.2.5 and Corollaries 9.2.2 and 9.2.4 we refer to [7] and [75].

9.3

Explicit maximal packings in four dimensions

In this section we realize most of the packing numbers listed in the previous section by explicit symplectic packings. Sometimes, we shall give two different maximal packings. It is known that for the 4-ball and CP2 and for ruled symplectic 4-manifolds, any two packings by k closed balls of equal size are symplectically isotopic, see [6], [58]. As we have seen in Lemmata 3.1.8 and 5.3.1, for each > 0 there exist explicit symplectic embeddings B 4 (a) → 2 (a + ) × 22 (1) and 2 (a) × 22 (1) → B 4 (a + ). We shall omit > 0 and think of B 4 (a) as 2 (a) × 22 (1). Variations of the map α in Figure 3.3 yield different shapes. We recall that α is represented by (α) in Figure 9.1. Let α be its mirror represented by (α), and let β, γ and δ be represented

199

9.3 Explicit maximal packings in four dimensions

(α)

(α)

(β)

(γ )

(δ)

Figure 9.1. The images of the disc.

by (β), (γ ) and (δ), respectively. Applying α to both the z1 - and the z2 -plane yields a shape whose x1 -x2 -shadow is (arbitrarily close to) the simplex [αα] in Figure 9.2, applying α to the z1 -plane and α to the z2 -plane yields [αα], and other combinations yield various other shapes.

[αα]

[αα]

[α α]

[ββ]

[γ α]

[δδ]

Figure 9.2. The x1 -x2 -shadows.

The symplectic 4-manifolds (M, ω) we shall consider contain a domain of equal volume which is explicitly symplectomorphic to U × 22 (1) ⊂ R2 (x) × R2 (y). In order to construct an explicit symplectic packing of (M, ω) by k equal balls it thus suffices to insert k disjoint x1 -x2 -shadows of equal width as in Figure 9.2 into U . Remark 9.3.1. Besides of being explicit, the 4-dimensional symplectic packings constructed in [41], [81] and in this section have yet another advantage over the packings found in [61], [7], [8]: The symplectic packings of (M, ω) by k balls obtained from the method in [61], [7], [8] are maximal in the following sense. For every > 0 there

200


exists a symplectic embedding ϕ :

1k

i=1 B

2n (a)

→ (M, ω) such that

Vol (Im ϕ , ω) (9.3.1) ≥ pk (M, ω) − . Vol (M, ω) Karshon’s symplectic packings of CP2 , ωSF by 2 and 3 balls B 4 π2 given by the map (9.2.1) and compositions of this map with coordinate permutations fill exactly 21 and 43 of CP2 , ωSF . Similarly, the 4-dimensional packings in [81] and in this section are maximal in the following sense: 1 There exists a symplectic embedding ϕ : ki=1 B 4 (a) → (M, ω) such that Vol (Im ϕ, ω) = pk (M, ω). Vol (M, ω)

(9.3.2)

1 Moreover, ϕ is explicit in the following sense: The image ki=1 ϕ B 4 (a) of ϕ is 1 explicit, and given a < a one can construct ϕ such that its restriction to ki=1 B 4 (a ) is given pointwise. Indeed, choose a sequence a < aj a. The packings in [81] and our packings 1 ϕ(aj ) : ki=1 B 4 (aj ) → (M, ω) can be chosen such that Im ϕ(aj ) ⊂ Im ϕ(aj +1 )

for all j.

The claim now follows from a result of McDuff, [55], stating that two symplectic embeddings of a closed ball into a larger ball are isotopic via a symplectic isotopy of the larger ball. 3 9.3.1 Maximal packings of the 4-ball and CP2 . In view of the symplectomorphism (9.2.1) and the identity (9.2.2) we only need to construct packings of the 4-ball. It follows from Table 9.1 that any k of the embeddings in Figure 9.3(a) yield a maximal packing of B 4 by k balls, k = 2, 3, 4, and that any k of the embeddings in Figure 9.3(b) yield a maximal packing by k = 5, 6 balls. Figure 9.3(c) shows a full packing by 9 balls. Explicit maximal packings of B 4 by k ≤ 6 balls were first constructed by Traynor in [81]. As the symplectic wrapping embeddings described in Section 7.1, her symplectic embeddings of a 4-ball into a larger 4-ball are of the form δE γ β αE . This time, β = diag π, π, π1 , π1 , and γ : R2 (x) × R2 (y) → T 2 × R2 (y) is of the form γ = (id τ ) (p M) × M ∗ where τ is a translation of R2 (y) and M ∈ SL(2; R). For k ≤ 5, the matrix M can be chosen in SL(2; Z), but for k = 6 also elements M ∈ SL(2; R) \ SL(2; Z) for which p M : 22 (π ) → T 2 is injective must be considered. 63 and p8 B 4 = 288 We cannot realize the packing numbers p7 B 4 = 64 289 by our packing method. This method as well as Traynor’s method and their combination only fill 79 and 89 of the 4-ball by 7 and 8 equal balls, respectively.

201

9.3 Explicit maximal packings in four dimensions x2

x2

x2

x1

x1 (a)

(b)

x1 (c)

Figure 9.3. Maximal packings of B 4 for k ≤ 6 and k = l 2 .

Question 9.3.2. Is there an explicit embedding of 7 or 8 equal balls into the 4-ball filling more than 79 and 89 of the volume?

9.3.2 Maximal packings of ruled symplectic 4-manifolds. Given a ruled symplectic 4-manifold (M, ωab ), let ck (a, b) be the supremum of those A for which 1k 2n (A) symplectically embeds into (M, ω ), so that B ab i=1 pk (M, ωab ) =

k ck2 (a, b) . 2 Vol(M, ωab )

(9.3.3)

We shall write c instead of ck (a, b) if (M, ωab ) and k are clear from the context. Maximal packings of S 2 (a) × S 2 (b). As in Proposition 9.2.1 we assume that a ≥ b. Represent the symplectic structure of S 2 (a) × S 2 (b) by a split form. Using Lemma 3.1.5 we symplectically identify S 2 (a) \ pt with ]0, a[×]0, 1[ and S 2 (b) \ pt with ]0, b[×]0, 1[. Then

2(a, b) × 22 (1) = S 2 (a) × S 2 (b) \ S 2 (a) × pt ∪ pt × S 2 (b) . Besides for k ∈ {6, 7}, we will construct the explicit maximal packings promised after Proposition 9.2.1 by constructing packings of 2(a, b) × 22 (1) which realize the packing numbers of S 2 (a) × S 2 (b) computed in Proposition 9.2.1 and hence are maximal. (It is, in fact, known that all packing numbers of 2(a, b) × 22 (1) and S 2 (a) × S 2 (b) agree, see [61, Remark 2.1.E]). To construct explicit maximal packings for all k with 2k ab ≤ 1 is a trivial matter. Figure 9.4 shows a maximal packing by 1 and 2 respectively 5 and 6 balls. Let now k = 3, 4 and ab ≥ 21 . Figure 9.5 shows maximal packings of S 2 (a)×S 2 (b) by k balls for ab = 21 , ab = 43 and ab = 1. For ab > 21 the (x1 , x2 )-coordinates of the

202

9 Packing symplectic manifolds by hand x2

x2

b

b

x1

x1

a

a

(a)

(b)

Figure 9.4. Maximal packings of S 2 (a) × S 2 (b) by k balls, 2k ab ≤ 1. x2

x2

b

x2

b

b

x1 b 1 a = 2

a

x1 3 b a = 4

a

x1 b a =1

a

Figure 9.5. Maximal packings of S 2 (a) × S 2 (b) by 3 and 4 balls, ab ≥ 21 .

vertices of the “upper left ball” are (0, c),

(a − c, b),

(c, c),

(a − c, b − c),

where c = a+b 3 . As in most of the subsequent figures of this chapter, the three pictures in Figure 9.5 should be seen as moments of a movie starting at ab = 21 and ending at b a = 1. Each ball in this movie moves in a smooth way. Next, let k = 5 and ab ≥ 13 . In order to construct a smooth family of maximal packings of S 2 (a) × S 2 (b) by 5 balls, we think of the maximal packing for ab = 13 rather as in Figure 9.6 than as in Figure 9.4(a). The x1 -width of all balls is a+2b 5 , and the “upper left ball” has 5 vertices for 13 < ab ≤ 43 and 7 vertices for ab > 43 . For k ∈ {6, 7}, we cannot realize the packing numbers pk S 2 (a) × S 2 (b) by directly packing rectangles as for k ≤ 4. We shall instead construct certain maximal packings of CP2 which correspond to maximal packings of S 2 (a)×S 2 (b). As noticed in [7], the correspondence between symplectic packings and the symplectic blow-up operation implies 1 Lemma 9.3.3. Packing S 2 (a) × S 2 (b) by k equal balls ki=1 B 4 (c) corresponds to 1 1 4 packing (CP2 , (a + b − c)ωSF ) by the k + 1 balls B 4 (a − c) B 4 (b − c) k−1 i=1 B (c).

203

9.3 Explicit maximal packings in four dimensions x2

x2 b b

x1

x1

a

b 1 a = 3

a

1 b a = 2

x2

x2 b

b

x1 b 3 a = 4

x1

a

b a =1

a

Figure 9.6. Maximal packings of S 2 (a) × S 2 (b) by 5 balls, ab ≥ 13 .

In order to make this correspondence plausible, we choose ab = 23 and c = 2 c6 (a, b) = a+2b 5 , and we think of (CP , (a + b − c)ωSF ) as the simplex of width a + b − c and of S 2 (a) × S 2 (b) as the rectangle of width a and length b. As Figure 9.7 x2

a−c b

c

b−c

x1 a a+b−c

Figure 9.7. (CP2 , (a + b − c)ωSF ) \ B 4 (a − c)

1 4 B (b − c) = S 2 (a) × S 2 (b) \ B 4 (c).

illustrates, the space obtained by removing a ball B 4 (c) from S 21 (a) × S 2 (b) coin4 cides with the space obtained by removing the balls B (a − c) B 4 (b − c) from (CP2 , (a + b − c)ωSF ).

204


2 Figures 9.8, 9.9 9.10 describe 1 and 1k−1 4explicit packings of (CP , (a + b − c)ωSF ) 4 4 by balls B (a−c) B (b−c) i=1 B (c) for k ∈ {6, 7} and c as in Proposition 9.2.1. The lower left triangle represents B 4 (a − c) and the black “ball” represents B 4 (b − c).

x2 x2

x2

a−c

a−c a−c x1 a+b−c b 1 a = 3

x1 a+b−c

x1 a+b−c 2 b a = 3

1 b a = 2

Figure 9.8. Maximal packings of S 2 (a) × S 2 (b) by 6 balls, 13 ≤ ab ≤ 43 . x2

x2

a−c

x2

a−c

a+b−c

a−c

x1

a+b−c

b 3 a = 4

x1

x1 a+b−c

4 b a = 5

b a =1

Figure 9.9. Maximal packings of S 2 (a) × S 2 (b) by 6 balls, 43 ≤ ab ≤ 1. x2

x2

a−c

x2

a−c

a+b−c b 1 a = 4

a−c

x1

x1 a+b−c 2 b a = 5

x1 a+b−c 3 b a = 5

Figure 9.10. Maximal packings of S 2 (a) × S 2 (b) by 7 balls, 41 ≤ ab ≤ 35 .


205

From these packings one obtains explicit packings of S 2 (a) × S 2 (b) as follows: First symplectically blow up (CP2 , (a + b − c)ωSF ) twice by removing the balls B 4 (a − c) and B 4 (b − c) and collapsing the remaining boundary spheres to exceptional spheres in homology classes D1 and D2 . The resulting manifold, which is symplectomorphic to S 2 (a) × S 2 (b) blown up at one point with weight c, still contains the k − 1 explicitly embedded balls B 4 (c), and according to [7, Theorem 4.1.A] the exceptional sphere in class L − D1 − D2 can be symplectically blown down with weight c to yield the k’th ball B 4 (c) in S 2 (a) × S 2 (b). Here and in the sequel, L = [CP1 ] denotes the class of a line in CP2 . Finally, the construction of full packings of S 2 (mb) × S 2 (b) by 2ml 2 balls for l, m ∈ N is also straightforward. Figure 9.11 shows such a packing for l = m = 2. x2 b

x1 2b Figure 9.11. A full packing of S 2 (2b) × S 2 (b) by 16 balls.

Maximal packings of (S 2 S 2 , ωab ). In order to describe our maximal packings of (S 2 S 2 , ωab ), it will be convenient to work with the parameters α = a − b2 , β = b, so that α > 0, β > 0 and ωab = βA + (α + β)F . According to [62, Example 7.4], 31 of CP2 at one the manifold S 2 S 2 is diffeomorphic to the complex blow-up N point via a diffeomorphism under which the class L of a line corresponds to A + F and the class D1 of the exceptional divisor corresponds to A. We can therefore view 31 endowed with the symplectic form in class (α + β)L − αD1 (S 2 S 2 , ωab ) as N obtained by symplectically blowing up (CP2 , (α + β)ωSF ) with weight α. Since symplectically blowing up with weight α corresponds to removing a ball B 4 (α) and collapsing the remaining boundary sphere to an exceptional sphere in class D1 , we can think of this symplectic manifold as the truncated simplex obtained by removing the simplex of width α from the simplex of width α + β. Denote by "x# the integer part of x ≥ 0. In the parameters α and β, the packings promised afterProposition 9.2.3 are explicit maximal packings of (S 2 S 2 , ωab ) k β for all k with 2 α ≤ 1, for k ≤ 5 and α, β > 0 arbitrary, and for k = 6 and β β l α ∈ ]0, 1] ∪ [4, ∞[. Moreover, given ωab with α = m−l for some l, m ∈ N with 2 m > l, we will construct explicit full packings of (S S 2 , ωab ) by l(2m − l) balls. Set ck = ck (a, b) = ck (S 2 S 2 , ωab ). Replacing a by α + β/2 and b by β in the list in Proposition 9.2.3 and using β(2α + β) = 2 Vol(S 2 S 2 , ωab ) and (9.3.3), we

206


find that

α+β c2 = c3 = β, 2 α + 2β c4 = β, 4 α + 2β 2α + 2β , c5 = β, 4 5 α + 3β 2α + 3β 2α + 2β , , c6 = β, 6 7 5 c1 = β,

on ] 0, 1, ∞ [,

1 on 0, , ∞ , 2 1 3 on 0, , , ∞ , 2 2 1 5 on 0, , , 4, ∞ . 3 3

β To construct packings with pk = k 2α+β for all k with 2k βα ≤ 1 is very easy. Figure 9.12(a) shows a maximal packing by 1 ball, and Figures 9.12(b1) and (b2) show maximal packings by 4 and 5 balls for βα = 21 and βα < 21 , respectively. x2

x2

x2

α

α α α+β

x1

α+β

(a)

x1

α+β

(b1)

(b2) kβ

Figure 9.12. Maximal packings of (S 2 S 2 , ωab ) by k balls, 2 α ≤ 1.

Figure 9.13 shows maximal packings for k = 2, 3 and

β α

≥ 1.

x2

α x1 α+β

Figure 9.13. Maximal packings of (S 2 S 2 , ωab ) by 2 and 3 balls,

β α

≥ 1.

x1

207


Also our maximal packings by 4 balls are easy to understand (Figure 9.14 and Figure 9.15(a)): 2 c4 = β + α2 just means that the two middle gray balls touch each other. As long as βα ≤ 23 , there is enough room for a fifth (black) ball between these two balls. If βα > 23 , there is enough space for a fifth ball if and only if the capacity c of (Figures 9.15(b1) and (b2)). the balls satisfies 2c + 2c ≤ α + β; hence c5 = 2α+2β 5 x2

x2

x2

α α α x1

x1

α+β

x1

α+β

β 1 α = 2

α+β β 3 α = 2

β α =1

Figure 9.14. Maximal packings of (S 2 S 2 , ωab ) by 4 and 5 balls, 21 ≤ βα ≤ 23 . x2

x2

x2

α

α

α α+β

x1

(a): βα = 3

α+β

x1

(b1): βα = 2

α+β

x1

(b2): βα = 6

Figure 9.15. Maximal packings of (S 2 S 2 , ωab ) by 4 and 5 balls, βα ≥ 23 .

Let now k = 6. Figure 9.16 shows maximal packings for 13 ≤ βα ≤ 1. For βα > 13 the vertices of the “lower middle ball” are α+β α+β α+β α+β (α+β−2c6 , c6 ), , , (α+β−c6 , c6 ), , − c6 . 2 2 2 2 Maximal packings for

β α

≥ 4 are illustrated in Figure 9.17.

208

9 Packing symplectic manifolds by hand x2

x2

α

x2

α α

x1 β 1 α = 2

x1

α+β

x1

α+β

β 2 α = 3

β α =1

α+β

Figure 9.16. Maximal packings of (S 2 S 2 , ωab ) by 6 balls, 13 ≤ βα ≤ 1. x2

x2

α x1

α

x1

α+β β α =4

α+β β α = 19

Figure 9.17. Maximal packings of (S 2 S 2 , ωab ) by 6 balls, βα ≥ 4.

Remark 9.3.4. It is not a coincidence that we were not able to construct maximal packings of (S 2 S 2 , ωab ) by 6 balls for all ratios βα > 0. Indeed, a maximal packing of (S 2 S 2 , ωab ) by 6 equal balls for βα = 53 corresponds to a maximal packing of the 4-ball by 7 equal balls. 3 l Finally, suppose that βα = m−l for some l, m ∈ N with m > l. We can then fill 2 2 (S S , ωab ) by l(2m − l) balls by decomposing S 2 S 2 into l shells and filling the i-th shell with 2m + 1 − 2i balls (see Figure 9.18, where l = 2 and m = 4).

Maximal packings of g (a)×S 2 (b) and ( g S 2 , ωab ) for g ≥ 1. Fix a > 0 and b > 0. We represent the symplectic structure of the product g (a) × S 2 (b) by a split form. Removing a wedge of 2g loops from (a) and a point from S 2 (b) we see that g (a) × S 2 (b) contains 2(a, b) × 22 (1). The explicit construction of the “standard


209

x2

α

x1 α+β

Figure 9.18. A full packing of (S 2 S 2 , ωab ), βα = 1, by 12 balls.

Kähler form” in class [ωab ] given in [57, Section 3] and [62, Exercise 6.14] shows that also (g S 2 , ωab ) endowed with this standard form contains 2(a, b) × 22 (1). The explicit maximal packings promised after Proposition 9.2.5 can thus be constructed as for S 2 (a) × S 2 (b), see Figures 9.4 and 9.11. 9.3.3 Explicit packings of g (a)× h (b) for g, h ≥ 1. We consider 4-manifolds of the form g × h with g, h ≥ 1. The space of symplectic structures on such manifolds in not understood, but no symplectic structure different from g (a)×h (b) for some a > 0, b > 0 is known. For g (a) × h (b), no obstructions to full packings are known. Recall from (9.0.2) that for ab ∈ Q,

P g (a) × h (b) := inf k0 ∈ N | pk g (a) × h (b) = 1 for all k ≥ k0 is finite. In fact, Biran showed in Corollary 1.B and Section 5 of [8] that P T 2 (1) × T 2 (1) ≤ 2 and that

(9.3.4)

if a, b ∈ N, (9.3.5) if a, b ∈ N \ {1}. If ab ∈ / Q or if 1 ≤ k < P g (a) × h (b) , there is not much known about pk g (a) × h (b) : We can assume without loss of generality that a ≥ b. Since the symplectic packing numbers of S 2 (a) × S 2 (b) and 2(a, b) × 22 (1) agree, and since 2(a, b) × 22 (1) symplectically embeds into g (a) × h (b), pk S 2 (a) × S 2 (b) ≤ pk g (a) × h (b) for all k ∈ N, (9.3.6) P g (a) × h (b) ≤

8ab 2ab

210


and Figures 9.4, 9.5, 9.6 and 9.11 describe some explicit packings of g (a)×h (b). A comparison of Corollary 9.2.2 with the estimates (9.3.4) and (9.3.5) and with Proposition 9.3.5 below shows, however, that in general the inequalities (9.3.6) are not equalities and that for g (a) × h (b) not all of the packings in Figures 9.4, 9.5 and 9.6 are maximal. Elaborating an idea of Polterovich, [62, Exercise 12.4], Jiang constructed in [40, Corollary 3.3 and3.4] explicit symplectic embeddings of one ball which improve the b estimate 2a ≤ p1 g (a) × h (b) from (9.3.6). Proposition 9.3.5 (Jiang). Let (a) be any closed surface of area a ≥ 1. (i) There exists a constant C > 0 such that p1 (a) × T 2 (1) ≥ C. (ii) If h ≥ 2, there exists a constant C = C(h) > 0 depending only on h such that wG ((a) × h (1)) ≥ C log a. In other words, p1 ((a) × h (1)) ≥

(C log a)2 . 2a

Notice that for = S 2 Biran’s result p1 S 2 (a) × h (1) = min 1, a2 stated in Proposition 9.2.5 is much stronger. We shall use Jiang’s embedding method to prove the following quantitative version of Proposition 9.3.5 (i). Proposition 9.3.6. If a ≥ 1, √ max{a + 1 − 2a + 1, 2} . p1 (a) × T (1) ≥ 4a

2

In particular, the constant C in Proposition 9.3.5 (i) can be chosen to be C = 1/8. Proof. Set R(a) = {(x, y) ∈ R2 | 0 < x < 1, 0 < y < a}, and consider the linear symplectic map ϕ : (R(a) × R(a), dx1 ∧ dy1 + dx2 ∧ dy2 ) → (R2 × R2 , dx1 ∧ dy1 + dx2 ∧ dy2 ), (x1 , y1 , x2 , y2 ) → (x1 + y2 , y1 , −y2 , y1 + x2 ). Let pr : R2 → T 2 = R/Z × R/Z be the projection onto the standard symplectic torus. Then (id2 × pr) ϕ : R(a) × R(a) → R2 × T 2 is a symplectic embedding. Indeed, given (x1 , y1 , x2 , y2 ) and (x1 , y1 , x2 , y2 ) with x1 + y2 y1 −y2 y 1 + x2

= x1 + y2 = y1 ≡ −y2 mod Z ≡ y1 + x2 mod Z

(9.3.7) (9.3.8) (9.3.9) (9.3.10)


211

equations (9.3.8) and (9.3.10) imply x2 ≡ x2 mod Z, whence x2 = x2 . Moreover, (9.3.9) and (9.3.7) show that y2 − y2 = x1 − x1 ≡ 0 mod Z, and so x1 = x1 and y2 = y2 . Next observe that

(id2 × pr) ϕ R(a) × R(a) ⊂ ] − a, 0[×] − a − 1, a + 1[×T 2 .

Thus R(a) × R(a) symplectically embeds into (2a(a + 1)) × T 2 (1), and since B 4 (a) symplectically embeds into R(a) × R(a) and B 4 (1) symplectically embeds , into (a) × T 2 (1) for any a ≥ 1, Proposition 9.3.6 follows. Remark 9.3.7. Gromov’s Nonsqueezing Theorem stated in Section 1.2 was generalized by Lalonde and McDuff in [48] to all (2n−2)-dimensional symplectic manifolds (V , ω). General Nonsqueezing Theorem. If the ball B 2n (a) symplectically embeds into (V × B 2 (A), ω ⊕ ω0 ), then A ≥ a. Gromov’s original proof of his Nonsqueezing Theorem combined with the existence of Gromov–Witten invariants for arbitrary closed symplectic manifolds shows the Nonsqueezing Theorem / 2 wG V × S , ωV ⊕ ωS 2 ≤ ωS 2 (9.3.11) S2

for any closed symplectic manifold (V , ω), see [63, Section 9.3] and [60, Proposition 1.18]. This Nonsqueezing Theorem can be generalized by either viewing S 2 as a fibre or as the base of a symplectic fibration. (i) A ruled symplectic manifold (M, ω) is a 2-sphere bundle S 2 → M → V over a closed manifold V endowed with a symplectic form ω which restricts to an area form on each fibre. Examples are ruled symplectic 4-manifolds and products (V × S 2 , ωV ⊕ ωS 2 ). The proof of (9.3.11) extends to this situation and shows the Nonsqueezing Theorem / wG (M, ω) ≤

ω,

(9.3.12)

S2

see again [63, Section 9.3]. Recall that for the spaces S 2 (a)×S 2 (b) we assume a ≥ b. Using wG = c1 and (9.3.3), we read off from Propositions 9.2.1, 9.2.3 and 9.2.5 that for ruled symplectic 4-manifolds, wG (S 2 (a) × S 2 (b)) = wG (S 2 S 2 , ωab ) = b,

(9.3.13) √ wG (g (a) × S 2 (b)) = wG (g S 2 , ωab ) = min{ 2ab, b} if g ≥ 1. (9.3.14) , Since S 2 ωab = b in all cases, we see that the upper bound for the Gromov width predicted by the Nonsqueezing Theorem (9.3.12) and the volume condition is sharp for ruled symplectic 4-manifolds. Explicit maximal embeddings were given for g = 0 in

212


Figures 9.4 (a) and 9.12 (a) and for g ≥ 1 and a ≥ b in Figure 9.4 (a), but no explicit maximal embedding is known for g ≥ 1 and a < b. (ii) A locally trivial fibration V → M → S 2 of a closed connected symplectic manifold (M, ω) is called a symplectic fibration over S 2 if all fibres (V , ω|V ) are symplectic and mutually symplectomorphic. The area of such a fibration is naturally defined as , ωn Vol(M, ω) area(M, ω) = = , M . Vol(V , ω|V ) n V (ω|V )n−1 In [60, Proposition 1.20 (i)] McDuff proved the Nonsqueezing Theorem wG (M, ω) ≤ area (M, ω) provided that [ω|V ] vanishes on π2 (V ). The symplectic fibrations (S 2 S 2 , ωab ), for which wG (S 2 S 2 , ωab ) = b in view of (9.3.13) and area(S 2 S 2 , ωab ) = a, show that this assumption cannot be omitted. (iii) The Nonsqueezing Theorem (9.3.11) does not remain valid if the factor S 2 is replaced by a closed surface with positive genus. This was first noticed by Lalonde, [47], and follows from (9.3.14) as well as from Propositions 9.3.5 and 9.3.6. 3 9.3.4 Maximal packings of 4-dimensional ellipsoids. We finally construct some explicit maximal packings of 4-dimensional ellipsoids E(π, a) with a ≥ π . Proposition 9.3.8. (i) For each k ∈ N the ellipsoid E(π, kπ) admits an explicit full symplectic packing by k balls. a (ii) p1 (E(π, a)) = πa and p2 (E(π, a)) = min 2π a , 2π , and these packing numbers can be realized by explicit symplectic packings. The statement (i) was proved in [81, Theorem 6.3 (2)], and (ii) was proved in [54, Corolary 3.11]. Their embeddings are different from ours. Proof of Proposition 9.3.8. (i) Recall from Lemma 5.3.1 that we can think of B 4 (π ) ) as ) × 22 (1). The linear symplectic as 2 (π ) × 22 (1) and of E(π, kπ map (π, kπ 1 2 (x1 , x2 , y1 , y2 ) → x1 , kx2 , y1 , k y2 maps (π ) × 22 (1) to (π, kπ) × 2 1, k1 , and it is clear how to insert k copies of this set into (π, kπ) × 22 (1). (ii) The estimates p1 (E(π, a)) ≤ πa and p2 (E(π, a)) ≤ 2π a follow from the inclusion E(π, a) ⊂ Z 4 (π) and from Gromov’s Nonsqueezing Theorem, and the a follows from E(π, a) ⊂ B 4 (a) and from Gromov’s estimate p2 (E(π, a)) ≤ 2π 4 result p2 B (a) ≤ 21 stated in (9.1.7). The inclusion B 4 (π ) ⊂ E(π, a) shows that p1 (E(π, a)) = πa , and explicit symplectic packings of E(π, a) by two balls realizing a p2 (E(π, a)) = min 2π , 2 a 2π can be constructed as in the proof of (i).

9.4 Maximal packings in higher dimensions

9.4

213

Maximal packings in higher dimensions

In dimensions 2n ≥ 6, only few maximal symplectic packings by equal balls are known. 1. Balls and (CPn , ωSF ). As in dimension 4 we denote by ωSF the unique U(n + 1)invariant Kähler form on CPn whose integral over CP1 equals 1. The embedding (9.2.1) generalizes to all dimensions, and pk B 2n = pk (CPn , ωSF ) for all k, see [61, Remark 2.1.E]. Recall from (9.1.7) and (9.1.8) that k pk B 2n = n 2 pl n B 2n = 1

for 2 ≤ k ≤ 2n , for all l ∈ N.

An explicit maximal packing of (CPn , ωSF ) by k ≤ n + 1 balls was found by Karshon in [41], and explicit full packings of B 2n by l n balls for each l ∈ N were given by Traynor in [81]. Taking l = 2, any k balls of such a packing yield a maximal packing by k balls. The following different construction of an explicit full packing of B 2n by from l n equal balls is mentioned in [81, Remark 5.13]. Recall Lemma 5.3.1 that we can think of B 2n (π) as n (π)×2n (1) and of B 2n πl as n πl ×2n (1). The matrix diag l, . . . , l, 1l , . . . , 1l ∈ Sp(n; R) maps n πl × 2n (1) to n (π ) × 2n 1l . It is clear how to insert l n copies of n (π ) × 2n 1l into n (π ) × 2n (1). + 2. Products of balls, complex projective spaces and surfaces. Set n = di=1 ni and let a1 , . . . , ad ∈ π N. According to [61, Theorem 1.5.A], the product n CP 1 × · · · × CPnd , a1 ωSF ⊕ · · · ⊕ ad ωSF nd n1 n! admits a full symplectic packing by n1 ! ··· nd ! a1 · · · ad equal 2n-dimensional balls. These full packings can be constructed in an explicit way. Indeed, explicit full packings of B 2ni (ai ) by aini equal balls as in 1. above can be used to construct explicit full packings of B 2n1 (a1 ) × · · · × B 2nd (ad ) nd n1 n! by n1 ! ··· nd ! a1 · · · ad balls, see [45, Section 3.2]. In particular, there are explicit full packings of the polydisc P (a1 , . . . , an ) and of the products of surfaces g1 (a1 ) × · · · × gn (an ) with ai ∈ πN by n! a1 · · · an equal balls, see also [81, Section 4.1], [54, Theorem 4.1], and Figure 9.11 above for the case n = 2. An explicit packing construction in [54, Theorem 1.21] yields the lower bounds

224 p7 C 6 (π) ≥ 375

and

9 p8 C 6 (π ) ≥ . 16

214


The technique in the proof of Proposition 9.3.6 can be used to generalize Proposition 9.3.5 (i): For any closed surface endowed with an area form σ and any constant symplectic form ω on the 2n-dimensional torus T 2n , there exists a constant C > 0 such that p1 ( × T 2n , aσ ⊕ ω) ≥ C for all a ≥ 1, see [40, Theorem 3.1]. 3. Ellipsoids. Generalizing the result of Proposition 9.3.8 (ii), the packing numbers n a1n 2 p1 (E(a1 , . . . , an )) = a1 ···a of a and p2 (E(a1 , . . . , an )) = a1 ···a min a1n , a2n n n 2n-dimensional ellipsoid were computed and realized by explicit symplectic packings in [54, Corollary 3.11]. Remark 9.4.1. Karshon’s explicit packing of (CPn , ωSF ) by k ≤ n + 1 balls is maximal in the sense of (9.3.1). Since in dimensions ≥ 6 it is not yet known whether the space of symplectic embeddings of a closed ball into a larger ball is connected, all other explicit (and non-explicit) maximal symplectic packings known in dimensions ≥ 6 are maximal only in the sense of (9.3.2). 3 We conclude this chapter with addressing two widely open problems. As before, we consider connected symplectic manifolds of finite volume. Question 9.4.2. Which connected symplectic manifolds (M, ω) of finite volume satisfy pk (M, ω) = 1 for all k ≥ 1? 2 blown up at Examples are 2-dimensional manifolds, (CP √ , ωSF ) 2symplectically N ≥ 9 points with weights close enough to 1/ N and S (1)√× S 2 (1) symplectically blown up at N ≥ 8 points with weights close enough to 1/ N (see [7, Section 5]), the ruled symplectic 4-manifolds g (a) × S 2 (b) and (g S 2 , ωab ) with g ≥ 1 and b ≥ 2a and their symplectic blow-ups (see Proposition 9.2.5 and [7, Theorem 6.A]), as well as certain closed symplectic 4-manifolds described in [7, Theorem 2.F] and their symplectic blow-ups. A related problem is

Question 9.4.3. Which connected symplectic manifolds (M, ω) of finite volume satisfy p1 (M, ω) = 1? Examples different from the above ones are the ball B 2n and (CPn , ωSF ), and, more generally, the complement (CPn \ , ωSF ) of a closed complex submanifold of CPn (see [61, Corollary 1.5.B]).

Appendix

A The Extension after Restriction Principle In this appendix we give a proof of the well-known Extension after Restriction Principle for symplectic embeddings of bounded starshaped domains. This principle was used to derive the Symplectic Hedgehog Theorem from Sikorav’s Nonsqueezing Theorem for the torus T n and in the proof of Theorem 1 to bring Ekeland–Hofer capacities into play. We shall also discuss an extension of this principle to symplectic embeddings of unbounded starshaped domains, which was proved in [71]. A domain U in R2n is called starshaped if U contains a point p such that for every point z ∈ U the straight line between p and z is contained in U . A Hamiltonian diffeomorphism of R2n is called compactly supported if it is generated by a compactly supported Hamiltonian function H : R × R2n → R. Proposition A.1. Assume that ϕ : U → R2n is a symplectic embedding of a bounded starshaped domain U ⊂ R2n . Then for any subset A ⊂ U whose closure in R2n is contained in U there exists a compactly supported Hamiltonian diffeomorphism A of R2n such that A |A = ϕ|A . Proof. We closely follow [20]. We first prove that there exists a symplectic isotopy φ : R × U → R2n such that φ 0 = idU and φ 1 = ϕ, and then cut off a Hamiltonian function generating this isotopy. Step 1. Alexander’s trick. We can assume that 0 is a star point of U and that ϕ(0) = 0 because translations of R2n are symplectically isotopic to the identity. We can also assume that dϕ(0) = id because the linear symplectic group Sp(n; R) retracts onto U(n) (see [62, Section 2.2]) and is hence connected. Since U is starshaped with respect to 0 we can use “Alexander’s trick” and define a continuous path ϕ t , t ∈ R, of symplectic embeddings ϕ t : U → R2n by setting z if t ≤ 0, t ϕ (z) := 1 (A.1) t ϕ(tz) if t ≥ 0. The path ϕ t is smooth except possibly at t = 0. In order to smoothen ϕ t , we define the smooth function η : R → R by 0 if t ≤ 0, η(t) := 2 −2/t (A.2) e e if t ≥ 0,

216

Appendix

where e denotes the Euler number, and for t ∈ R and z ∈ U we set φ t (z) := ϕ η(t) (z).

(A.3)

Then φ t , t ∈ R, is a smooth path of symplectic embeddings φ t : U → R2n such that φ 0 = idU and φ 1 = ϕ. Step 2. Cutting off the isotopy. Since U is starshaped, it is contractible, and so the same holds true for all the open sets φ t (U ), t ∈ R. We therefore find a smooth time-dependent Hamiltonian function 0 H: {t} × φ t (U ) → R (A.4) t∈R

generating the path φ t , i.e., φ t is the solution of the Hamiltonian system t d t z ∈ U, t ∈ R, dt φ (z) = J ∇H t, φ (z) , φ 0 (z) = z,

z ∈ U.

(A.5)

Here, J denotes the standard complex structure defined by ω0 (z, w) = &J z, w',

z, w ∈ R2n .

Fix now a subset A of U whose closure A in R2n is contained in U . Since U is bounded, the set A is compact, and so the set 0 K= {t} × φ t (A) ⊂ R × R2n t∈[0,1]

is also compact and hence bounded. We therefore find a bounded - neighbourhood V of K which is open in R × R2n and is contained in the set t∈R {t} × φ t (U ). By Whitney’s Theorem, there exists a smooth function f on R × R2n which is equal to 1 on K and vanishes outside V . Since V is bounded, the function f H : R × R2n → R has compact support, and so the Hamiltonian system associated with f H can be solved for all t ∈ R. We define A to be the resulting time-1-map. Then A is a globally defined symplectomorphism of R2n with compact support, and since f ≡ 1 on K we have A |A = φ 1 |A = ϕ|A . The proof of Proposition A.1 is thus complete. 2

Addendum: An extension of the Extension after Restriction Principle to unbounded starshaped domains. In Proposition A.1 we assumed that U is bounded. In this addendum we discuss how essential this assumption is. Definition A.2. Consider a symplectic embedding ϕ : U → R2n of a starshaped domain U ⊂ R2n . We say that the pair (U, ϕ) has the extension property if for each subset A ⊂ U whose closure in R2n is contained in U there exists a Hamiltonian diffeomorphism A of R2n such that A |A = ϕ|A .

A The Extension after Restriction Principle

217

By Proposition A.1 the pair (U, ϕ) has the extension property whenever U is bounded. As the following example shows, this is not true for all symplectic embeddings of starshaped domains. Example A.3. We let U ⊂ R2 be the strip ]1, ∞[ × ] − 1, 1[. The symplectic em2 2 bedding ϕ : U → R defined by ϕ(x, y) = 1/x, −x y maps (k, 0) to (1/k, 0), k = 2, 3, . . . , and so there does not exist any subset A of U containing the set 3 {(k, 0) | k = 2, 3, . . . } for which ϕ|A extends to a diffeomorphism of R2 . Observe that if (U, ϕ) has the extension property, then ϕ is proper in the sense that each subset A ⊂ U whose closure in R2n is contained in U and whose image ϕ(A) is bounded is bounded. The map ϕ in Example A.3 is not proper in this sense. However, the map ϕ in the following example is proper in this sense, and still (U, ϕ) does not have the extension property. Example A.4. Let U ⊂ R2 be the strip R × ] − 1, 0[, and let

A = (x, y) ∈ U | y + 21 ≤ f (x) where f : R → 0, 21 is a smooth function such that / 1 2 − f (x) dx < ∞, R

(A.6)

cf. Figure A.1. Using the method used in Step 4 of Section 3.2 we find a symplectic embedding ϕ : U → R2 such that ϕ(x, y) = (x, y) if x ≥ 1

and ϕ(x, y) = (−x, −y) if x ≤ −1,

cf. Figure A.1. In view of the estimate (A.6) the component C of R2 \ ϕ(A) which contains the point (1, 0) has finite volume. Any symplectomorphism A of R2 such that A |A = ϕ|A would map the “upper” component of R2 \ A, which has infinite volume, to C. This is impossible. 3 Example A.4 shows that the assumption (A.7) on ϕ in Theorem A.6 below cannot be omitted. For technical reasons in the proof of Theorem A.6 we shall also impose a mild convexity condition on the starshaped domain U . The length of a smooth curve γ : [0, 1] → R2n is defined by / 1 γ (s) ds. length(γ ) := 0

On any domain U ⊂

R2n

we define a distance function dU : U × U → R by dU (z, z ) := inf {length(γ )}

where the infimum is taken over all smooth curves γ : [0, 1] → U with γ (0) = z and γ (1) = z . Then z − z ≤ dU (z, z ) for all z, z ∈ U .

218

Appendix y −1

0

1

U x A

−1

ϕ y

1 1 x −1

Figure A.1. A pair (U, ϕ) which does not have the extension property.

Definition A.5. We say that a domain U ⊂ R2n is a Lipschitz domain if there exists a constant λ > 0 such that dU (z, z ) ≤ λ|z − z |

for all z, z ∈ U.

Each convex domain U ⊂ R2n is a Lipschitz domain with Lipschitz constant λ = 1. Figure A.2 shows a starshaped domain in R2 with smooth boundary which is not a Lipschitz domain.

Figure A.2. A starshaped domain with smooth boundary which is not a Lipschitz domain.

B Flexibility for volume preserving embeddings

219

Theorem A.6. Assume that ϕ : U → R2n is a symplectic embedding of a starshaped Lipschitz domain U ⊂ R2n such that there exists a constant L > 0 satisfying ϕ(z) − ϕ(z ) ≥ L z − z for all z, z ∈ U. (A.7) Then the pair (U, ϕ) has the extension property. Outline of the proof. One first easily verifies that one can assume that 0 is a star point of U and that ϕ(0) = 0 and dϕ(0) = id. Then apply Alexander’s trick as in Step 1 of the proof of Proposition A.1 and consider a function H as in (A.4) satisfying (A.5). Fix a subset A of U whose closure in R2n is contained in U . If the set U is not bounded, then A does not need to be relatively compact, and so there might be no cut off f H of H whose Hamiltonian flow exists for all t ∈ [0, 1]. One therefore needs to extend the Hamiltonian H more carefully. One first verifies that assumption (A.7) on ϕ implies that ∇H is linearly bounded. Since it is not clear how to extend a linearly bounded gradient field to a linearly bounded gradient field, one then passes to the function H (t, w) G(t, w) = g (|w|) where g (|w|) = |w| for |w| large. The assumption that U is a Lipschitz domain implies that G is Lipschitz continuous in w and can hence be extended to a continuous ˆ in w ˆ on [0, 1]×R2n which is Lipschitz continuous in w. After smoothing G function G 3 3 3 to G one obtains a continuous extension H (t, w) = g (|w|) G(t, w) of H which may not be smooth in t but is smooth in w and has linearly bounded gradient. The Hamil3 can therefore be solved for all t ∈ [0, 1], and the tonian system associated with H resulting time-1-map A is a symplectomorphism of R2n such that A |A = ϕ|A . We refer to [71] for the detailed construction of A . In order to find a smooth Hamiltonian function generating A one proceeds as in the proof of Proposition A.1: Applying Alexander’s trick one obtains a smooth path t , t ∈ R, of symplectomorphisms of R2n such that 0 = id and 1 = A , and since R2n is contractible, there exists a 2 smooth Hamiltonian function H : R × R2n → R generating this path.

B

Flexibility for volume preserving embeddings

We endow each open subset U of Rn with the volume form 0

= dx1 ∧ · · · ∧ dxn .

A volume form on an arbitrary smooth n-dimensional manifold M is a nowhere vanishing differential n-form on M. A smooth embedding ϕ : U → M is called volume preserving if ϕ ∗ = 0 . The volume , forms 0 and orient, each open n subset of R and M. We write Vol(U, 0 ) = U 0 and Vol(M, ) = M . This appendix, which is taken from [72], contains a proof of

220

Appendix

Theorem B.1. Consider an open subset U of Rn and a connected n-dimensional manifold M endowed with a volume form . Then there exists a volume preserving embedding ϕ : U → M if and only if Vol(U, 0 ) ≤ Vol(M, ). If U is a bounded subset whose boundary has zero measure and if Vol(U, 0 ) < Vol(M, ), Theorem B.1 is an easy consequence of Moser’s deformation method, for which we refer to [66] or [39, Chapter 1.3]. Moreover, if U is a ball and M is compact, Theorem B.1 has been proved by Katok in [43]. Our point is that Theorem B.1 holds true for an arbitrary open subset of Rn and an arbitrary connected manifold even in case that the volumes are equal or infinite. Proof of Theorem B.1. Assume first that ϕ : U → M is a smooth embedding such that ϕ ∗ = 0 . Then / / / / ∗ Vol(U, 0 ) = ϕ = ≤ = Vol(M, ). 0 = U

U

ϕ(U )

M

Assume now that Vol(U, 0 ) ≤ Vol(M, ). We are going to construct a smooth embedding ϕ : U → M such that ϕ ∗ = 0 . We abbreviate the Lebesgue measure of a measurable subset V of Rn by |V |, so that |V | = Vol (V , 0 ) if V is open, and we write V for the closure of V in Rn . Moreover, we denote by Br the open ball in Rn of radius r centred at the origin. Proposition B.2. Assume that V is a non-empty open subset of Rn . Then there exists a smooth embedding σ : V → Rn such that |Rn \ σ (V )| = 0. Proof. We choose an increasing sequence V1 ⊂ V2 ⊂ · · · ⊂ Vk ⊂ Vk+1 ⊂ · · · of non-empty open subsets of V such that Vk ⊂ Vk+1 for k ≥ 1 and ∞ k=1 Vk = V . To fix the ideas, we assume that the Vk have smooth boundaries. Let σ1 : V2 → Rn be a smooth embedding such that σ1 (V1 ) ⊂ B1 and |B1 \ σ1 (V1 )| ≤ 2−1 . Since V1 ⊂ V2 and σ1 (V1 ) ⊂ B1 ⊂ B2 , we find a smooth embedding σ2 : V3 → Rn such that σ2 |V1 = σ1 |V1 and σ2 (V2 ) ⊂ B2 and |B2 \ σ2 (V2 )| ≤ 2−2 . Arguing by induction we find smooth embeddings σk : Vk+1 → Rn such that σk |Vk−1 = σk−1 |Vk−1 and σk (Vk ) ⊂ Bk and |Bk \ σk (Vk )| ≤ 2−k ,

(B.1)


221

k ≥ 1. The map σ : V → Rn defined by σ |Vk = σk |Vk is a well defined smooth embedding of V into Rn . Moreover, the inclusions σk (Vk ) ⊂ σ (V ) and the estimates (B.1) imply that |Bk \ σ (V )| ≤ |Bk \ σk (Vk )| ≤ 2−k , and so

n R \ σ (V ) = lim |Bk \ σ (V )| = 0. k→∞

This completes the proof of Proposition B.2.

2

Our next goal is to construct a smooth embedding of Rn into the connected n-dimensional manifold M such that the complement of the image has measure zero. If M is compact, such an embedding has been obtained by Ozols [68] and Katok [43, Proposition 1.3]. While Ozols combines an engulfing method with tools from Riemannian geometry, Katok successively exhausts a smooth triangulation of M. Both approaches can be generalized to the case of an arbitrary connected manifold M, and we shall follow Ozols. We abbreviate R>0 = {r ∈ R | r > 0} and R>0 = R>0 ∪ {∞}. We endow R>0 with the topology whose base of open sets consists of the intervals ]a, b[ ⊂ R>0 and the subsets of the form ]a, ∞] = ]a, ∞[ ∪ {∞}. We denote the Euclidean norm on Rn by ( · ( and the unit sphere in Rn by S1 . Proposition B.3. Endow Rn with its standard smooth structure, let µ : S1 → R>0 be a continuous function and let x n S = x ∈ R 0 ≤ (x( < µ (x( be the starshaped domain associated with µ. Then S is diffeomorphic to Rn . Remark B.4. The diffeomorphism guaranteed by Proposition B.3 may be chosen such that the rays emanating from the origin are preserved. 3 Proof of Proposition B.3. If µ(S1 ) = {∞}, there is nothing to prove. In the case that µ is bounded, Proposition B.3 has been proved by Ozols [68]. In the case that neither µ(S1 ) = {∞} nor µ is bounded, Ozols’s proof readily extends to this situation. Using his notation, the only modifications needed are: Require in addition that r0 < 1 and that 1 < 2, and define continuous functions µ˜ i : S1 → R>0 by

µ˜ i = min i, µ − i + δ2i . With these minor adaptations the proof in [68] applies word-by-word.

2

In the following we shall use some basic Riemannian geometry. We refer to [44] for basic notions and results in Riemannian geometry. Consider an n-dimensional complete Riemannian manifold (N, g). We denote the cut locus of a point p ∈ N by C(p).

222

Appendix

Corollary B.5. The maximal normal neighbourhood N \ C(p) of any point p in an n-dimensional complete Riemannian manifold (N, g) is diffeomorphic to Rn endowed with its standard smooth structure. Proof. Fix p ∈ N. We identify the tangent space (Tp N, g(p)) with Euclidean space Rn by a (linear) isometry. Let expp : Rn → N be the exponential map at p with respect to g, and let S1 be the unit sphere in Rn . We define the function µ : S1 → R>0 by µ(x) = inf{t > 0 | expp (tx) ∈ C(p)}. (B.2) Since the Riemannian metric g is complete, the function µ is continuous [44, VIII, Theorem 7.3]. Let S ⊂ Rn be the starshaped domain associated with µ. In view of Proposition B.3 the set S is diffeomorphic to Rn , and in view of [44, VIII, Theorem 7.4 (3)] we have expp (S) = N \ C(p). Therefore, N \ C(p) is diffeomorphic to Rn . 2 A main ingredient of our proof of Theorem B.1 are the following two special cases of a theorem of Greene and Shiohama [29]. Proposition B.6. (i) Assume that 1 is a volume form on the connected open subset U of Rn such that Vol(U, 1 ) = |U | < ∞. Then there exists a diffeomorphism ψ of U such that ψ ∗ 1 = 0 . (ii) Assume that 1 is a volume form on Rn such that Vol(Rn , there exists a diffeomorphism ψ of Rn such that ψ ∗ 1 = 0 .

1)

= ∞. Then

End of the proof of Theorem B.1. Let U ⊂ Rn and (M, ) be as in Theorem B.1. After enlarging U , if necessary, we can assume that |U | = Vol(M, ). We set N = M \ ∂M. Then |U | = Vol(M, ) = Vol(N, ). (B.3) Since N is a connected manifold without boundary, there exists a complete Riemannian metric g on N . Indeed, according to a theorem of Whitney [86], N can be embedded as a closed submanifold in some Rm . We can then take the induced Riemannian metric. A direct and elementary proof of the existence of a complete Riemannian metric was given by Nomizu and Ozeki in [67]. Fix a point p ∈ N. As in the proof of Corollary B.5 we identify (Tp N, g(p)) with n R and define the function µ : S1 → R>0 as in (B.2). Using polar coordinates on Rn we see from Fubini’s Theorem that the set 3 C(p) = {µ(x)x | x ∈ S1 } ⊂ Rn 3 has measure zero, and so C(p) = expp C(p) also has measure zero (see [13, VI, Corollary 1.14]). It follows that Vol(N \ C(p), ) = Vol(N, ).

(B.4)


223

According to Corollary B.5 there exists a diffeomorphism δ : Rn → N \ C(p). After composing δ with a reflection of Rn , if necessary, we can assume that δ is orientation preserving. In view of (B.3) and (B.4) we then have |U | = Vol(Rn , δ ∗ ).

(B.5)

Case 1. |U | < ∞. Let U1 , U2 , . . . be the countably many components of U . Then 0 < |Ui | < ∞ for each i. Given numbers a and b with −∞ ≤ a < b ≤ ∞ we abbreviate the “open strip” Sa,b = {(x1 , . . . , xn ) ∈ Rn | a < x1 < b}. In view of the identity (B.5) we have . |Ui | = |U | = Vol(Rn , δ ∗ ). i≥1

We can therefore inductively define a0 = −∞ and ai ∈ ] − ∞, ∞] by Vol(Sai−1 ,ai , δ ∗ ) = |Ui | . Abbreviating Si = Sai−1 ,ai we then have Rn = i≥1 Si . For each i ≥ 1 we choose an orientation preserving diffeomorphism τi : Rn → Si . In view of Proposition B.2 we find a smooth embedding σi : Ui → Rn such that Rn \σi (Ui ) has measure zero. After composing σi with a reflection of Rn , if necessary, we can assume that σi is orientation preserving. Using the definition of the volume, we can now conclude that Vol(Ui , σi∗ τi∗ δ ∗ ) = Vol(σi (Ui ), τi∗ δ ∗ ) = Vol(Rn , τi∗ δ ∗ ) = Vol(Si , δ ∗ ) = |Ui |. In view of Proposition B.6 (i) we therefore find a diffeomorphism ψi of Ui such that = 0. (B.6) ψi∗ σi∗ τi∗ δ ∗ We define ϕi : Ui → M to be the composition of diffeomorphisms and smooth embeddings ψi

σi

τi

δ

→ Rn − → Si ⊂ Rn − → N \ C(p) ⊂ M. Ui −→ Ui − The identity (B.6) implies that ϕi∗ = 0 . The smooth embedding 2 2 ϕ= ϕi : U = Ui → M therefore satisfies ϕ ∗

=

0.

224

Appendix

Case 2. |U | = ∞. In view of (B.5) we have Vol(Rn , δ ∗ ) = ∞. Proposition B.6 (ii) shows that there exists a diffeomorphism ψ of Rn such that ψ ∗δ∗

=

0.

(B.7)

We define ϕ : U → M to be the composition of inclusions and diffeomorphisms ψ

δ

→ Rn − → N \ C(p) ⊂ M. U ⊂ Rn − The identity (B.7) implies that ϕ ∗

C

=

0.

The proof of Theorem B.1 is complete. 2

Symplectic capacities and the invariants cB and cC

In the first two sections of this appendix we collect and prove those results on symplectic capacities which are relevant for this book. We refer to [4], [16], [20], [21], [36], [37], [38], [39], [48], [62], [77], [78], [82] for more information on symplectic capacities. In Section C.3 we compare the invariants cB and cC studied in Chapters 2 to 7 with symplectic capacities. In the last two sections we compare cB with the symplectic diameter and interpret cC as a symplectic projection invariant. C.1 Intrinsic and extrinsic symplectic capacities on R2n . We say that a subset S ⊂ R2n symplectically embeds into T ⊂ R2n if there exists a symplectic embedding ϕ of an open neighbourhood of S into R2n such that ϕ(S) ⊂ T . We again write Z 2n (a) for the symplectic cylinder D(a) × R2n−2 . Recall the Definition C.1. An intrinsic symplectic capacity on (R2n , ω0 ) is a map c associating with each subset S of R2n a number c(S) ∈ [0, ∞] in such a way that the following axioms are satisfied. A1. Monotonicity: c(S) ≤ c(T ) if S symplectically embeds into T . A2. Conformality: c(λS) = λ2 c(S) for all λ ∈ R \ {0}. A3. Nontriviality: 0 < c(B 2n (π)) and c(Z 2n (π )) < ∞. An intrinsic symplectic capacity c on R2n is normalized if A3 . Normalization: c(B 2n (π)) = c(Z 2n (π)) = π . Remark C.2. In view of the monotonicity axiom, symplectic embedding results lead to constraints on the space of intrinsic symplectic capacities on R2n . We refer to [16] for more information on this application of symplectic embedding results. 3

C Symplectic capacities and the invariants cB and cC

225

For any subset S of R2n we define the Gromov width wG (S) and the cylindrical capacity z(S) by wG (S) := sup{a | B 2n (a) symplectically embeds into S }, z(S) := inf {a | S symplectically embeds into Z 2n (a)}. It follows from Gromov’s Nonsqueezing Theorem stated in Section 1.2.2 that both wG and z are normalized intrinsic symplectic capacities on R2n . Another example of a normalized intrinsic symplectic capacity on R2n is the Hofer–Zehnder capacity, [39]. Proposition C.3. For any subset S of R2n and any normalized intrinsic symplectic capacity c on R2n we have wG (S) ≤ c(S) ≤ z(S). Proof. If B 2n (a) → S is a symplectic embedding, we conclude from monotonicity, conformality and normalization that a = B 2n (a) ≤ c(S). Taking the supremum, we find wG (S) ≤ c(S). Similarly, if S → Z 2n (a) is a symplectic embedding, we conclude from the axioms that c(S) ≤ c(Z 2n (a)) = a. Taking the infimum we find c(S) ≤ z(S). 2 If S is an open connected subset of R2 , the inequalities in Proposition C.3 are equalities, see Corollary C.10 (i). For n ≥ 2, however, the Symplectic Hedgehog Theorem stated in Section 1.2.2 shows that there exist bounded starshaped domains S ⊂ R2n with arbitrarily small Gromov width and z(S) = 1. Still, for an ellipsoid and, more generally, a convex Reinhardt domain all normalized symplectic capacities on R2n coincide, [37]. It is conjectured that this holds for any bounded convex domain, [85]. The subsequent proposition due to Viterbo, [85], confirms this conjecture up to a constant. Proposition C.4 (Viterbo). For any two normalized intrinsic symplectic capacities c and c on (R2n , ω0 ) and any bounded convex domain K ⊂ R2n we have c (K) ≤ n2 c(K). Moreover, c (K) ≤ n c(K) if K is centrally symmetric. Proof. The proof is similar to the proof of Proposition 8.2.2. Let E be John’s ellipsoid of K satisfying n1 E ⊂ K ⊂ E. According to [39, p. 40], there exists a linear symplectomorphism of R2n mapping E to an ellipsoid E = E(a1 , . . . , an ), and Gromov’s Nonsqueezing Theorem and Proposition C.3 show that c(E ) = c (E ), so that c(E) = c (E). Together with the monotonicity and conformality axioms we conclude c (K) ≤ c (E) = c(E) = n2 c n1 E ≤ n2 c(K).

226

Appendix

If K is centrally symmetric, then

√1 E n

⊂ K ⊂ E, so that c (K) ≤ n c(K).

2

As in Chapter 2 we denote by D(n) the group of symplectomorphisms of (R2n , ω0 ). We also recall the Definition C.5. An extrinsic symplectic capacity on (R2n , ω0 ) is a map c associating with each subset S of R2n a number c(S) ∈ [0, ∞] in such a way that the following axioms are satisfied. A1. Monotonicity: c(S) ≤ c(T ) if there exists ϕ ∈ D(n) such that ϕ(S) ⊂ T . A2. Conformality: c(λS) = λ2 c(S) for all λ ∈ R \ {0}. A3. Nontriviality: 0 < c(B 2n (π)) and c(Z 2n (π )) < ∞. An extrinsic symplectic capacity c on R2n is normalized if A3 . Normalization: c(B 2n (π)) = c(Z 2n (π)) = π . Comparing Definitions C.1 and C.5 we see that any intrinsic symplectic capacity on R2n is an extrinsic symplectic capacity on R2n . The converse is not true as we shall see in Proposition C.7. For any subset S of R2n we define wˆ G (S) := sup{a | there exists ϕ ∈ D(n) such that ϕ(B 2n (a)) ⊂ S }, zˆ (S) := inf{a | there exists ϕ ∈ D(n) such that ϕ (S) ⊂ Z 2n (a)}. It follows again from Gromov’s Nonsqueezing Theorem that both wˆ G and zˆ are normalized extrinsic symplectic capacities on R2n . Other examples of normalized extrinsic symplectic capacities on R2n are the first Ekeland–Hofer capacity c1 , which was introduced in [20] and used in the proof of Theorem 2.2.4, and the displacement energy e introduced in [38]. The simply connected hull Sˆ of a bounded subset S of R2n is the union of its closure S and the bounded components of R2n \ S. Proposition C.6. For any subset S of R2n and any normalized extrinsic symplectic capacity c on R2n we have wˆ G (S) ≤ c(S) ≤ zˆ (S). ˆ Moreover, wˆ G (S) = wG (S) and zˆ (S) = zˆ (S). Proof. The inequalities wˆ G (S) ≤ c(S) ≤ zˆ (S) are proved in the same way as Proposition C.3. The inequality wˆ G (S) ≤ wG (S) is obvious. In order to prove wˆ G (S) ≥ wG (S) we fix > 0 and assume that ϕ symplectically embeds B 2n (a) into S. According to Proposition A.1 there exists ∈ D(n) such that |B 2n (a−) = ϕ|B 2n (a−) . In


227

particular, B 2n (a − ) ⊂ S, and so wˆ G (S) ≥ a − . Since > 0 was arbitrary, we conclude that wˆ G (S) ≥ a. Taking the supremum over a we find wˆ G (S) ≥ wG (S). ˆ In view of the monotonicity of zˆ we have zˆ (S) ≤ zˆ (S). In order to prove 2n ˆ ≤ zˆ (S) we assume that ϕ ∈ D(n) embeds S into Z (a). Then ϕ(S) ⊂ Z 2n (a). zˆ (S) Moreover, if U is a bounded component of R2n \S, then ϕ(U ) is a bounded component ˆ ⊂ Z 2n (a), and hence of R2n \ ϕ(S), and so ϕ(U ) ⊂ Z 2n (a). It follows that ϕ(S) ˆ ≤ a. Taking the infimum over a we conclude that zˆ (S) ˆ ≤ zˆ (S). The proof of zˆ (S) Proposition C.6 is complete. 2 The next proposition shows that neither the first Ekeland–Hofer capacity c1 nor the displacement energy e nor the outer cylindrical capacity zˆ are intrinsic symplectic capacities on R2n . Let S 1 = (u, v) | u2 + v 2 = 1 be the unit circle. Proposition C.7. For the standard Lagrangian torus T n = S 1 × · · · × S 1 in R2n we have c1 (T n ) = e(T n ) = zˆ (T n ) = π while c(T n ) = 0 for any intrinsic symplectic capacity c on R2n . Proof. According to Théorème (b) on page 43 of [78] or [79] we have c1 (T n ) = π , Theorem 1.6 (i) in [38] implies c1 (T n ) ≤ e(T n ), and Proposition C.6 shows that e(T n ) ≤ zˆ (T n ). Since clearly zˆ (T n ) ≤ π , we conclude that π = c1 (T n ) ≤ e(T n ) ≤ zˆ (T n ) ≤ π and so the first assertion follows. In order to prove the second assertion we assume that c is an intrinsic symplectic capacity on R2n . Fix > 0. In view of Theorem C.8 (ii) below there exists a symplectic embedding ϕ : S 1 → D(). The product ϕ × · · · × ϕ : T n → D() × · · · × D() ⊂ Z 2n () symplectically embeds T n into Z 2n (), and so c(T n ) ≤ c Z 2n () = c Z 2n (π ) . π Since > 0 was arbitrary and since c Z 2n (π ) < ∞ it follows that c(T n ) = 0 as claimed. 2 Notice that for n ≥ 2, Proposition C.7 implies the remarkable Nonsqueezing Theorem mentioned in Section 1.2.2 and implying the Symplectic Hedgehog Theorem: Even though the volume of T n in R2n vanishes and T n does not bound an open set, there is no symplectomorphism of R2n mapping T n into Z 2n (π ).

228

Appendix

C.2 Symplectic capacities on R2 . If n ≥ 2 the computation of a capacity of a subset of R2n is usually a difficult problem. In this section we show that for subsets of R2 the situation is different. Since B 2 (π ) = Z 2 (π) we can assume that any (intrinsic or extrinsic) symplectic capacity c on R2 is normalized. Given a subset S of R2 we denote by Int S and S its interior and its closure and by Si the components of Int S. If S is bounded, Sˆ denotes again the simply connected hull of S. The outer Lebesgue measure of S is denoted by µ(S), and if S is measurable, µ(S) denotes its Lebesgue measure. Theorem C.8. Consider a subset S of R2 . (i) wG (S) = supi µ (Si ). (ii) z(S) = µ(S). ˆ If S is unbounded, zˆ (S) = ∞. (iii) If S is bounded, zˆ (S) = µ(S). Proof. (i) We abbreviate s := supi µ (Si ). If ϕ symplectically embeds D(a) into S, then ϕ(D(a)) ⊂ Si for some i, and so wG (S) ≤ s. In order to prove the reverse inequality we can assume that s ∈ ]0, ∞]. If s ∈ ]0, ∞[ we fix ∈ ]0, s[. Then there exists Si such that µ(Si ) > s − . According to Proposition 1 in Section 1.2.2 we find a symplectic embedding of D(s − ) into Si , and so wG (S) ≥ s − . Since ∈ ]0, s[ was arbitrary we conclude that wG (S) ≥ s. A similar argument shows that wG (S) = ∞ if s = ∞. Assertion (i) thus follows. (ii) If ϕ symplectically embeds S into D(a), then µ(S) = µ (ϕ(S)) ≤ µ(D(a)) = a. Taking the infimum over a we find that µ(S) ≤ z(S). In order to prove the reverse inequality we can assume that µ(S) < ∞. Fix > 0. In view of the definition of µ(S) there exists an open neighbourhood U of S such that µ(U ) ≤ µ(S) + . By Proposition 1 stated in Section 1.2.2 we find a symplectic embedding ϕ of U into D µ(S) + . Therefore z(S) ≤ µ(S) + . Since > 0 was arbitrary we conclude that z(S) ≤ µ(S). Assertion (ii) thus follows. (iii) Using Proposition C.6, the identity in (ii) and the fact that Sˆ is measurable we find ˆ ≥ z(S) ˆ = µ(S) ˆ = µ(S). ˆ zˆ (S) = zˆ (S) ˆ we therefore need to show zˆ (S) ˆ ≤ µ(S). ˆ By construcIn order to prove zˆ (S) ≤ µ(S) 2 ˆ ˆ tion, S is compact and R \ S is connected. Therefore, each path-component of Sˆ is ˆ is finite. Fix > 0. simply connected. Since Sˆ is compact, µ(S) Lemma C.9. There exists a simply connected open neighbourhood W of Sˆ such that ˆ + 2. µ(W ) ≤ µ(S)


229

Proof. We say that a subset of R2 is elementary if it is the union of finitely many open rectangles in R2 . By definition of µ and since Sˆ is compact we find an elementary ˆ + . Let U be one of the finitely subset R of R2 such that Sˆ ⊂ R and µ(R) ≤ µ(S) many components of R. The elementary set U is diffeomorphic to an open disc from which k closed discs have been removed. Since Sˆ ∩ U is a compact subset of U all of whose path-components are simply connected, the set U \ Sˆ is open and connected. We therefore find k curves γi in U \ Sând elementary neighbourhoods Ni of γi such that the elementary subset V := U \ Ni is a simply connected neighbourhood of ˆ Applying this construction to each component of R we obtain disjoint simply U ∩ S. ˆ We finally connected elementary subsets V1 , . . . , Vl of R whose union contains S. choose l − 1 curves γ i in R2 \ Vi and elementary neighbourhoods N i of γ i such that the elementary set W :=

l 0 i=1

Vi ∪

l−1 0

Ni

i=1

- i is simply connected and such that µ N ≤ . Then W is a simply connected ˆ neighbourhood of S and 0

µ(W ) ≤ µ

Lemma C.9 thus follows.

0 ˆ + 2. Vi + µ N i ≤ µ(R) + ≤ µ(S) 2

Let W be a simply connected open neighbourhood of Sˆ as guaranteed by Lemma C.9. Then W is diffeomorphic to D (µ(W )). According to Proposition B.6 (i) we ˆ is a compact therefore find a symplectomorphism ϕ : W → D (µ(W )). Since ϕ(S) subset of D (µ(W )), we find a ∈ ]0, µ(W )[ such that ϕ(S) ⊂ D(a). According to Proposition A.1 we find a symplectomorphism of R2 such that |D(a) = ϕ −1 |D(a) . The map −1 is then a symplectomorphism of R2 which embeds Sˆ into D(a). It ˆ ≤ a ≤ µ(W ) ≤ µ(S) ˆ + 2. Since > 0 was arbitrary we conclude follows that zˆ (S) ˆ ≤ µ(S). ˆ This completes the proof of the first claim in assertion (iii). that zˆ (S) If S is unbounded, then ϕ(S) is unbounded for any diffeomorphism ϕ of R2 , and so zˆ (S) = ∞. Assertion (iii) thus follows, and so the proof of Theorem C.8 is complete. 2

Corollary C.10. (i) Assume that S is a subset of R2 such that Int S is connected and µ (Int S) = µ(S). Then c(S) = µ(S) for any normalized intrinsic symplectic capacity c on R2 . If in addition µ (Int S) is finite, then S is measurable and so c(S) = µ(S). (ii) Assume that S is a bounded subset of R2 such that Int S is connected and ˆ Then S is measurable and c(S) = µ(S) for any normalized µ (Int S) = µ(S). extrinsic symplectic capacity c on R2 .

230

Appendix

Proof. Let c be any normalized intrinsic symplectic capacity on R2 . In view of Proposition C.3, (i) and (ii) in Theorem C.8 and the assumptions on S we have wG (S) ≤ c(S) ≤ z(S) = µ (S) = µ (Int S) = wG (S) and so c(S) = µ(S). If µ (Int S) is finite, the identities µ (Int S) = µ(S) = inf { µ(U ) | S ⊂ U, U open } , the monotonicity of µ and the additivity of µ imply that µ (S \ Int S) = 0. Therefore S = Int S ∪ S \ Int S is measurable, and so c(S) = µ(S). ˆ is finite. This and the inclusion S ⊂ Sˆ (ii) Since S is bounded, µ (Int S) = µ(S) yield ˆ − µ(Int S) = 0. µ(S \ Int S) ≤ µ(Sˆ \ Int S) = µ(S) It follows that S is measurable and that µ(S) = µ (Int S). Let now c be any normalized extrinsic symplectic capacity on R2 . In view of Proposition C.6, (i) and (iii) in Theorem C.8 and the assumptions on S we have ˆ = µ (Int S) = wG (S) wG (S) = wˆ G (S) ≤ c(S) ≤ zˆ (S) = µ(S) and so c(S) = µ(Int S) = µ(S) as claimed.

2

We recall that Gromov’s Nonsqueezing Theorem states that a ball B 2n (a) symplectically embeds into a cylinder Z 2n (A) only if A ≥ a. Corollary C.11. Fix a ∈ ]0, π ] and a normalized extrinsic symplectic capacity c on R2 . Then the Nonsqueezing Theorem is equivalent to the identity inf c p ϕ(B 2n (a)) = a (C.1) ϕ

where ϕ varies over all symplectomorphisms of R2n which embed B 2n (a) into Z 2n (π ), and where p : Z 2n (π) → B 2 (π) is the projection. Proof. Assume first that the Nonsqueezing Theorem does not hold. Then there exist a > 0 and A < a and a symplectic embedding ϕ : B 2n (a ) → Z 2n (A). According to Proposition A.1 we find a ∈ ]A, a [ and a symplectomorphism of R2n such that B 2n (a ) ⊂ Z 2n (A).

Notice that aa A < a ≤ π. We denote the dilatation z → aa z of R2n by α. The 2n 2n composition α −1 a α is2na symplectomorphism of R which embeds B (a) into 2n the subset Z a A of Z (π), i.e., a p (α −1 α) B 2n (a) ⊂ D A . a


231

The axioms for the normalized symplectic capacity c on R2 imply that a c p (α −1 α) B 2n (a) ≤ A. a a a A

< a we conclude that (C.1) does not hold. Assume now that (C.1) does not hold. Since c p B 2n (a) = c(D(a)) = a we then find a symplectomorphism ϕ of R2n and > 0 such that (C.2) c p ϕ B 2n (a) ≤ a − 2. Since

We abbreviate U := p ϕ B 2n (a) . Since U is open and connected, from Corollary C.10 (i), Proposition C.6 and the estimate (C.2) it follows that z(U ) = µ(U ) = wG (U ) = wˆ G (U ) ≤ c(U ) ≤ a − 2. We therefore find a symplectic embedding ψ : U → D(a − ). The composition (ψ × id2n−2 ) ϕ symplectically embeds B 2n (a) into Z 2n (a − ) and so the Nonsqueezing Theorem does not hold. 2 According to Proposition C.7 we have c(S 1 ) = 0 for any intrinsic symplectic capacity c on R2 and c(S 1 ) = π for the first Ekeland–Hofer capacity, the displacement energy and the outer cylindrical capacity. The following result, which was pointed out to me by David Hermann, shows that there exist other normalized extrinsic symplectic capacities on R2 . Proposition C.12 (D. Hermann). For any a ∈ [0, π ] there exists a normalized extrinsic symplectic capacity c on R2 such that c(S 1 ) = a. Proof. Fix a ∈ [0, π ]. For r ≥ 0 we set c (rD(π )) := r 2 π

and

c(rS 1 ) := r 2 a,

and given an arbitrary subset S ⊂ R2 we set

c(S) := max wˆ G (S), σ (S) where

wˆ G (S) := sup r 2 π | there exists ϕ ∈ D(2) such that ϕ (rD(π )) ⊂ S ,

σ (S) := sup r 2 a | there exists ϕ ∈ D(2) such that ϕ(rS 1 ) ⊂ S . Then c is a well-defined normalized extrinsic symplectic capacity on R2 , and 2 c(S 1 ) = a.

232

Appendix

C.3 Comparison of cB and cC with symplectic capacities. R2n we define the symplectic invariants cB and cC by

For any subset S of

cB (S) := inf{a | S symplectically embeds into B 2n (a)},

(C.3)

cC (S) := inf{a | S symplectically embeds into C (a)}.

(C.4)

2n

For n = 1, the invariants cB and cC coincide with the cylindrical capacity z. For n ≥ 2, the invariants cB and cC fulfil all axioms of a normalized intrinsic symplectic capacity on R2n except that they are infinite for the symplectic cylinder. Recall from Proposition C.3 that the cylindrical capacity z is the largest normalized intrinsic symplectic capacity on R2n . Proposition C.13. For any subset S of R2n we have z(S) ≤ cC (S) ≤ cB (S). Proof. The claim follows from the inclusions B 2n (a) ⊂ C 2n (a) ⊂ Z 2n (a).

2

Remark C.14. If n ≥ 2, the inequalities in Proposition C.13 are in general not equalities: For a polydisc P = P 2n (π, . . . , π, a) with a > π we have z(P ) = π < cC (P ). Moreover, (2.2.3) and (2.3.1) imply cC (C 2n (π )) = π < cB (C 2n (π )) = nπ . (This example is extremal in the sense that for any S ⊂ R2n we have cB (S) ≤ n cC (S).) 3 C.4 Comparison of cB with the symplectic diameter. Recall that for any subset S of R2n the diameter d(S) is defined by

d(S) := sup |z − z | | z, z ∈ S . Symplectifying the Euclidean invariant d(S) we obtain the symplectic invariant ds (S) defined by

ds (S) := inf d ϕ(S) | ϕ symplectically embeds S into R2n . For convenience we shall work with the symplectic invariant δ(S) defined by ds (S) 2 δ(S) := π 2

(C.5)

rather than with ds (S). Theorem C.15. For any subset S of R2n we have δ(S) ≤ cB (S) ≤ and if n = 1, then δ(S) = cB (S) = µ(S).

4n δ(S) 2n + 1

(C.6)


233

Proof. Fix S ⊂ R2n . The circumradius R(S) of S is the radius of the smallest ball containing S, i.e.,

R(S) := inf R | there exists w ∈ R2n such that τw (S) ⊂ B 2n (π R 2 ) where τw denotes the translation z → z + w of R2n . Symplectifying the Euclidean invariant R(S) we obtain the symplectic invariant

Rs (S) := inf R(ϕ(S)) | ϕ symplectically embeds S into R2n . Notice that each translation τw of R2n is symplectic. Comparing the definition (C.3) of the invariant cB (S) with the definitions of R(S) and Rs (S) we therefore find that cB (S) = π (Rs (S))2 . The main ingredient of the proof of Theorem C.15 are the inequalities : 1 n d(T ) ≤ R(T ) ≤ d(T ) 2 2n + 1

(C.7)

(C.8)

valid for any subset T of R2n . While the left inequality in (C.8) is obvious, the right inequality is the main content of Jung’s Theorem [14, Chapter 2, Theorem 11.1.1]. Applying (C.8) to all symplectic images ϕ(S) of S in R2n and taking the infimum we find that : n 1 ds (S) ≤ Rs (S) ≤ ds (S). 2 2n + 1 These inequalities, the definition (C.5) and the identity (C.7) imply (C.6). Assume now S ⊂ R2 . The left inequality in (C.6) and Theorem C.8 (ii) show that δ(S) ≤ cB (S) = z(S) = µ(S). In order to show that δ(S) ≥ µ(S) we assume that ϕ symplectically embeds S into R2 and denote the convex hull of ϕ(S) by conv ϕ(S). Applying the Bieberbach inequality [14, Chapter 2, Theorem 11.2.1] to ϕ(S) we find d (ϕ(S)) 2 π ≥ µ (conv ϕ(S)) ≥ µ (ϕ(S)) = µ(S). 2 Since ϕ was arbitrary it follows that δ(S) ≥ µ(S). We conclude that δ(S) = cB (S) = µ(S). The proof of Theorem C.15 is complete. 2 Remarks C.16. 1. Assume that n≥ 2. It follows from inequality [14, 2n the Bieberbach 2n Chapter 2, Theorem 11.2.1] that δ B (π) = cB B (π ) and so the left inequality in (C.6) is sharp. We do not know, however, whether the right inequality in (C.6) is sharp. 2 2. The invariants δ = π d2s and cB = π(Rs )2 are examples of “Euclidean invariants for symplectic domains” as studied in [76]. 3

234

Appendix

C.5 The invariant cC as a symplectic projection invariant. As was pointed out in [26, p. 580], symplectic capacities on R2n measure to some extent the area of two dimensional symplectic projections of a set. We are now going to make this point precise for the invariants z and cC . For i = 1, . . . , n we set

Ei = z ∈ Cn | zj = 0 for j = i , and we denote the orthogonal projection R2n → Ei by pi . The Lebesgue measure on Ei is denoted µi , and the outer Lebesgue measure on Ei is denoted µi . Given any subset S of R2n we set sm (S) = inf min µi (pi (ϕ(S))) , ϕ 1≤i≤n

sM (S) = inf max µi (pi (ϕ(S))) , ϕ 1≤i≤n

where ϕ varies over all symplectic embeddings of S into R2n . Remarks C.17. 1. If S ⊂ R2n is Lebesgue measurable, then so are the sets pi (ϕ(S)) ⊂ Ei , i = 1, . . . , n, and so sm (S) = inf min area (pi (ϕ(S))) , ϕ 1≤i≤n

sM (S) = inf max area (pi (ϕ(S))) . ϕ 1≤i≤n

2. Composing an “infimal” embedding ϕ : S → R2n in the definition of sm (S) with the linear symplectomorphism (z1 , z2 , . . . , zi−1 , zi , zi+1 , . . . , zn ) → (zi , z2 , . . . , zi−1 , z1 , zi+1 , . . . , zn ) we find that for all subsets S of R2n , sm (S) = inf µ1 (p1 (ϕ(S))) . ϕ

In view of the Extension after Restriction Principle Proposition A.1, the restriction of sm to Ks2n thus agrees with the invariant πs considered in Section 8.2. 3 Theorem C.18. For any subset S of R2n we have z(S) = sm (S) and cC (S) = sM (S). Proof. We show z(S) ≤ sm (S). We may assume that sm (S) is finite. Fix > 0. By Remark C.17.2 there exists a symplectic embedding ϕ : S → R2n such that µ1 (p1 (ϕ(S))) ≤ sm (S) + . According to Theorem C.8 (ii) we find a symplectic

D Computer programs

235

embedding ψ of p1 (ϕ(S)) into D (sm (S) + 2). The composition (ψ × id2n−2 ) ϕ symplectically embeds S into Z 2n (sm (S) + 2). Since > 0 was arbitrary, we conclude z(S) ≤ sm (S). The reverse inequality sm (S) ≤ z(S) is obvious. 2 The equality cC (S) = sM (S) is proved in the same way as z(S) = sm (S). Remark C.19. The invariants sm = z and sM = cC are special symplectic projection invariants as studied in [76]. 3

D

Computer programs

This appendix contains computer programs computing the functions fEB and fEC defined in 4.3.1 and 4.4.1 and drawn in Figures 1.1 and 7.1 and in Figure 4.13. The Mathematica programs below can be found under ftp://ftp.math.ethz.ch/pub/papers/schlenk/folding.m For convenience, in the programs (but not in the text) both the u-axis and the capacityaxis are rescaled by the factor 1/π . D.1 The estimate fEB . We fix a and u1 and try to embed the ellipsoid E(π, a) into the ball B 4 (2π + (1 − 2π/a)u1 ) by the multiple folding procedure specified in 4.3.1. If u1 is admissible, we set A(a, u1 ) = 2π + (1 − 2π/a)u1 and A(a, u1 ) = a otherwise. A[a_, u1_] := Block[{A=2+(1-2/a)u1}, i = 2; ui = (a+1)/(a-1)u1-a/(a-1); ri = a-u1-ui; li = ri/a; While[True, Which[EvenQ[i], If[ri < ui, Return[A], If[ui ui+1 and ui+1 ≤ 2 li+1 . We denote the infimum of the admissible u1 ’s again by u0 = u0 (a). Then A(a, u1 ) = a for u1 < u0 , and A(a, u1 ) = 2π + (1 − 2π/a)u1 is a linear increasing function for u1 > u0 . Since, by (4.3.7), we can assume that u0 ≤ a/2, we have A(a, u0 ) ≤ A(a, a/2) = π + a/2 < a. Therefore, u0 is found up to accuracy acc/2 by the following bisectional algorithm. u0[a_, acc_] := Block[{}, b = a/(a+1); c = a/2; u1 = (b+c)/2; While[(c-b)/2 > acc/2, If[A[a,u1] < a, c=u1, b=u1]; u1 = (b+c)/2 ]; Return[u1] ] Here the choice b = aπ/(a + π ) is also based on (4.3.7). Up to accuracy acc, the resulting estimate fEB (a) is given by fEB[a_, acc_] := 2 + (1-2/a)u0[a,acc]. D.2 The estimate fEC . We fix a > π. As we have seen in Remark 4.4.1.1, folding only once does not yield an optimal embedding. We therefore fold at least twice and

D Computer programs

237

aπ a , 2 . We first calculate the height of the image of T (a, π ) hence choose u1 ∈ a+π determined by a and u1 . The following program is best understood by looking at Figure 4.12. h[a_, u1_] := Block[{l1=1-u1/a}, i = 2; ui = (a+1)/(a-1)u1-a/(a-1); ri = a-u1-ui; li = ri/a; hi = 2l1; While[ri > u1+l1 - li, i++; ui = (a+1)/(a-1)ui; ri = ri-ui; lj = li; li = ri/a; If[EvenQ[i], hi = hi+2lj] ]; Which[EvenQ[i], hi = hi+li, OddQ[i], hi = hi+Max[lj,2li] ]; Return[hi] ] As explained in 4.4.1, the optimal folding point u0 (a) is the u-coordinate of the unique intersection point of the graphs of h(a, u1 ) and w(a, u1 ). It can thus be found by a bisectional algorithm. u0[a_, acc_] := Block[{}, b = a/(a+1); c = a/2; u1 = (b+c)/2; While[(c-b)/2 > acc/2, If[h[a,u1] > 1+(1-1/a)u1, b=u1, c=u1]; u1 = (b+c)/2 ]; Return[u1] ] Up to accuracy acc, the resulting estimate fEC (a) is given by fEC[a_, acc_] := 1+(1-1/a)u0[a,acc].

238

Appendix

E

Some other symplectic embedding problems

In this book we studied only some few symplectic embedding problems. In this appendix we briefly describe some other embedding problems in symplectic geometry and topology. 1. Local Nonsqueezing. Recall from the introduction that Z 2n (π ) denotes the open cylinder D(π) × R2n−2 in (R2n , ω0 ), and let Z 2n (π ) be its closure. Assume that K ⊂ Z 2n (π ) is a compact subset with smooth boundary ∂K such that ∂K ∩ ∂Z 2n (π ) is non-empty. Can K be mapped into Z 2n (π) via a symplectic isotopy? This problem was almost fully solved in [50], and interesting variants were studied in [69]. 2. Embeddings of Lagrangian submanifolds. Let K be a closed n-dimensional manifold, and let (M, ω) be a 2n-dimensional symplectic manifold. By the Whitney Embedding Theorem, K smoothly embeds into M. An n-dimensional submanifold L of (M, ω) is called Lagrangian if ω vanishes on T L. Does K embed as a Lagrangian submanifold into (M, ω)? This is a famous problem in symplectic topology, and for the state of the art we refer to [1], [11], [63, Section 9.3] and the references therein. 3. Neighbourhoods of Lagrangian submanifolds. Consider a closed Lagrangian submanifold L of a 2n-dimensional symplectic manifold (M, ω). By a theorem of Weinstein, there exists a symplectomorphism from a neighbourhood of L in (M, ω) to a neighbourhood of the+ zero-section 0L of the cotangent bundle (T ∗ L, ωcan ) mapping L to 0L . Here, ωcan = ni=1 dpi ∧ dqi is the canonical symplectic form on T ∗ L. How large can this neighbourhood be chosen? Here, the size of a neighbourhood can be measured in many ways. (i) If L comes with a Riemannian metric g, one can look at the ball bundles ∗ Tr L = {(q, p) ∈ T ∗ L | gq (p, p) < r} and study s sizeg (M, L) = sup π r 2 | (Tr∗ L, 0L → (M, L)}. (ii) The size of a neighbourhood of L as above can be measured by any symplectic capacity as defined in [39]. (iii) A more local size of a neighbourhood of L in M is given by the relative Gromov width s

wG (M, L) = sup a | B 2n (a), B 2n (a) ∩ Rn (x) → (M, L) considered recently by Barraud and Cornea in [3].

E Some other symplectic embedding problems

239

4. The isotopy problem. Given a natural number k, a compact subset K of (R2n , ω0 ) homeomorphic to a closed 2n-dimensional ball, and a 2n-dimensional connected symplectic manifold (M, ω), let Embk (K, M, ω) be the space of symplectic embeddings of k disjoint copies of K into (M, ω) endowed with the C ∞ -topology. Is Embk (K, M, ω) connected? (i) We say that K is starshaped if its interior is starshaped. The following proposition for k = 1 is due to Banyaga, [2], and for arbitrary k was pointed out to me together with the proof by Paul Biran. Proposition E.1. If K is starshaped, then Embk (K, R2n , ω0 ) is connected for every k. The proof is given below. (ii) The camel space in R2n with eye of width a is the subset C 2n (a) = {x1 < 0} ∪ {x1 > 0} ∪ B 2n (a) of (R2n , ω0 ). If b > a, the “symplectic camel” B 2n (b) cannot pass the eye of width a, so that Embk B 2n (b), C 2n (a), ω0 has at least 2k components, see [22], [65], [84]. (iii) For some closed symplectic 4-manifolds such as CP2 and ruled symplectic 4-manifolds and for the 4-ball it is known that Embk B 4 (a), M, ω is connected for all k, see [58]. (iv) Denote by P (a, b) the closure of the polydisc P (a, b) = D(a) × D(b). If a + b > π and a, b < π , then Emb1 P (a, b), P (π, π ), ω0 has at least two components, see [26]. No similar result is known in higher dimensions. (v) Nothing is known for ellipsoids if (M, ω) is closed or a bounded subset of (R2n , ω0 ), nor is anything known for any K if (M, ω) is a closed symplectic manifold of dimension ≥ 6. Proof of Proposition E.1. Let ϕ=

k 2 i=1

ϕi :

k 2

K → R2n

i=1

be a symplectic embedding of k copies of the starshaped set K. By definition, the embeddings ϕi extend to disjoint symplectic embeddings of an open neighbourhood of K. As is easy to see, this neighbourhood can be chosen starshaped. We can assume that 0 is a star point of K. Since the linear symplectic group Sp(n; R) is connected, we can also assume that dϕ1 (0) = id. Using Alexander’s trick as in Step 1 of the proof of Proposition A.1 and extending the thereby obtained symplectic isotopy to R2n as in Step 2 of that proof, we obtain a (compactly supported) symplectic isotopy t , t ∈ [0, 1], of R2n such that 0 = id and φ1 := 1 ϕ1 = idK .

(E.1)

240

Appendix

If k = 1 we are done. If k ≥ 2, set φi = 1 ϕi and pi = φi (0), Ki = φi (K). After slightly translating Ki for i = 3, . . . , k, if necessary, we can assume that the open rays C(pi ) = {t pi | t > 0} are disjoint for i = 2, . . . , k. For i = 1, . . . , k we consider now the symplectic isotopy φit (z) =

1 φi (1 − t)z , 1−t

z ∈ K, t ∈ [0, 1[,

starting at φi . Since the embeddings ϕi are disjoint, the embeddings φit are disjoint for each t. In view of (E.1) the isotopy φ1t is constant. To describe the other isotopies, we set Kit = φit (K) and consider the punctured cones

C Kit := tz | t > 0, z ∈ Kit . For t → 1, the cone C Kit “converges” to the ray C(pi ). Since these rays are disjoint, we find t0 ∈ [0, 1[ such that the cones Ci := C Kit0 are mutually disjoint for i = 2, . . . , k. Notice that the cones Ci have non-empty interior since K was 1 assumed to be homeomorphic to a 2n-ball. Set qi = φit0 (0) = 1−t pi . Using 0 again that Sp(n; R) is connected and applying Alexander’s trick, we find for each i = 2, . . . , k a symplectic isotopy ψi : [0, 1] × K → R2n such that ψit (0) = qi for all t ∈ [0, 1] and ψi0 (z) = φit0 (z)

and

ψi1 (z) = z + qi

for all z ∈ K.

Choose ρi so large that the image ψi ([0, 1] × K) is contained in the ball Bρi (qi ) of radius ρi centred at qi . Since Ci has non-empty interior, we find si > 0 such that the translate Bρi (qi + si qi ) of Bρi (qi ) is contained in Ci and is disjoint from K. Translating Kit0 by si qi and finally applying the symplectic isotopy [0, 1] × K → R2n ,

(t, z) → ψit (z) + si qi ,

whose image lies in Bρi (qi + si qi ), we end up with a symplectic isotopy ϕt =

k 2 i=1

ϕit : [0, 1] ×

k 2

K → R2n

i=1

starting at ϕ 0 = ϕ and ending at ϕ 1 with ϕ11 = id and ϕi1 (z) = z + qi + si qi for i = 2, . . . , k. Since ϕ11 (K) = K and since the sets ϕi1 (K) for i = 2, . . . , k lie in embedding ψ 1 obtained in the mutually disjoint cones Ci , it is clear that a symplectic 1k this way from another symplectic embedding ψ : i=1 K → R2n is symplectically 2 isotopic to ϕ 1 . The proof of Proposition E.1 is thus complete.

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Index

C 0 -stability for symplectomorphisms, 5 action of a closed characteristic, 10 action spectrum, 11 simple, 27 adjusting the fibres, 46, 56 admissible family of loops, 32 Alexander’s trick, 215, 239 area preserving embedding construction of, 32–37, 40–42, 45–47, 97–98, 116, 117, 137, 140–141, 170–172, 176 Arnold conjecture for the torus, 5 Arnold, V., 5 Banyaga, A., 239 Barraud, J. F., 238 Basic Problem, 6 Betke, U., 192 Biran, P., 21, 188–190, 194, 209, 239 bisectional algorithm, 236, 237 Blichfeldt, H. F., 193 camel space, 239 canonical transformation, 3 characteristic foliation, 24 characteristic line bundle, 10, 24 Cieliebak, K., 24 closed characteristic, 10 Cohn, H., 193 computer calculations, estimates suggested by, 70, 76, 157, 158 computer programs, 235–237 Conley, C., 5, 190 contact type boundary, 24

convex domain, 165–168 Cornea, O., 238 cylindrical capacity, 167, 225, 228, 234 Darboux’s Theorem, 18 domain, 6 convex, 165–168 Lipschitz, 218 starshaped, 6–8, 24, 166, 215, 219, 225, 239 very connected, 147 Ekeland, I., 8–11, 25, 29 Ekeland–Hofer capacities, 9, 14, 15, 25 of symplectic ellipsoids, 11, 25 of symplectic polydiscs, 29 Eliashberg, Y., 5, 24, 239 Elkies, N., 193 Euclidean packing, 191 density, highest, 192–193 Extension after Restriction Principle, 7, 8, 24, 215 for unbounded starshaped domains, 216 extension property, 216, 219 Fejes Tóth, L., 192 flexibility for symplectic embeddings of skinny shapes, 19 for symplectic immersions, 6 for volume preserving embeddings, 7, 18, 219–224 Floer, A., 9, 27, 239 floor, 52, 56, 57, 174 folding in the base, 44, 55

248 full symplectic packing, 188 of (S 2 S 2 , ω(2m−l)2l ) by l(2m − l) balls, 208 of g (m) × h (1) by 2ml 2 balls, 205 of B 4 by l 2 balls, 200 of B 2n by l n balls, 213 of E(π, kπ) by k balls, 212 Gauss, C. F., 192 Greene, R., 222 Gritzmann, P., 192 Gromov width, 189, 225, 228 higher, 189, 190 Gromov’s Alternative, 5 Gromov’s Nonsqueezing Theorem, 7 Gromov, M., 5, 7, 12, 21, 188, 190, 239 Hales, T., 192 Hamiltonian diffeomorphism of R2n , 2, 6 diffeomorphism of R2n compactly supported, 215 energy of, 187 flow, 2 system, 2 vector field, 2, 180 Henk, M., 192 Hermann, D., 8, 225, 231 Hofer distance, 187 Hofer, H., 8–11, 24, 25, 27, 29, 227, 239 Hug, D., 166 intersection invariant, 166 symplectic, 167, 168 isotopy problem, 239 Jiang, M.-Y., 210, 214 John’s ellipsoid, 166, 225 Jung, H. W., 233 Karshon, Y., 21, 189, 195, 213 Katok, A., 220, 221

Index

Keppler conjecture, 192 Kruglikov, B., 21, 189, 213 Kuperberg, W., 191 Lagrangian submanifold, 238 Lalonde, F., 14, 31, 161, 164, 195, 211, 212, 238 lift, 13 lifting the fibres, 42, 55 Liouville’s Theorem, 3 Maley, F. M., 21, 189, 212–214 Mastrangeli, J., 21, 189, 212–214 McDuff, D., 14, 19–21, 27, 31, 124, 161, 164, 188–190, 193–196, 200, 211, 213, 214, 238, 239 Moser, J., 220 Nagata conjecture, 194 Nomizu, K., 222 Nonsqueezing Theorem, 7, 8, 20, 211, 227, 230 generalizations, 211–212 local, 161, 238 optimal world, 35, 37 Ozeki, H., 222 Ozols, V., 221 packing numbers Euclidean, 191 of a ball, 191–193 symplectic, see symplectic packing numbers volume preserving, 191 packing obstruction, 188 superrecurrence via, 190 partially bounded subset of Z 2n (π ), 178, 187 Paternain, G., 238 Polterovich, L., 19–21, 124, 167, 186, 188–190, 193, 194, 210, 213, 214, 238 preparing the fibres, 41, 54

Index

projection invariant, 166 symplectic, 167, 168, 234 Question, 14, 16, 30, 64, 70, 161, 165, 178, 187, 189, 193, 200, 214, 225, 233, 238–239 Rankin, R. A., 191 ruled surface, 195 symplectic 4-manifold, 195, 211, 239 symplectic manifold, 211 sausage, 192 conjecture, 192 separating smaller fibres from larger ones, 40, 54 Seshadri constants, 194 Shiohama, K., 222 Siburg, K. F., 238 Sikorav’s Nonsqueezing Theorem, 8 Sikorav, J.-C., 8, 227 simple action spectrum, 27 simply connected hull, 178, 226, 228 stairs, 56, 57, 174 starshaped domain, 6–8, 24, 166, 215, 219, 225, 239 symplectic capacity, 9, 190, 224–232, 234–235 cylindrical, 225, 228, 234 Ekeland–Hofer, 9, 14, 15, 25 extrinsic on R2 , 162, 163, 165, 178, 187, 229–231 on R2n , 25, 226 intrinsic on R2 , 178, 187, 229 on R2n , 224, 225, 227 symplectic circumradius, 233 symplectic cylinder, 7, 162, 163, 178, 187, 230 symplectic diameter, 232 symplectic ellipsoid, 9, 24, 31

249 as fibration over a rectangle, 35, 91 as Lagrangian product of a simplex and a cube, 90, 150 symplectic embedding, 6, 18 symplectic embeddings E(π, a) → B 4 (A), 58–66, 154, 157–158, 235–236 E(π, a) → C 4 (A), 73–76, 236–237 E 2n (π, . . . , π, a) → B 2n (A), 31, 90–106, 147, 149, 154, 155, 158–159 E 2n (π, . . . , π, a) → C 2n (A), 147 P (π, a) → B 4 (A), 66–70 P (π, a) → C 4 (A), 77–79, 154, 159–160 P 2n (π, . . . , π, a) → B 2n (A), 147 2n P (π, . . . , π, a) → C 2n (A), 84–89, 147, 149, 155, 160–161 λE 2n (π, . . . , π, a) → (M, ω), 123–148 λP 2n (π, . . . , π, a) → (M, ω), 107–122, 147–148 elementary, 12 of many balls into (M, ω), 123 of many balls into a cube, 124 of many cubes into (M, ω), 108 via the h-principle, 12 symplectic fibration over S 2 , 212 area of, 212 symplectic folding construction, 15, 31, 39–51, 157 in higher dimensions, 82–83 modification of, 52 multiple, 53–57, 171–173 procedure, 58–60, 66, 73–74, 77–78, 84–87, 93 optimality of, 61, 67, 74, 78, 87 symplectic form, 17

250 compatible with a ruling, 195 on CP2 , 194 on g S 2 , 196 on g × S 2 , 195 on a ruled surface, admissible, 195 standard on R2n , 2 Symplectic Hedgehog Theorem, 8, 166 symplectic lifting construction, 177, 179–184, 187 symplectic manifold, 17 examples, 18 symplectic map, 5 symplectic packing numbers, 21, 188 and algebraic geometry, 194 of (g S 2 , ωab ), 198 of CP2 , 194 of g (a) × h (b), 209 of g (a) × S 2 (b), 198 of ( S 2 S 2 , ωab ), 197 of B 4 , 194 of B 2n , 213 of E(π, a), 212 of E(a1 , . . . , an ), 214 of S 2 (a) × S 2 (b), 196 of T 2 (1) × T 2 (1), 209 symplectic packing problem, 21, 188 motivations for, 189–194 symplectic packings of (S 2 S 2 , ωab ), 205 of (g S 2 , ωab ), 208–209 of (S 2 S 2 , ωab ), 208 of CP2 , 200–201 of g (a) × h (b), 209–211

Index

of g (a) × S 2 (b), 208–209 of B 4 , 200–201 of B 2n , 213 of E(π, a), 212 of S 2 (a) × S 2 (b), 201–205 symplectic polydisc, 16, 29, 66, 150 symplectic wrapping construction, 13, 149–153, 157 symplectomorphism of R2n , 3 Taubes, C., 194 Theorem 1, 9 Theorem 2, 14 Theorem 3, 19 Theorem 4, 20 Thue, A., 192 Traynor’s Wrapping Theorem, 13 Traynor, L., 13, 21, 23, 149, 189, 195, 200, 212–214, 239 Viterbo, C., 165, 225, 239 volume condition, 7 volume form, 219 volume preserving diffeomorphism of R2n , 3 embedding, 7, 18, 219, 220 packing, 191 Whitney, H., 222 Wrapping Theorem, 13 Wysocki, K., 9, 27, 239 Zehnder, E., 5, 190

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