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de Gruyter Expositions in Mathematics
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EmbeddingProblemsin
SymplecticGeometry
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 capaci-
ties, geometric constructions, Hamiltonian systems, rigidity and flexibility
Printed on acid-free paper which falls within the guidelines
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
516.316dc22
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 athttp://dnb.ddb.de.
Copyright 2005 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
Allrights 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.

Tomy parents

Preface
Symplectic geometry is the geometry underlying Hamiltonian dynamics, and sym-
plectic 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 preserv-
ing 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 radiusrinto the inﬁnite cylinderB
2
(1)×R
2n−2
ifr>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.
Wefeel 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
butnot 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 McDuff and Leonid Polterovich. I am in partic-
ular indebted to Dusa McDuff who explained to me symplectic folding, a technique
basic for the whole book.
Doing symplectic geometry at ETH has been greatly facilitated through the ex-
istence 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, Hel-
mut 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 con-
tinuous 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 Introduction 1
1.1 From classical mechanics to symplectic geometry 1
1.2 Symplectic embedding obstructions 4
1.3 Symplectic embedding constructions 11
2Proof of Theorem 1 23
2.1 Comparison of the relations≤
i 23
2.2 Rigidity for ellipsoids 24
2.3 Rigidity for polydiscs ? 28
3Proof of Theorem 2 31
3.1 Reformulation of Theorem 2 31
3.2 The folding construction 39
3.3 End of the proof 47
4 Multiple symplectic folding in four dimensions 52
4.1 Modiﬁcation of the folding construction 52
4.2 Multiple folding 53
4.3 Embeddings into balls 57
4.4 Embeddings into cubes 73
5 Symplectic folding in higher dimensions 82
5.1 Four types of folding 82
5.2 Embedding polydiscs into cubes 84
5.3 Embedding ellipsoids into balls 90
6Proof of Theorem 3 107
6.1 Proof of lim
a→∞p
P
a
(M, ω)=1 107
6.2 Proof of lim
a→∞p
E
a
(M, ω)=1 123
6.3 Asymptotic embedding invariants 147

x Contents
7 Symplectic wrapping 149
7.1 The wrapping construction 149
7.2 Folding versus wrapping 157
8Proof of Theorem 4 162
8.1 A more general statement 162
8.2 A further motivation for Problemζ 165
8.3 Proof by symplectic folding 168
8.4 Proof by symplectic lifting 177
9Packing symplectic manifolds by hand 188
9.1 Motivations for the symplectic packing problem 189
9.2 The packing numbers of the 4-ball andCP
2
and of ruled symplectic
4-manifolds 194
9.3 Explicit maximal packings in four dimensions 198
9.4 Maximal packings in higher dimensions 213
Appendix 215
A The Extension after Restriction Principle 215
B Flexibility for volume preserving embeddings 219
C Symplectic capacities and the invariantsc
Bandc C 224
D Computer programs 235
E Some other symplectic embedding problems 238
References 241
Index 247

Chapter 1
Introduction
In the ﬁrst section of this introduction we recall how symplectomorphisms ofR
2n
arise in classical mechanics, and introduce such notions as “Hamiltonian”, “symplec-
tic” 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 someR
n
subject to a force ﬁeldF. According
to Newton’s second law of motion, the acceleration of the particle is equal to the
force acting upon it,¨x=F.Inmany classical problems, such as those in celestial
mechanics, the force ﬁeldFis 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 variablesy=˙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 potentialsU,itisahopeless task to solve (1.1.1) explicitly. One
can, however, obtain some quantitative insight as follows. The structure of the sys-
tem (1.1.1) is not very beautiful. Notice, though, that (1.1.1) is “skew-coupled” in the
sense that the derivative ofxdepends onyonly and vice versa. We capitalize on this
by considering the function
H(x,y,t)=
y
2
2
−U(x,t), (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 theHamiltonian
system
˙x(t)=
∂H
∂y
(x, y,t),
˙y(t)=−
∂H
∂x
(x, y,t).







(1.1.3)
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
ω
0=
n
.
i=1
dxi∧dyi (1.1.4)
onR
2n
.Itiscalled thestandard symplectic formonR
2n
. It’sn’thexterior product is
ω
n
0
=ω0∧···∧ω 0=n! 0, where the volume form

0=dx1∧dy1∧···∧dx n∧dyn
agrees with the Euclidean volume formdx 1∧···∧dx n∧dy1∧···∧dy nup to the
factor(−1)
n(n−1)
2.Itfollows thatω 0is a non-degenerate 2-form, so that the equation
ω
0(XH(z, t),·)=dH(z,t) (1.1.5)
of 1-forms has a unique solutionX
H(z, t)for eachz=(x, y)∈R
2n
and eacht∈R.
The time-dependent vector ﬁeldX
His calledHamiltonian vector ﬁeldofH. Notice
now thatX
H=
∗
∂H
∂y
,−
∂H
∂x
⊕
,sothat the Hamiltonian system (1.1.3) takes the compact
form
˙z(t)=X
H(z(t), t). (1.1.6)
Under suitable assumptions on the potentialUthis ordinary differential equation can
be solved for all initial valuesz(0)=z∈R
2n
and for all timest. The resulting ﬂow
{ϕ
t
H
}deﬁned by
d
dt
ϕ
t
H
(z)=X H
∗
ϕ
t
H
(z), t
⊕
,
ϕ
0
H
(z)=z, z∈R
2n
,





(1.1.7)
is called theHamiltonian ﬂowofH. Each diffeomorphismϕ
t
H
is called aHamiltonian
diffeomorphism. More generally, any time-t -mapϕ
t
H
obtained in this way via a smooth
functionH:R
2n
×R→R, not necessarily of the form (1.1.2), is called a Hamiltonian
diffeomorphism.
The above formal manipulations were primarily motivated by aesthetic consider-
ations. As the following facts show, there are important pay-offs, however.
Fact 1.IfHis time-independent, thenHis preserved by the ﬂowϕ
t
H
.

1.1 From classical mechanics to symplectic geometry 3
Proof.SinceHis time-independent and in view of deﬁnitions (1.1.7) and (1.1.5),
d
dt
H
α
ϕ
t
H
β
=dH
α
ϕ
t
H
β
d
dt
ϕ
t
H
=dH
α
ϕ
t
H
β
X
H
α
ϕ
t
H
β
=ω
0
α
X
H,XH
β
ˇϕ
t
H
,
which vanishes becauseω
0is skew-symmetric. α
Historically, this fact was the main reason for working in the Hamiltonian formal-
ism. For us, two other pay-offs will be more important.
Fact 2.Hamiltonian diffeomorphisms preserve the symplectic formω
0.
Proof.Using Cartan’s formulaL
XH
=dιXH
+ιXH
d, deﬁnition (1.1.5) anddω 0=0,
we compute
L
XH
ω0=dιXH
ω0+ιXH
dω0=ddH+0=0 for allt.
Therefore,
d
dt
α
ϕ
t
H
β
∗
ω0=
α
ϕ
t
H
β
∗
LXH
ω0=0 for allt,sothat
α
ϕ
t
H
β
∗
ω0=ω0.α
A diffeomorphismϕofR
2n
is calledsymplectic diffeomorphismorsymplecto-
morphismif it preserves the symplectic formω
0,
ϕ
∗
ω0=ω0.
In classical mechanics, symplectomorphisms play the role of those coordinate trans-
formations which preserve the class of Hamiltonian vector ﬁelds, and are thus called
canonical transformations. Symplectic geometry of (R
2n
,ω0)is the study of its au-
tomorphisms, which are the symplectomorphisms. By Fact 2, the set of Hamiltonian
diffeomorphisms is embedded in this geometry.
A diffeomorphismϕofR
2n
is calledvolume preservingif 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
0.
Proof.ϕ
∗
0=ϕ
∗
α
1
n!
ω
n
0
β
=
1
n!
(ϕ
∗
ω0)
n
=
1
n!
ω
n
0
= 0. α
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 inR
n
generated
by the Newtonian system (1.1.1).
Summarizing, we have
Hamiltonian⇒symplectic⇒volume preserving.

4 1 Introduction
It is well-known that symplectomorphisms ofR
2n
are Hamiltonian diffeomorphisms
(see Appendix A), so that “Hamiltonian” and “symplectic” is the same. In dimen-
sion 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.Weonly 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 vol-
ume preserving diffeomorphisms ofR
2n
ifn≥2: The linear symplectic group has
dimension 2n
2
+n, while the group of matrices with determinant 1 has dimension
4n
2
−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 ofR
2n
is also much smaller than the set of volume preserving
diffeomorphisms from a global point of view: Every time-dependent compactly sup-
ported function onR
2n
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 var-
ious 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-formω
0in Fact 2
and made several seminal and fruitful conjectures in symplectic geometry, whose
proofs in particular would have demonstrated that both Hamiltonian and symplec-
tic 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 torusR
2n
/Z
2n
endowed
with the induced symplectic formω
0by(T
2n
,ω0).
Arnold conjecture for the torus.Every Hamiltonian diffeomorphism of the standard
torus(T
2n
,ω0)must have at least2n+1ﬁxed points.
It in particular follows that volume preserving (or symplectic) diffeomorphisms of
(T
2n
,ω0)cannot beC
0
-approximated by Hamiltonian diffeomorphisms in general.
Indeed, translations demonstrate that this global ﬁxed point theorem is a truly Hamil-
tonian result, which does not hold for all volume preserving or symplectic diffeomor-
phisms. For a Hamiltonian diffeomorphism which is generated by a time-independent
Hamiltonian or which isC
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 ofR
2n
wasfound by Gro-
mov and Eliashberg. In the early 1970s, Gromov proved the following alternative.
Gromov’s Alternative.The group of symplectomorphisms ofR
2n
is eitherC
0
-closed
in the group of all diffeomorphisms(hardness),or itsC
0
-closure is the group of volume
preserving diffeomorphisms(softness).
Notice that “symplectic” is aC
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 ofR
2n
isC
0
-closed in the group of all diffeomorphisms.
It follows that volume preserving diffeomorphisms ofR
2n
cannot beC
0
-approxi-
mated 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 mapϕ:U→R
2n
de-
ﬁned on an open (not necessarily connected) subsetUofR
2n
is calledsymplecticif
ϕ
∗
ω0=ω0. Locally, symplectic maps are embeddings. Indeed, symplectic maps
preserve the volume form
0=
1
n!
ω
n
0
and are thus immersions.

6 1 Introduction
Flexibility for symplectic immersions.Forevery open setVinR
2n
there exists a
symplectic immersion ofR
2n
intoV.
Proof.Since translations are symplectic, we can assume thatVcontains the origin.
LetDbe an open disc inR
2
centred at the origin whose radiusris so small that
D×···×D⊂V.Weshall construct a symplectic immersionϕofR
2
intoD. The
productϕ×···×ϕwill then symplectically immerseR
2n
intoV.
Step1.Choose a diffeomorphismf:R→]0,r
2
/2[. Then the map
R
2
→]0,r
2
/2[×R,(x, y)−
ρ
f(x),
y
f
τ
(x)
σ
, (1.2.1)
is a symplectomorphism.
Step2.The map
]0,r
2
/2[×R→D, (x,y)−
α√
2xcosy,
√
2xsiny
β
,
is a symplectic immersion. α
A symplectic mapϕ:U→R
2n
is called asymplectic embeddingif it is injective.
In view of the above result, we shall only consider symplectic embeddings from now
on.
AdomainVinR
2n
is a non-empty connected open subset ofR
2n
. The basic
problem addressed in this book is
Basic Problem.Consider an open setUinR
2n
and a domainVinR
2n
. Does there
exist a symplectic embedding ofUintoV?
Areader with a more physical or dynamical background may rather ask for em-
beddings ofUintoVinduced byHamiltoniandiffeomorphisms ofR
2n
. This is not
the same problem in general:
Example.LetUbe the annulus
U=

(x, y)∈R
2
|1<x
2
+y
2
<2

of areaπand letVbe the disc of areaa.Asiseasy to see (or by Proposition 1
below),Usymplectically embeds intoVif and only ifa≥π.Onthe other hand,U
symplectically embeds intoVvia a Hamiltonian diffeomorphism ofR
2
only ifa≥2π
since such an embedding must map the whole disc of radius
√
2 intoV. β
Foralarge class of domainsUinR
2n
,however, ﬁnding a symplectic or a Hamilto-
nian embedding is almost the same problem. A domainUinR
2n
is calledstarshaped
ifUcontains a pointpsuch that for every pointz∈Uthe straight line betweenpand
zis contained inU.

1.2 Symplectic embedding obstructions 7
Extension after Restriction Principle.Assume thatϕ:Uı→R
2n
is a symplectic
embedding of a bounded starshaped domainU⊂R
2n
. Then for any subsetA⊂U
whose closure inR
2n
is contained inUthere exists a Hamiltonian diffeomorphism
˚
AofR
2n
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,Uwill 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
0and are injective, they
preserve the total volume Vol(U)=
,
U
0.Anecessary condition for the existence
of a symplectic embedding ofUintoVis therefore
Vol(U)≤Vol(V ). (1.2.2)
Forvolume preserving embeddings, this necessary condition is also sufﬁcient.
Proposition 1.An open setUinR
2n
embeds into a domainVinR
2n
by a volume
preserving embedding if and only ifVol(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 radiusr
B
2n
α
πr
2
β
=
6
(x, y)∈R
2n

. n
i=1
x
2
i
+y
2
i
<r
2
7
and the open 2n-dimensionalsymplectic cylinder
Z
2n
(π)=

(x, y)∈R
2n
|x
2
1
+y
2
1
<1

.
While the ballB
2n
(a)has ﬁnite volume for eacha, the symplectic cylinderZ
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 ballB
2n
(a)symplectically embeds into the
cylinderZ
2n
(π)if and only ifa≤π.
Remarks.1. Proposition 1 shows that forn≥2 the wholeR
2n
embeds intoZ
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)−(≥ x
1,≥
−1
x2,x3,...,xn,≥y1,≥
−1
y2,y3,...,yn)

8 1 Introduction
ofR
2n
embeds the ball of radius≥
−1
intoZ
2n
(π).
2. The “symplectic cylinder”Z
2n
(π)in the Nonsqueezing Theorem cannot be
replaced by the “Lagrangian cylinder”

(x, y)∈R
2n
|x
2
1
+x
2
2
<1

.
Indeed, fora=
√
2/2 then-fold product of the map (1.2.1) symplectically embeds
the wholeR
2n
into this cylinder. The linear symplectomorphism
(x,y)−(≥x, ≥
−1
y) (1.2.3)
ofR
2n
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 theC
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. β
Gromov deduced his Nonsqueezing Theorem from his compactness theorem for
pseudo-holomorphic spheres. In his proof, the obstruction to a symplectic embedding
ofB
2n
(a)intoZ
2n
(π)fora>πis the symplectic area
,
S
ω0of a holomorphic curve
SinB
2n
(a)passing through the centre, which is at leasta>π.
Using Gromov’s compactness theorem for pseudo-holomorphic discs with La-
grangian boundary conditions, Sikorav found another amazing Nonsqueezing Theo-
rem. LetS
1
be the unit circle inR
2
(x, y), and consider the torusT
n
=S
1
×···×S
1
inR
2n
. Sikorav proved in [79] thatthere does not exist a symplectomorphism ofR
2n
which mapsT
n
intoZ
2n
(π).Notice that the volume ofT
n
inR
2n
vanishes, and that
T
n
does not bound any open set! In Sikorav’s proof, the obstruction is the symplectic
area
,
D
ω0of a closed holomorphic discD⊂R
2n
with boundary onT
n
, which is at
leastπ.
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.Forn≥2,nostarshaped domain inR
2n
containing
the torusT
n
symplectically embeds into the cylinderZ
2n
(π).
A more martial reader may prefer calling it Symplectic Flail Theorem. Notice that
there are starshaped domains containingT
n
of arbitrarily small volume!
Weﬁnally describe a symplectic non-embedding result which will lead to the
search for interesting symplectic embedding constructions. Given an open subsetU
ofR
2n
and a numberλ>0we set
λU={λz|z∈U}.

1.2 Symplectic embedding obstructions 9
Our Basic Problem can be reformulated as
Problem UV.Consider an open setUinR
2n
and a domainVinR
2n
. What is the
smallestλsuch thatUsymplectically embeds intoλV?
In the two theorems above, the symplectic embedding realizing the smallestλ
wassimply the identity embedding. In these theorems, the setUwasaround ball
B
2n
(a)witha>πand a starshaped domain containing the “round” torusT
n
, the set
Vwasthe “long and thin” cylinderZ
2n
(π), and the outcome was that these “round”
setsUcannot be symplectically “squeezed” into the “long but thinner” setV.In
order to formulate a symplectic embedding problem in which we have a chance to
ﬁnd interesting symplectic embeddings, we therefore take nowU“long and thin” and
V“round”. Our hope is then thatUcan be symplectically “folded” or “wrapped”
intoV.Toﬁxthe ideas, we takeVto be a ball andUto be an ellipsoid. Using
complex notationz
i=(xi,yi),wedeﬁne the opensymplectic ellipsoidwith radii
√
ai/πas
E(a
1,...,an)=
6
(z 1,...,zn)∈C
n

. n
i=1π|zi|
2
ai
<1
7
.
Here,|·|denotes the Euclidean norm inR
2
. Notice thatE(a,...,a)=B
2n
(a).
Since a permutation of the symplectic coordinate planes is a (linear) symplectic map,
we may assumea
1≤a2≤··· ≤a n. If, for instance,a 1=··· =a n−1anda nis
much larger thana
1, thenE(a 1,...,an)is indeed “long and thin”. With these choices
forUandV, Problem UV specializes to
Problem EB.What is the smallest ballB
2n
(A)into whichE(a 1,...,an)symplecti-
cally embeds?
Of course, the inclusion symplectically embedsE(a
1,...,an)intoB
2n
(A)if
A≥a
n. The following rigidity result shows that one cannot do better if the ellipsoid
is still “quite round”.
Theorem 1.Assumea
n≤2a 1. Then the ellipsoidE(a 1,...,an)does not symplecti-
cally embed into the ballB
2n
(A)ifA<a n.
In the casen=2, Theorem 1 was proved in [26] as an application of symplectic
homology. Our proof is simpler and works in all dimensions. It uses then’thEkeland–
Hofer capacity. Symplectic capacities are special symplectic invariants prompted
by Gromov’s work [31] and introduced by Ekeland and Hofer in [20]. Deﬁnitions
and a discussion of properties relevant for this book can be found in Chapter 2 and
Appendix C, and a thorough exposition of symplectic capacities is given in the book
[39]. For now, it sufﬁces to know that with starshaped domainsUandVinR
2n
a
symplectic capacitycassociates numbersc(U)andc(V )in[0,∞]in such a way that

10 1 Introduction
A1. Monotonicity:c(U)≤c(V )ifUsymplectically embeds intoV.
A2. Conformality:c(λU)=λ
2
c(U)for allλ∈R\{0}.
A3. Nontriviality:0<c
α
B
2n
(π)
β
andc
α
Z
2n
(π)
β
<∞.
A symplectic capacitycisnormalizedif
A3
δ
. Normalization:c
α
B
2n
(π)
β
=c
α
Z
2n
(π)
β
=π.
In view of the monotonicity axiom, symplectic capacities can be used to detect sym-
plectic embedding obstructions. Indeed, the existence of any normalized symplectic
capacity implies Gromov’s Nonsqueezing Theorem at once. It therefore cannot be
easy to construct a symplectic capacity.
From Gromov’s work on pseudo-holomorphic curves one can extract normalized
symplectic capacities, and the afore mentioned proofs of the Nonsqueezing Theorem
and the Symplectic Hedgehog Theorem can be formulated in terms of these capacities.
These normalized symplectic capacities are useless for Problem EB, however. Indeed,
B
2n
(a1)⊂E(a 1,...,an)⊂Z
2n
(a1):=B
2
(a1)×R
2n−2
,
so thatc(E(a
1,...,an))=a 1for any normalized symplectic capacity. Given a sym-
plectic embeddingE(a
1,...,an)ı→B
2n
(A), such a symplectic capacity therefore
only yieldsa
1≤A,aninformation already covered by the volume condition (1.2.2).
Shortly after the appearance of Gromov’s work, Ekeland and Hofer found a way
to construct symplectic capacities via Hamiltonian dynamics. In order to give the idea
of their approach, we consider a bounded starshaped domainU⊂R
2n
with smooth
boundary∂U.Aclosed characteristicon∂Uis an embedded circle in∂Utangent to
thecharacteristic line bundle
L
U={(x, ξ)∈T∂U|ω 0(ξ, η)=0 for allη∈T x∂U}.
If∂Uis represented as a regular energy surface{x∈R
2n
|H(x)=const}of a
smooth functionHonR
2n
, then the Hamiltonian vector ﬁeldX Hrestricted to∂U
is a section ofL
Uin view of its deﬁnition (1.1.5), and so the traces of the periodic
orbits ofX
Hon∂Uare the closed characteristics on∂U. TheactionA(γ)of a closed
characteristicγon∂Uis deﬁned as
A(γ)=

/
γ
λ

,
whereλis any primitive ofω
0.Inview of Stokes’ Theorem,A(γ )is the symplectic
area

,
D
ω0

of a closed discD⊂R
2n
with boundaryγ. The set
˛(U)=

kA(γ)|k=1,2,...; γis a closed characteristic on∂U

1.3 Symplectic embedding constructions 11
is called theaction spectrumofU.In[20], Ekeland and Hofer associated withU
a numberc
1(U)deﬁned via critical point theory applied to the classical action func-
tional of Hamiltonian dynamics, and in this way obtained a symplectic capacityc
1
satisfyingc 1(U)∈˛(U).IfUis convex, thenc 1(U)is the smallest number in˛(U).
Therefore,c
1(E(a1,...,an))=a 1. From this, the Nonsqueezing Theorem follows
at once, and also the Symplectic Hedgehog Theorem can be proved by using the sym-
plectic capacityc
1,see [79] and [83]. But again,c 1is useless for Problem EB. In [21],
then, Ekeland and Hofer repeated their construction from [20] in anS
1
-equivariant
setting and usedS
1
-equivariant cohomology to obtain a whole familyc 1≤c2≤···
of symplectic capacities satisfyingc
j(U)∈˛(U). Besidesc 1, these capacities are
not normalized, and for an ellipsoidE=E(a
1,...,an)they are given by
{c
1(E)≤c 2(E)≤···}={ka i|k=1,2,...; i=1,...,n}.
ForanellipsoidE(a
1,...,an)witha n≤2a 1and for a ballB
2n
(A)we therefore ﬁnd
c
n(E(a1,...,an))=a nandc n
α
B
2n
(A)
β
=A,
so that Theorem 1 follows in view of the monotonicity ofc
n.InChapter 2 we shall
prove a stronger result. For further symplectic non-embedding results we refer to
Chapters 4 and 9, Appendix E, and the references given therein.
Summarizing, we have seen that there are various symplectic non-embedding
theorems, and that the methods used in their proofs are quite different. The obstructions
found, though, have a common feature: Once it is the symplectic area of a holomorphic
curve through the centre of a ball, once it is the symplectic area of a holomorphic disc
with Lagrangian boundary conditions, and once it is the symplectic area of a disc
whose boundary is a closed characteristic.
1.3 Symplectic embedding constructions
What, then, can be done by symplectic embeddings? The main characters of this
book are various symplectic embedding constructions. They are best motivated, de-
scribed and understood when applied to speciﬁc symplectic embedding problems. In
this section we describe the results thus obtained and only give a vague idea of the
constructions. They will be carried out in detail in Chapters 3 to 9. These symplec-
tic embedding constructions are all elementary and explicit. The need for explicit
symplectic embedding constructions could be sufﬁciently motivated by purely math-
ematical curiosity alone. More importantly, these constructions will shed some light
on the nature of symplectic rigidity. Sometimes, they show that the known symplec-
tic embedding obstructions are sharp. More often, they yield symplectic embedding
results which are not known to be optimal or known to be not optimal, and thereby
prompt new questions on symplectic embeddings. Certain non-explicit symplectic
embeddings can be obtained via the so-calledh-principle for symplectic embeddings
of codimension at least 2 and via the symplectic blow-up operation. While the former

12 1 Introduction
method is addressed only brieﬂy right below, the latter will be important in Chapter 9,
which is devoted to symplectic packings by balls.
1.3.1 From rigidity to ﬂexibility.The following result, which is due to Gromov
[32, p. 335] and is taken from [26, p. 579], gives a partial answer to Problem EB and
shows that the assumptiona
n≤2a 1in Theorem 1 cannot be omitted.
Symplectic embeddings via theh-principle.Foranya>0there exists an≥>0
such that the2n-dimensional ellipsoidE(≥,...,≥,a√symplectically embeds into
B
2n
(π).
Proof.This is an immediate consequence of Gromov’sh-principle for symplectic
embeddings of codimension at least 2. Indeed, choose asmoothembeddingϕ
0of the
closed disc
B
2
(a)intoB
2n
(π). According to [32, p. 335] or [24, Theorem 12.1.1],
arbitrarilyC
0
-close toϕ 0there exists a symplectic embeddingϕ 1:
B
2
(a) ı→B
2n
(π),
meaning thatϕ
∗
1
ω0=ω0. Using the Symplectic Neighbourhood Theorem, we ﬁnd
≥>0 such thatϕ
1extends to a symplectic embedding ofB
2n−2
(≥√×
B
2
(a)and in
particular to a symplectic embedding ofE(≥,...,≥,a√. α
Since theC
0
-small perturbationϕ 1of the smooth embeddingϕ 0provided 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≥.Weﬁrst 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
ofR
2n
contains transformations of the form (1.2.3) and, more generally, of the form
(x, y)−
α
Ax,
α
A
T
β
−1
y
β
(1.3.1)
whereAis 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 inR
2
into another domain inR
2
is symplectic, and by Propo-
sition 1 there are plenty of such embeddings. An example are the “inverse symplectic
polar coordinates”
(x, y)−
α√
2xcosy,
√
2xsiny
β
(1.3.2)
embedding
δ
0,a/2π
γ
×]0,2π[intoB
2
(a), which we met before. As we shall see in
Section 3.1, symplectic embeddings of domains inR
2
can be described in an almost

1.3 Symplectic embedding constructions 13
explicit way. Taking products, we obtain almost explicit symplectic embeddings of
domains inR
2n
.
3. Lifts.Forconvenience we write(u, v,x,y)=(x
1,y1,x2,y2). Using deﬁ-
nition (1.1.3) we ﬁnd that the Hamiltonian vector ﬁeld of the Hamiltonian function
(u, v,x,y)−→−xis(0,0,0,1),sothat the time-1-map of the Hamiltonian ﬂow is the
translation(u, v,x,y)−(u, v, x, y+1). Choose a smooth functionf:R→[0,1]
such that
f(s)=0ifs≤0 andf(s)=1ifs≥1.
The Hamiltonian vector ﬁeld of the Hamiltonian function(u, v,x,y)−→−f (u)xis
α
0,f
τ
(u)x,0,f(u)
β
,sothat its time-1-map
(u, v, x, y)−
α
u, v+f
τ
(u)x,x,y+f (u)
β
(1.3.3)
ﬁxes the half space{u≤0}and lifts the space{u≥1}by1inthey -direction. Choosing
fsuch that
f(s)=0ifs≤0ors≥3 andf(s)=1ifs∈[1,2]
and looking at the time-1-map generated by the Hamiltonian function
(u, v, x, y)−→−f (u)f (v)x (1.3.4)
we ﬁnd “true” lifts (called elevators in the US, I guess). β
At ﬁrst glance, these elementary symplectic embeddings look useless for Prob-
lem EB. Indeed, none of them embeds the ellipsoidE(a
1,...,an)into a ballB
2n
(A)
withA<a
n.However, these elementary symplectic embeddings will serve as build-
ing 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. Givenλ>0
the ellipsoidλE(a
1,a2)symplectically embeds into the ballλB
4
(A)if and only if
E(a
1,a2)symplectically embeds intoB
4
(A).We can thus assume without loss of
generality thata
1=π.
Traynor’s Wrapping Theorem.There exists a symplectic embedding
E
α
π, k(k−1)π
β
ı→B
4
(kπ+≥√
for every integerk≥2and 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 prod-
uctŁ×∗ of a simplex and a square inR
2
+
(x)×R
2
+
(y), and then uses a map of
the form (1.3.1) to wrap this product around the torusT
2
(y)=R
2
(y)/2πZ
2
in
R
2
+
(x)×T
2
(y). The point is then that the product of the area preserving map (1.3.2)
extends to a symplectic embedding ofR
2
+
(x)×T
2
(y)intoR
2
(x)×R
2
(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 functionw
EBon[π,∞[drawn in Figure 1.1 below, in which
we again assumea
1=πand writea=a 2.Weinparticular see thatw EB(a)<a
only fora>3π,sothat 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)
areass
1(U)ands 2(U)of the projections of a domainUinR
4
to the coordinate
planesR
2
(x1,y1)andR
2
(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 tos
1in both the Nonsqueezing Theorem and the Symplectic
Hedgehog Theorem and equal tos
2in Theorem 1. Consider now an ellipsoidE=
E(a
1,a2). When we “foldEappropriately” toE
τ
, the smaller projection will double,
s
1(E
τ
)=2a 1, while the larger projection should decrease,s 2(E
τ
)<a2.Ifa 2≤2a 1,
thens
1(E
τ
)=2a 1≥a2=s1(B
4
(a2)),sothatE
τ
does not ﬁt into a ballB
4
(A)with
A<a
2,aspredicted by Theorem 1. Ifa 2>2a 1, thens 1(E
τ
)=2a 1<a2and
s
2(E
τ
)<a2,however, so that we can hope that folding can be achieved in such a way
thatE
τ
ﬁts into a ballB
4
(A)withA<a 2. This can indeed be done in a symplectic
way.
Theorem 2.Assumea
n>2a 1. Then there exists a symplectic embedding of the
ellipsoidE(a
1,...,a1,an)into the ballB
2n
(an−δ)for everyδ∈
≤
0,
an
2
−a1
≡
.
The reader might ask why we look at “skinny” ellipsoids witha
n−1=a1in
Theorem 2. The reason is that for “ﬂat” ellipsoids an analogous embedding result does
not hold in general. For instance, the third Ekeland–Hofer capacityc
3implies that for
n≥3 the “ﬂat” 2n-dimensional ellipsoidE(a,3a,...,3a) does not symplectically
embed into the ballB
2n
(A)ifA<3a . The second Ekeland–Hofer capacityc 2implies
that the “mixed” ellipsoidE(a,2a,3a)does not symplectically embed into the ball
B
6
(A)ifA<2 a,but we do not know the answer to
Question 1.Does the ellipsoidE(a,2a,3a)symplectically embed intoB
6
(A)for
someA<3a ?
Symplectic folding was invented by Lalonde and McDuff in [48] in order to prove
the General Nonsqueezing Theorem stated in Remark 9.3.7 as well as an inequal- ity between Gromov width and displacement energy implying that the Hofer metric

1.3 Symplectic embedding constructions 15
on the group of compactly supported Hamiltonian diffeomorphisms is always non-
degenerate. In the same work [48] Lalonde and McDuff also observed that symplectic
folding can be used to prove Theorem 2 in the casen=2. A reﬁnement of their sym-
plectic 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 ellipsoidE(a
1,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 ofB
2
(a2)intoR
2
, 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 assumea
1=πand writea=a 2. The
optimal valuesAfor the embedding problemsE(,a) ı→B
4
(A)are encoded in the
“characteristic function”χ
EBon[π,∞[deﬁned by
χ
EB(a)=inf

A|E(π,a)symplectically embeds intoB
4
(A)

.
Weillustrate the results with the help of Figure 1.1. In view of Theorem 1 we have
χ
EB(a)=afora∈[π,2π].Fora>2π, the second Ekeland–Hofer capacityc 2
still implies thatχ EB(a)≥2π. This information is vacuous ifa≥4π, since the
volume condition Vol
α
E(π,a)
β
≤Vol
α
B
4
(χEB(a))
β
translates toχ EB(a)≥
√
πa.
The estimateχ
EB(a)≤a/2+πstated in Theorem 2 is obtained by folding once. It
will turn out that fora>2πand for eachk≥1, foldingk+1 times embedsE(π,a)
into a strictly smaller ball than foldingktimes. The functionf
EBon]2π,∞[deﬁned
by
f
EB(a)=inf

A|E(π,a)embeds intoB
4
(A)by multiple symplectic folding

is therefore obtained by folding “inﬁnitely many times”. The graph of the functionf
EB
is computed by a computer program. The functionw EBencoding Traynor’s Wrapping
Theorem is alternatingly larger and smaller thanf
EB.
Weare particularly interested in the behaviour ofχ
EB(a)asa→2π
+
and as
a→∞.Weshall prove that
lim sup
≥→0
+
fEB(2π+≥√−2π
≥
≤
3
7
,
and so the same estimate holds forχ
EB.
Question 2.How doesχ
EB(a)look like neara=2π?Inparticular,
lim sup
≥→0
+
χEB(2π+≥√−2π
≥
<
3
7
?

16 1 Introduction
We havef EB(a)<w EB(a)for alla∈]2π,5.1622π]. The computer program for
f
EByields the particular values
f
EB(3π)≈2.3801π andf EB(4π)≈2.6916π.
Wedo not expect thatχ
EB(3π)=f EB(3π)andχ EB(4π)=f EB(4π).
Question 3.Is it true thatχ
EB(3π)=χ EB(4π)=2π?
The differencew
EB(a)−
√
πabetweenw EBand the volume condition is bounded
by(3−
√
3)π.Weshall also prove thatf EB(a)−
√
πais bounded. It follows that
χ
EB(a)−
√
πais bounded. We in particular have
lim
a→∞
Vol(E(π,a))
Vol
α
B
4
(χEB(a))
β=1. (1.3.5)
This means that the embedding obstructions encountered for smallamore and more
disappear asa→∞.
2π
2π
3π
4π
4π
5π
6π
6π8π 12π 15π 20π 24π
a
A
f
EB(a)
w
EB(a)
χ
EB(a)?
A=aA =
a
2
+π
A=
√
πa
c
2
Figure 1.1. What is known about the embedding problemE(, a) ı→B
4
(A).
Denote byD(a)=B
2
(a)the open disc inR
2
of areaacentred at the origin. The
opensymplectic polydiscP(a
1,...,an)inR
2n
is deﬁned as
P(a
1,...,an)=D(a 1)×···×D(a n).

1.3 Symplectic embedding constructions 17
The “n-cube” P
2n
(a,...,a)will be denoted byC
2n
(a).Uptonow,our model sets
were ellipsoids, which we tried to symplectically embed into small balls. Starting
from Sikorav’s Nonsqueezing Theorem for the torusT
n
and noticing that (the closure
of)C
2n
(π)is the convex hull ofT
n
,wecould equally well have taken polydiscs and
cubes.
Problem PC.What is the smallest cubeC
2n
(A)into whichP(a 1,...,an)symplec-
tically embeds?
Forthis problem, no interesting obstructions are known, however. The reason
is that symplectic capacities only see the size min{a
1,...,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 ellip-
soids and polydiscs into balls and cubes. While embedding an open setUinto a
minimal ball is related to minimizing its diameter (a 1-dimensional, metric quantity),
embeddingUinto a minimal cube amounts to minimizing the areas of its projections
to the symplectic coordinate planes (a 2-dimensional, “more symplectic” quantity).
Werefer 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 setUinR
2n
and a domainVinR
2n
. What is the
largestλsuch thatλUsymplectically embeds intoV?
As before, we shall eventually specializeUto an ellipsoid or a polydisc, but
this time we takeVto be an arbitrary domain inR
2n
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 inR
2n
of ﬁnite volume. As will become clear in Chapter 6 not much is lost
thereby.
Adifferential 2-formωon a smooth manifoldMis calledsymplecticifωis
non-degenerate and closed. The pair(M, ω)is then called asymplectic manifold.
The non-degeneracy ofωimplies thatMis even-dimensional, dimM=2n, and
that =
1
n!
ω
n
is a volume form onM,sothatMis orientable. The non-degeneracy
together with the closedness ofωimply that(M, ω)is locally isomorphic to(R
2n
,ω0)
withω
0as in (1.1.4):

18 1 Introduction
Darboux’s Theorem.Forevery pointp∈Mthere exists a coordinate chart
ϕ
U:U→R
2n
such thatϕ U(p)=0andϕ
∗
U
ω0=ω.
Therefore, a symplectic manifold is a smooth 2n-dimensional manifold admitting
an atlas{(U,ϕ
U)}such that all coordinate changes
ϕ
Vˇϕ
−1
U
:ϕU(U∩V)→ϕ V(U∩V)
are symplectic. Examples of symplectic manifolds are open subsets of(R
2n
,ω0), the
torusR
2n
/Z
2n
endowed with the induced symplectic form, surfaces equipped with an
area form, Kähler manifolds like complex projective spaceCP
n
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.
Werefer to the book [62] for more information on symplectic manifolds.
As before, we endow each open subsetUofR
2n
with the standard symplectic
formω
0.Asmooth embeddingϕ:Uı→(M, ω)is calledsymplecticif
ϕ
∗
ω=ω 0.
Problem UV generalizes to
Problem UM.Consider an open setUinR
2n
and a connected2n-dimensional
symplectic manifold(M, ω). What is the largest numberλsuch thatλUsymplectically
embeds into(M, ω)?
A smooth embeddingϕ:Uı→(M, ω)is calledvolume preservingif
ϕ
∗
= 0
where as before 0=
1
n!
ω
n
0
and =
1n!
ω
n
.Ofcourse, every symplectic embedding
is volume preserving. A necessary condition for a symplectic embedding ofUinto
(M, ω)is therefore
Vol(U)≤Vol(M,ω)
where we set Vol(M,ω)=
1
n!
,
M
ω
n
.For volume preserving embeddings, this obvi-
ous condition is the only one in view of the following generalization of Proposition 1,
aproof of which can again be found in Appendix B.
Proposition 2.An open setUinR
2n
embeds into(M, ω)by a volume preserving
embedding if and only ifVol(U)≤Vol(M, ω).
Forsymplectic embeddings of “round” shapesU⊂R
2n
into(M, ω),how ever,
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

1.3 Symplectic embedding constructions 19
the product(M,ω)=(S
2
×S
2
,σ⊕kσ), whereσis an area form on the 2-sphereS
2
of area
,
S
2σ=πand wherek≥1. Then the ballB
4
(a)symplectically embeds into
(M,ω)if and only ifa≤π.For skinny shapes, though, the situation for symplectic
embeddings is not too different from the one for volume preserving embeddings:
WechooseUto be a 2n-dimensional ellipsoidE(π,...,π,a)ora2n-dimensional
polydiscP(π,...,π,a), consider a connected 2n-dimensional symplectic manifold
(M, ω)of ﬁnite volume Vol(M,ω)=
1
n!
,
M
ω
n
, and deﬁne for eacha≥πthe
numbers
p
E
a
(M, ω)=sup
λ
Vol
α
λE(π,...,π,a)
β
Vol(M,ω)
,
p
P
a
(M, ω)=sup
λ
Vol
α
λP(π,...,π,a)
β
Vol(M,ω)
,
where the supremum is taken over all thoseλfor whichλE(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
a→∞
p
E
a
(M, ω)=1andlim
a→∞
p
P
a
(M, ω)=1.
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 ofMwith 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 McDuff and Polterovich on ﬁlling a cube
by balls.
1.3.3 A vanishing theorem.The Basic Problem and its variations asked for sym-
plectic embeddings which were not required to have any additional properties. We
now look at symplectic embeddingsϕof the ballB
2n
(a)into the symplectic cylinder
Z
2n
(π)whose imageϕ(B
2n
(a))⊂Z
2n
(π)has a speciﬁc property. By Gromov’s
Nonsqueezing Theorem there does not exist a symplectic embedding ofB
2n
(a)into
Z
2n
(π)ifa>π.Soﬁxa∈]0,π]. The simply connected hullˆTof a subsetTofR
2
is the union of its closure
Tand the bounded components ofR
2
\T.Wedenote by
µthe Lebesgue measure onR
2
, and we abbreviateˆµ(T )=µ(ˆT).For the unit circle

20 1 Introduction
S
1
inR
2
we haveµ(S
1
)=0<π=ˆµ(S
1
).Asiswell-known, the Nonsqueezing
Theorem is equivalent to each of the identities
a=inf
ϕ
µ
α
p

ϕ
α
B
2n
(a)
βλβ
,
a=inf
ϕ
ˆµ
α
p

ϕ
α
B
2n
(a)
βλβ
,
whereϕvaries over all symplectomorphisms ofR
2n
which embedB
2n
(a)intoZ
2n
(π)
and wherep:Z
2n
(π)→B
2
(π)is the projection, see [22] and Appendix C.2. Fol-
lowing McDuff, [59], we considersectionsof the imageϕ(B
2n
(a))instead of its
projection, and deﬁne
ζ(a):=inf
ϕ
sup
x
µ
α
p

ϕ(B
2n
(a))∩D x
λβ
,
ˆζ(a):=inf
ϕ
sup
x
ˆµ
α
p

ϕ(B
2n
(a))∩D x
λβ
,
whereϕagain varies over all symplectomorphisms ofR
2n
which embedB
2n
(a)into
Z
2n
(π), and whereD x⊂Z
2n
(π)denotes the disc
D
x=B
2
(π)×{x},x∈R
2n−2
.
Clearly,
ζ(a)≤ˆζ(a)≤a.
It is also well-known that the Nonsqueezing Theorem is equivalent to the identity
ˆζ(π)=π. (1.3.6)
Indeed, the Nonsqueezing Theorem implies that for every symplectomorphismϕ
ofR
2n
which embedsB
2n
(π)intoZ
2n
(π)there exists anxinR
2n−2
such that
ϕ(B
2n
(π))∩D xcontains the unit circleS
1
×{x}.Onher search for symplectic
rigidity phenomena beyond the Nonsqueezing Theorem, McDuff therefore posed the
following problem.
Problemζ.Find a non-trivial lower bound for the functionζ(a).Inparticular, is it
true thatζ(a)→πasa→π?
A further motivation for this problem comes from convex geometry and from the
fact that on bounded convex subsets of(R
2n
,ω0)normalized symplectic capacities
agree up to a constant. It was known to Polterovich thatζ(a)/a→0asa→0.The
following result answers the question in Problemζin the negative and completely
solves Problemζ.
Theorem 4.ζ(a)=0for alla∈]0,π]andˆζ(a)=0for alla∈]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.

1.3 Symplectic embedding constructions 21
1.3.4 Symplectic packings.Weﬁnally look at thesymplectic packing problem.
Given a connected 2n-dimensional symplectic manifold(M, ω)of ﬁnite volume and
given a natural numberk, this problem asks
Problem kBM.What is the largest numberafor which the disjoint union ofkequal
ballsB
2n
(a)symplectically embeds into(M, ω)?
Equivalently, one studies thek’thsymplectic packing number
p
k(M, ω)=sup
a
kVol
α
B
2n
(a)
β
Vol(M,ω)
where the supremum is taken over all thoseafor which
1
k
i=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 thatp
k(B
2n
(π), ω0)≤
k
2
nfor 2≤k≤2
n
. 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 ofk.Forlargek,however, it was shown in [61] that
lim
k→∞
pk(M,ω)=1
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 numberk
0
such thatp k(M, ω)=1 for allk≥k 0. 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 ofk,
explicitmaximal 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
p
k(M, ω)for those symplectic 4-manifolds(M, ω)and numberskconsidered in [41],
[81], [45], [54] as well as for ruled symplectic 4-manifolds and small values ofk.For
example, we shall see that maximal packings of the standard 4-ball by 5 or 6 balls
and of the productS
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 productsα
1×α2of suitable area
preserving diffeomorphisms between a disc and a rectangle. Takingn-fold products
we shall also construct a full packing of the standard 2n-ball byl
n
balls for eachl∈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 ofS
2
×S
2
by 5 balls.
Contrary to the symplectic embedding results described before, none of our sym-
plectic 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 ofkfor which the explicit construc-
tions 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 satisfac-
tory 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
Problemζand 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 Ap-
pendix 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 embed-
dings 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 theC
∞
-category, i.e., all manifolds and dif-
feomorphisms are assumed to beC
∞
-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.
Wedenote byO(n)the set of bounded domains inR
2n
diffeomorphic to a ball. Each
U∈O(n)is endowed with the standard symplectic structureω
0=
+
n
i=1
dxi∧dyi.
OrientingUby the volume form
0=
1
n!
ω
n
0
we write|U|=
,
U
0for the usual
volume ofU. LetD(n)be the group of all symplectomorphisms ofR
2n
and let
Sp(n;R)be its subgroup of linear symplectomorphisms ofR
2n
. Deﬁne the following
relations onO(n):
U≤
1V⇐⇒There exists aϕ∈Sp(n;R)withϕ(U)⊂V.
U≤
2V⇐⇒There exists aϕ∈D(n)withϕ(U)⊂V.
U≤
3V⇐⇒There exists a symplectic embeddingϕ:Uı→V.
2.1 Comparison of the relations≤ i
Clearly,≤ 1⇒≤2⇒≤3.Itis, however, well known that the relations≤ lare different.
Proposition 2.1.1.The relations≤
lare all different.
Proof.In order to show that the relations≤
1and≤ 2are different it sufﬁces to
ﬁnd an area and orientation preserving diffeomorphismϕof
α
R
2
,ω0
β
mapping the
unit discD(π)to a set which is not convex. The existence of such aϕfollows, for
instance, from Lemma 3.1.5 below. For the productsU=D(π)×···×D(π)and
V=ϕ(D(π))×···×ϕ(D(π))we then haveUωδ
1VandU≤ 2V.
The construction of setsUandV∈O(n)withU≤
3VbutUωδ 2Vrelies on the
following simple observation. Suppose thatU≤
2Vand in addition that|U|=|V|.
Ifϕis a map realizingU≤
2V,nopoint ofR
2n
\Ucan then be mapped toV, and
we conclude thatϕis a homeomorphism from the boundary∂UofUto the boundary
∂VofV.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 setU=C
2n
(π)=D(π)×···×D(π)andV=SD(π)×···×SD(π).
By Proposition 1 of the introduction,D(π)≤
3SD(π), see also Lemma 3.1.5 below.
Therefore,U≤
3V, and clearly|U|=|V|. But∂Uand∂Vare not homeomorphic.α
Wewish to mention that forn≥2 more interesting examples showing that≤
2and
≤
3are different were found by Eliashberg and Hofer, [23], and by Cieliebak, [15].
In order to describe their examples, we assume that∂Uis smooth. Recall that the
characteristic line bundleL
Uof∂Uis deﬁned as
L
U={(x, ξ)∈T∂U|ω 0(ξ, η)=0 for allη∈T x∂U}. (2.1.1)
The unparametrized integral curves ofL
Uare calledcharacteristicsand form the
characteristic foliationof∂U. Moreover,∂Uis said to beof contact typeif on a
neighbourhood of∂Uthere exists a smooth vector ﬁeldXwhich is transverse to∂U
and meetsL
Xω0=dιXω0=ω0. E.g., the radial vector ﬁeldX=
1
2
∂
∂r
onR
2n
meetsL Xω0=ω0, and so all starshaped domains inR
2n
are of contact type. If
U≤
2V, then the characteristic foliations of∂Uand∂Vare isomorphic, and∂U
is of contact type if and only if∂Vis of contact type. Theorem 1.1 in [23] and its
proof show that there exist convexU,V∈O(n)with smooth boundaries such thatU
andVare symplectomorphic andC
∞
-close to the ballB
2n
(π),but the characteristic
foliation of∂Ucontains an isolated closed characteristic while the one of∂Vdoes not.
And Corollary A in [15] and its proof imply that given anyU∈O(n),n≥2, with
smooth boundary∂Uof contact type, there exists a symplectomorphic andC
0
-close
setV∈O(n)whose boundary is not of contact type. We in particular see that even
forUbeing a ball,U≤
3Vdoes not implyU≤ 2V.
2.2 Rigidity for ellipsoids
Proposition 2.1.1 shows that in order to detect some rigidity via the relations≤ lwe
must pass to a small subcategory of sets: LetE(n)be the collection of symplectic
ellipsoids deﬁned in Section 1.2.2,
E(n)={E(a)=E(a
1,...,an)},a =(a 1,...,an),
and writeα
lfor the restrictions of the relations≤ ltoE(n).Notice again that
α
1⇒α 2⇒α 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 relationsα
1
andα 2are different. The relationsα 2andα 3are very similar: Since ellipsoids are
starshaped, the Extension after Restriction Principle implies
E(a)α
3E(a
τ
)⇒E(δa)α 2E(a
τ
)for allδ∈]0,1[. (2.2.2)

2.2 Rigidity for ellipsoids 25
It is, however, not known whetherα
2andα 3are the same: While Theorem 2.2.4
proves this under an additional condition, the folding construction of Section 3.2
suggests thatα
2andα 3are different in general. But let us ﬁrst prove a general and
common rigidity property of these relations:
Proposition 2.2.1.The relationsα
lare partial orderings onE(n).
Proof.The relations are clearly reﬂexive and transitive, so we are left with identitivity,
i.e.,
α
E(a)α
lE(a
τ
)andE(a
τ
)αlE(a)
β
⇒E(a)=E(a
τ
).
Of course, the identitivity ofα
3implies the one ofα 2which, in turn, implies the one
ofα
1.Toprove the identitivity ofα 3we use Ekeland–Hofer capacities introduced
in [21].
Deﬁnition 2.2.2.Anextrinsic symplectic capacity on(R
2n
,ω0)is a mapcassociating
with each subsetSofR
2n
a numberc(S)∈[0,∞]in such a way that the following
axioms are satisﬁed.
A1. Monotonicity:c(S)≤c(T )if there existsϕ∈D(n)such thatϕ(S)⊂T.
A2. Conformality:c(λS)=λ
2
c(S)for allλ∈R\{0}.
A3. Nontriviality:0<c(B
2n
(π))andc(Z
2n
(π)) <∞.
The Ekeland–Hofer capacities form a countable family{c
j},j≥1, of extrinsic
symplectic capacities onR
2n
.For a symplectic ellipsoidE=E(a 1,...,an)these
invariants are given by the identity of sets
{c
1(E)≤c 2(E)≤...}={ka i|k=1,2,...; i=1,...,n}, (2.2.3)
see [21, Proposition 4]. Observe that for anyl=1,2,3 andλ>0
E(a)α
lE(a
τ
)⇒E(λa)α lE(λa
τ
). (2.2.4)
This is seen by conjugating the given mapϕwith the dilatation byλ
−1
. Recalling
(2.2.2) we conclude that for anyδ
1,δ2∈]0,1[the postulated relations
E(a)α
3E(a
τ
)α3E(a)
imply
E(δ
2δ1a)α2E(δ1a
τ
)α2E(a).
Now the monotonicity property (A1) of the capacities and the set of relations in (2.2.3)
immediately imply thata=a
τ
. This completes the proof of Proposition 2.2.1.α

26 2 Proof of Theorem 1
Remark 2.2.3.In the above proof we derived the identitivity ofα 1andα 2from the
one ofα
3.Weﬁnd it instructive to give direct proofs.
It is well known from linear symplectic algebra [39, p. 40] that
E(a)α
1E(a
τ
)⇐⇒a i≤a
τ
i
for alli, (2.2.5)
in particularα
1is identitive.
In order to give an elementary proof of the identitivity ofα
2we look at the char-
acteristic foliation on the boundary and at the actionsA(γ )of closed characteristics.
Tocompute the characteristic foliation of∂E(a)recall thatE(a)=H
−1
(1), where
H(z)=
+
n
i=1
π|z i|
2
ai
. Using deﬁnition (1.1.3) we ﬁnd
X
H(z)=−2πJ
α
1
a1
z1,...,
1
an
zn
β
whereJ=
√
−1∈C=R
2
(x, y)is the standard complex structure. The char-
acteristic on∂E(a)throughz=z(0)can therefore by parametrized asz(t)=
(z
1(t),...,zn(t)), where
z
i(t)=e
−2πJt/ai
zi(0), i=1,...,n.
If thennumbersa
1,...,anare linearly independent overZ, then the only periodic
orbits are(0,...,0 ,z
i(t),0,...,0 )with
z
i(t)=e
−2πJt/ai
zi(0)andπ|z i(0)|
2
=ai.
In general, the traces of the closed characteristics on∂E(a)form the disjoint union
∂E(a
1
)∪···∪∂E(a
d
)
wherea
1
∪···∪a
d
is the partition ofa=(a 1,...,an)into maximal linearly dependent
subsets. Recall that the action of a closed characteristicγis deﬁned asA(γ )=

,
γ
λ

,
wheredλ=ω
0. Denoting bya(γ)the smallest subset ofasuch thatγ⊂∂E(a(γ)),
and choosingλ=
+
n
i=1
xidyi,wereadily compute thatA(γ )is the least common
multiple of the elements ina(γ),
A(γ )=lcm(a(γ)). (2.2.6)
Assume now thatE(a)α
2E(b)andE(b)α 2E(a). Then there exists a sym-
plectomorphismϕofR
2n
such thatϕ(E(a))=E(b), and we see as in the proof of
Proposition 2.1.1 thatϕ(∂E(a))=∂E(b).Itfollows easily from the deﬁnition (2.1.1)
ofL
E(a)andL E(b)thatϕmaps the characteristic foliation on∂E(a)to the one of
∂E(b). Moreover, the actions of closed characteristics are preserved. Indeed, ifγ
is a closed characteristic on∂E(a),wechoose a smooth closed discD⊂R
2n
with
boundaryγand ﬁnd
/
ϕ(γ )
λ=
/
ϕ(D)
ω0=
/
D
ϕ
∗
ω0=
/
D
ω0=
/
γ
λ,

2.2 Rigidity for ellipsoids 27
so thatA(ϕ(γ))=A(γ ). Denoting thesimple action spectrumofE(a)by
σ(E(a))={A(γ)|γis a closed characteristic on∂E(a)}
we in particular haveσ(E(a))=σ(E(b)).
Ifa
1,...,anare linearly independent overZ, then∂E(a)carries only thenclosed
characteristics∂E(a
i)with actiona i,sothat
{a
1,...,an}=σ(E(a))=σ(E(b))={b 1,...,bn},
provingE(a)=E(b). The proof of the general case is not much harder: Of course,
a
1=min{σ(E(a))}=min{σ(E(b))}=b 1.
Arguing by induction we assume thata
i=bifori=1,...,k−1. Suppose that
a
k<bk.Wethen consider the subsets˙(a|a k)⊂∂E(a)and˙(b|a k)⊂∂E(b)
formed by those closed characteristics whose action dividesa
k.Bythe above discus-
sion,ϕ(˙(a|a
k))=˙(b|a k),sothat˙(a|a k)and˙(b|a k)must be homeomorphic.
On the other hand, let{a
i1
,...,ail
}be the set of thosea iin{a1,...,ak−1}which di-
videa
k.Wethen read off from (2.2.6) that˙(b|a k)=∂E(a i1
,...,ail
). Similarly,
˙(a|a
k)=∂E(a i1
,...,ail
,ak,...,ak+mk−1)wherem kis the multiplicity ofa kina.
In particular, dim˙(b|a
k)<dim˙(a|a k). This contradiction shows thata k≥bk.
Interchangingaandbwe also ﬁnda
k≤bk,sothata k=bk. This completes the
induction, and the identitivity ofα
2is proved in an elementary way. β
Recall thatα
2does not implyα 1in general. However, a suitable pinching condi-
tion guarantees that “linear” and “non linear” coincide:
Theorem 2.2.4.Letκ∈
δ
b
2
,b
γ
. Then the following statements are equivalent:
(i)B
2n
(κ)α 1E(a)α 1E(a
τ
)α1B
2n
(b),
(ii)B
2n
(κ)α 2E(a)α 2E(a
τ
)α2B
2n
(b),
(iii)B
2n
(κ)α 3E(a)α 3E(a
τ
)α3B
2n
(b).
Weshould mention that forn=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 Theorem2.2.4. 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)α 2E(a)α 2B
2n
(b).

28 2 Proof of Theorem 1
Hence, by the monotonicity of the ﬁrst Ekeland–Hofer capacityc 1we obtain
κ≤a
1≤b, (2.2.7)
and by the monotonicity ofc
n
κ≤c n(E(a))≤b. (2.2.8)
The estimates (2.2.7) andκ>b/2 imply 2a
1>b, whence the only Ekeland–Hofer
capacities ofE(a)possibly smaller thanbarea
1,...,an.Itfollows therefore from
(2.2.8) thata
n=cn(E(a)), whence c i(E(a))=a ifori=1,...,n. Similarly we
ﬁndc
i(E(a
τ
))=a
τ
i
fori=1,...,n, and fromE(a)α 2E(a
τ
)we concludea i≤a
τ
i
.
(iii)⇒(i) now follows by a similar reasoning as in the proof of the identitivity of
α
3. Indeed, starting from
B
2n
(κ)α 3E(a)α 3E(a
τ
)α3B
2n
(b),
the implication (2.2.2) shows that for anyδ
1,δ2,δ3∈]0,1[
B
2n
(δ3δ2δ1κ)α2E(δ2δ1a)α2E(δ1a
τ
)α2B
2n
(b).
Choosingδ
1,δ2,δ3so large thatδ 3δ2δ1κ>b/2wecan apply the already proved
implication to see
B
2n
(δ3δ2δ1κ)α1E(δ2δ1a)α1E(δ1a)α1B
2n
(b),
and sinceδ
1,δ2,δ3can 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. α
In Section 1.2.2 we gave a direct proof of Theorem 1. Here, we show how Theo-
rem 1 follows from Theorem 2.2.4. In the notation of this section, Theorem 1 reads
Theorem 2.2.5.Assume thatE(a
1,...,an)α3B
2n
(A)for someA<a n. Then
a
n>2a 1.
Proof.Arguing by contradiction we assumeE(a
1,...,an)α3B
2n
(A)for some
A<a
nanda n≤2a 1.Avolume comparison showsa 1<A. Hence,a 1∈
δ
A
2
,A
γ
.
Therefore,B
2n
(a1)α3E(a1,...,an)α3B
2n
(A), Theorem 2.2.4 and the equiva-
lence (2.2.5) imply thata
n≤A. This contradiction showsa n>2a 1,asclaimed.α
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

2.3 Rigidity for polydiscs ? 29
thatP(a
1,...,an)denotes the open symplectic polydisc. We may again assume
a
1≤a2≤···≤a n. The Ekeland–Hofer capacities of a polydisc are given by
c
j(P (a1,...,an))=ja 1,j=1,2,..., (2.3.1)
[21, Proposition 5], and so they only see the smallest areaa
1. 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 thatP(a
1,...,an)embeds into
P(A
1,...,An)by a linear symplectomorphism if and only ifa i≤Aifor alli,asthe
following example shows.
Lemma 2.3.1.Assumer>1+
√
2. Then there existsA<πr
2
such that the poly-
discP
2n
(π,...,π,πr
2
)embeds into the cubeC
2n
(A)=P
2n
(A,...,A)by a linear
symplectomorphism.
Proof.It is enough to prove the lemma forn=2. Consider the linear symplecto-
morphism given by
(z
1,z2)−(z
τ
1
,z
τ
2
)=
1
√
2
(z
1+z2,z1−z2).
For(z
1,z2)∈P(π,πr
2
)andi=1,2wehave

z
τ
i

2
≤
1
2
α
|z
1|
2
+|z2|
2
+2|z 1||z2|
β
<
1
2
+
r
2
2
+r. (2.3.2)
The right hand side of (2.3.2) is strictly smaller thanr
2
provided thatr>1+
√
2.α
Moreover, it is not known whether the full analogue of Proposition 2.2.1 for polydiscs
instead of ellipsoids holds true. LetP(n)be the collection of polydiscs
P(n)={P(a
1,...,an)}
and writeζ
lfor the restrictions of the relations≤ ltoP(n),l=1,2,3. Againζ 2and
ζ
3are very similar, and again all the relationsζ lare clearly reﬂexive and transitive.
Furthermore, the smooth part of the boundary∂P(a
1,...,an)is foliated by closed
characteristics with actionsa
1,...,an,sothat the identitivity ofζ 2and hence the one
ofζ
1follows at once. The identitivity ofζ 2also follows from a result proved in [26]
by using symplectic homology:Symplectomorphic polydiscs are equal.Forn=2,
the identitivity ofζ
3follows 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. Forn≥3, however, we do not know
whether the relationζ
3is identitive. In particular, we have no answer to the following
question.

30 2 Proof of Theorem 1
Question 2.3.2.Assume that there exist symplectic embeddings
P(a
1,a2,a3)ı→P(a 1,a
τ
2
,a
τ
3
)andP(a 1,a
τ
2
,a
τ
3
)ı→P(a 1,a2,a3).
Is it then true thata
2=a
τ
2
anda 3=a
τ
3
?
Wealso do not know whether the polydisc-analogue of Theorem 1 or of Theo-
rem 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 con-
struction 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 Chap-
ter 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 ellipsoidE(a 1,...,an)is deﬁned by
E(a
1,...,an)=
6
(z 1,...,zn)∈C
n

. n
i=1π|zi|
2
ai
<1
7
. (3.1.1)
Theorem 2 in Section 1.3.1 clearly can be reformulated as follows.
Theorem 3.1.1.Assumea>2π. ThenE
2n
(π,...,π,a)symplectically embeds into
B
2n
α
a
2
+π+≥
β
for every≥>0.
The symplectic folding construction of Lalonde and McDuff 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 dimen-
sional in nature. We will reﬁne the method in such a way that it allows us to prove
Theorem 3.1.1 for everyn≥2.
Weshall conclude Theorem 3.1.1 from the following proposition in dimension 4.
Proposition 3.1.2.Assumea>2π. Given≥>0there exists a symplectic embedding
˚:E(a,) ı→B
4

a
2
+π+≥

satisfying
π|˚(z
1,z2)|
2
<
a
2
+≥+
π
2
|z1|
2
a
+π|z
2|
2
for all(z 1,z2)∈E(a,π).

32 3 Proof of Theorem 2
Werecall that|·|denotes the Euclidean norm. Postponing the proof, we ﬁrst show
that Proposition 3.1.2 implies Theorem 3.1.1.
Corollary 3.1.3.Assume that˚is as in Proposition3.1.2. Then the composition
of the permutationE
2n
(π,...,π,a)→E
2n
(a,π,...,π)with the restriction of
˚×id
2n−4toE
2n
(a,π,...,π)embedsE
2n
(π,...,π,a)intoB
2n
∗
a
2
+π+≥
⊕
.
Proof.Letz=(z
1,...,zn)∈E
2n
(a,π,...,π).ByProposition 3.1.2 and the
deﬁnition (3.1.1) of the ellipsoid,
π|˚×id
2n−4(z)|
2
=π

|˚(z 1,z2)|
2
+
n
.
i=3
|zi|
2

<
a
2
+≥+
π
2
|z1|
2
a
+π
n
.
i=2
|zi|
2
=
a
2
+≥+π

π|z
1|
2
a
+
n
.
i=2
π|zi|
2
π

<
a
2
+≥+π,
as claimed. ∗
It remains to prove Proposition 3.1.2. In order to do so, we start with some
preparations.
The ﬂexibility of 2-dimensional area preserving maps is crucial for the construction
of the map˚.Wenowmake sure that we can describe such a map by prescribing it
on an exhausting and nested family of embedded loops. Recall thatD(a)denotes the
open disc of areaacentred at the origin, and that|U|denotes the area of a domain
U⊂R
2
.
Deﬁnition 3.1.4.AfamilyLof loops in a simply connected domainU⊂R
2
is called
admissibleif there is a diffeomorphismβ:D(|U|)\{0}→U \{p}for some point
p∈Usuch that
(i) concentric circles are mapped to elements ofL,
(ii) in a neighbourhood of the originβis a translation.
Lemma 3.1.5.LetUandVbe bounded and simply connected domains inR
2
of equal
area and letL
UandL Vbe admissible families of loops inUandV,respectively.
Then there is a symplectomorphism betweenUandVmapping loops to loops.
Remark 3.1.6.The regularity condition (ii) imposed on the families taken into con-
sideration can be weakened. Some condition, however, is necessary. Indeed, ifL
Uis

3.1 Reformulation of Theorem 2 33
afamily of concentric circles andL
Vis a family of rectangles with smooth corners
and width larger than a positive constant, then no bijection fromUtoVmapping
loops to loops is continuous at the origin. β
Proof of Lemma3.1.5. Denote the concentric circle of radiusrbyC(r).Wemay
assume thatL
U={C(r)},0<r<R. Letβbe the diffeomorphism parametrizing
(V\{p},L
V). After reparametrizing ther-variable by a diffeomorphism of]0,R[
which is the identity near 0 we may assume thatβmaps the loopC(r)of radiusr
to the loopL(r)inL
Vwhich encloses the domainV(r)of areaπr
2
.Wedenote the
Jacobian ofβatre
iϕ
byβ
τ
(re
iϕ
). Sinceβis a translation near the origin andUis
connected, detβ
τ
(re
iϕ
)>0.By our choice ofβ,
πr
2
=|V(r)|=
/
D(πr
2
)
detβ
τ
=
/
r
0
ρdρ
/
2π
0
detβ
τ
(ρe
iϕ
)dϕ.
Differentiating inrwe obtain
2π=
/
2π
0
detβ
τ
(re
iϕ
)dϕ. (3.1.2)
Deﬁne the smooth functionh:]0,R[×R→Ras the unique solution of the initial
value problem
d
dt
h(r, t)=1/detβ
τ
(re
ih(r,t)
), t∈R
h(r, t)=0,t =0
˘
(3.1.3)
depending on the parameterr.Weclaim that
h(r, t+2π)=h(r, t)+2π. (3.1.4)
It then follows, since the functionhis strictly increasing in the variablet, that for
everyrﬁxed the maph(r,·):R→Rinduces a diffeomorphism of the circleR/2πZ.
In order to prove the claim (3.1.4) we denote byt
0(r) >0 the unique solution of
h(r, t
0(r))=2π. Substitutingϕ=h(r, t)into formula (3.1.2) we obtain, using
detβ
τ
(re
ih(r,t)
)·
d
dt
h(r, t)=1, that
2π=
/
t0(r)
0
dt=t 0(r).
Henceh(r,2π)=2π. Therefore, the two functions int,h(r, t+2π)−2πandh(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},re
iϕ
−β(re
ih(r,ϕ)
).
It is area preserving. Indeed, representingαas the composition
re
iϕ
−(r, ϕ)−(r, h(r, ϕ))−re
ih(r,ϕ)
−β(re
ih(r,ϕ)
)

34 3 Proof of Theorem 2
we obtain for the determinant of the Jacobian
1
r
·
∂h
∂ϕ
(r, ϕ)·r·detβ
τ
(re
ih(r,ϕ)
)=1,
where we again have used (3.1.3). Finally,αis a translation in a punctured neighbour-
hood of the origin and thus smoothly extends to the origin. This ﬁnishes the proof of
Lemma 3.1.5. α
Consider a bounded domainU⊂Cand a continuous functionf:U→R
>0.
The setF(U, f )inC
2
deﬁned by
F(U, f )=

(z
1,z2)∈C
2
|z1∈U, π|z 2|
2
<f(z1)

is the trivial ﬁbration overUhaving as ﬁbre overz
1the disc of capacityf(z 1).
Given two such ﬁbrationsF(U, f )andF(V, g),a symplectic embeddingϕ:Uı→V
deﬁnes a symplectic embeddingϕ×id:F(U, f ) ı→F(V, g)if and only iff(z
1)≤
g(ϕ(z
1))for allz 1∈U.
Examples 3.1.7.1. The ellipsoidE(a,b)can be represented as
E(a,b)=F
ρ
D(a),f(z
1)=b
ρ
1−
π|z
1|
2
a
σσ
.
2. Deﬁne the open trapezoidT(a,b)byT(a,b)=F(R(a), g), where
R(a)={z
1=(u, v)|0<u<a,0<v<1 }
is a rectangle andg(z
1)=g(u)=b(1−u/a).We setT
4
(a)=T(a,a). The
example is inspired by [49, p. 54]. It will be very useful to think ofT(a,b)as
depicted in Figure 3.1. β
a
b
u
ﬁbre capacity
Figure 3.1. The trapezoidT(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”.

3.1 Reformulation of Theorem 2 35
Lemma 3.1.8.Assume≥>0. Then
(i)E(a,b)symplectically embeds intoT(a+≥, b+≥√,
(ii)T
4
(a)symplectically embeds intoB
4
(a+≥√.
Proof.(i) Set≥
τ
=a≥
2
/(ab+a≥+b≥√.Weare going to use Lemma 3.1.5 to construct
an area preserving diffeomorphismα:D(a)→R(a)such that for the ﬁrst coordinate
in the imageR(a),
u(α(z
1))≤π|z 1|
2
+≥
τ
for allz 1∈D(a), (3.1.5)
see Figures 3.2 and 3.3.
1
v
u
a
1
2
≥
τ
4a
≥
τ
4
≥
τ
2
3≥
τ
4
L0
L1
2
L1
Figure 3.2. Constructing the embeddingα.
In an “optimal world” we would choose the loopsˆL u,0<u<a,inthe image
R(a)as the boundaries of the rectangles with corners(0,0),(0,1),(u,0),(u,1).If
the familyˆL=

ˆL
u

induced a mapˆα,wewould then haveu
α
ˆα(z
1)
β
≤π|z 1|
2
for
all(z
1,z2)∈R(a). The non admissible familyˆLcan be perturbed to an admissible
familyLin such a way that the induced mapαsatisﬁes the estimate (3.1.5). Indeed,
choose the translation disc appearing in the proof of Lemma 3.1.5 as the disc of radius
≥
τ
/8 centred at(u 0,v0)=
α
≥
τ
2
,
1
2
β
.Forr<≥
τ
/8 the loopsL(r)are therefore the circles
centred at(u
0,v0).Inthe 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
L0be the boundary of the rectangle with corners
α
≥
τ
4
,
≥
τ
4a
β
and
α
3≥
τ
4
,1−
≥
τ
4a
β
, and letL1be the boundary ofR(a).Wedeﬁne a family of loopsLs
by linearly interpolating betweenL0andL1, i.e.,Lsis the boundary of the rectangle
with corners
ρ
(1−s)
≥
τ
4
,(1−s)
≥
τ
4a
σ
and
ρ
u
s,1−
≥
τ
4a
+
≥
τ
4a
s
σ
,s∈[0,1],

36 3 Proof of Theorem 2
whereu s=
3≥
τ
4
+s
α
a−
3≥
τ
4
β
. Sinceu
s<a, the area enclosed by
Lsis estimated
from below by
ρ
u
s−
≥
τ
4
σρ
1−2
≥
τ
4a
σ
>u
s−
3≥
τ
4
. (3.1.6)
Let{L
s},s∈[0,1[,bethe smooth family of smooth loops obtained from{
Ls}by
smoothing the corners as indicated in Figure 3.2. By choosing the smooth corners of
L
smore and more rectangular ass→1, we can arrange that the set
1
0<s<1
Lsis the
domain bounded byL
0and
L1. Moreover, by choosing all smooth corners rectangular
enough, we can arrange that the area enclosed byL
sand
Lsis less than≥
τ
/4. In view
of (3.1.6), the area enclosed byL
sis then at leastu s−≥
τ
. Complete the families{L(r)}
and{L
s}to an admissible familyLof loops inR(a)and letα:D(a)→R(a)be the
map deﬁned byL. Fix(z
1,z2)∈D(a).Ifα(z 1)lies on a loop inL\{L s}
0<s<1,
thenu(α(z
1))<
3≥
τ
4
≤π|z 1|
2
+≥
τ
, and so the required estimate (3.1.5) is satisﬁed.
Ifα(z
1)∈L sfor somes∈]0,1[, then the area enclosed byL sisπ|z 1|
2
, and so
π|z
1|
2
+≥
τ
>us≥u(α(z 1)),whence (3.1.5) is again satisﬁed. This completes the
construction of a symplectomorphismα:D(a)→R(a)satisfying (3.1.5). In the
sequel, we will illustrate a map likeαby a picture like in Figure 3.3.
Tocontinue the proof of (i) we shall show that(α(z
1), z2)∈T(a+≥, b+≥√
for every(z
1,z2)∈E(a,b),sothat the symplectic mapα×id embedsE(a,b)into
T(a+≥, b+≥√.Take(z
1,z2)∈E(a,b). Then, using the deﬁnition (3.1.1) ofE(a,b),
the estimate (3.1.5) and the deﬁnition of≥
τ
we ﬁnd
π|z
2|
2
<b
ρ
1−
π|z
1|
2
a
σ
≤b
ρ
1−
u(α(z
1))
a
+
≥
τ
a
σ
<b
ρ
1−
u(α(z
1))a+≥
σ
+b
≥
τ
a
=b
ρ
1−
u(α(z
1)) a+≥
σ
+≥−
≥
a+≥
α
a+≥
τ
β
≤b
ρ
1−
u(α(z
1))
a+≥
σ
+≥−
≥
a+≥
u(α(z
1))
=(b+≥√
ρ
1−
u(α(z
1)) a+≥
σ
.
It follows that
(α(z
1), z2)∈T(a+≥, b+≥√=F
ρ
R(a+≥√,(b+≥√
ρ
1−
u
a+≥
σσ
as claimed.
In order to prove (ii) we shall construct an area preserving diffeomorphismωfrom
a rectangular neighbourhood ofR(a)having smooth corners and areaa+≥toD(a+≥√
such that
π|ω(z
1)|
2
≤u+≥for allz 1=(u, v)∈R(a). (3.1.7)

3.1 Reformulation of Theorem 2 37
Such a mapωcan again be obtained with the help of Lemma 3.1.5. In an “optimal
world” we would choose the loopsˆL
uin the domainR(a)as before. This time, we
perturb this non admissible family to an admissible familyLof loops as illustrated in
Figure 3.3. If the smooth corners of all those loops inLwhich enclose an area greater
than≥/2 lie outsideR(a)and if the upper, left and lower edges of all these loops are
close enough, then the induced mapωwill satisfy (3.1.7).D(a)
D(a+≥√
z 1
z1
α
ω
1
1
v
v
u
u
a
a
Figure 3.3. The ﬁrst and the last base deformation.
RestrictingωtoR(a)we obtain a symplectic embeddingω×id:T
4
(a) ı→R
4
.
For(z
1,z2)∈T
4
(a)we haveπ|z 2|
2
<a(1−u/a), wherez 1=(u, v)∈R(a).In
view of (3.1.7) we conclude that
π
α
|ω(z
1)|
2
+|z2|
2
β
<u+≥+a

1−
u
a

=u+≥+a−u
=a+≥,
and so(ω×id)(z
1,z2)∈B
4
(a+≥√for all(z 1,z2)∈T
4
(a). α
Lemma 3.1.8 allows us to reformulate Proposition 3.1.2 as follows.

38 3 Proof of Theorem 2
Proposition 3.1.9.Assumea>2π. Given≥>0, there exists a symplectic embed-
ding
ﬁ:T(a,) ı→T
4

a
2
+π+≥

,(z
1,z2)−(z
τ
1
,z
τ
2
),
z
1=(u, v)andz
τ
1
=(u
τ
,v
τ
), satisfying
u
τ
+π|z
τ
2
|
2
<
a
2
+≥+
πu
a
+π|z
2|
2
for all(u, v, z 2)∈T(a,π). (3.1.8)
Postponing the proof, we ﬁrst show that Proposition 3.1.9 implies Proposition 3.1.2.
Corollary 3.1.10.Assume the statement of Proposition3.1.9holds true. Then there
exists a symplectic embedding˚:E(a,) ı→B
4
α
a
2
+π+≥
β
satisfying
π|˚(z
1,z2)|
2
<
a
2
+≥+
π
2
|z1|
2
a
+π|z
2|
2
for all(z 1,z2)∈E(a,π).(3.1.9)
Proof.Let≥
τ
>0besosmall thatca+≥
τ
>2π, wherec=1−≥
τ
/π.Asinthe
proof of Lemma 3.1.8 we can construct a symplectic embedding
α×id:E(ca,c) ı→T(ca+≥
τ
,cπ+≥
τ
)=T(ca+≥
τ
,π)
satisfying the estimate
u(α(z
1))≤π|z 1|
2
+
a(≥
τ
)
2
caπ+a≥
τ
+≥
τ
for allz 1∈D(ca) (3.1.10)
and another symplectic embedding
ω×id:T
4

ca
2
+π+≥
τ

ı→B
4

ca
2
+π+2≥
τ

satisfying
π|ω(z
1)|
2
≤u+≥
τ
for allz 1=(u, v)∈R

ca
2
+π+≥
τ

.(3.1.11)
Sinceca+≥
τ
>2π, Proposition 3.1.9 applied toca+≥
τ
replacingaand≥
τ
/2 replacing
≥guarantees a symplectic embedding
ﬁ:T(ca+≥
τ
,) ı→T
4

ca
2
+π+≥
τ

,
(z
1,z2)−(ﬁ 1(z1,z2), ﬁ2(z1,z2)), satisfying
u(ﬁ
1(α(z1), z2))+π|ﬁ 2(α(z1), z2)|
2
<
ca
2
+≥
τ
+
πu(α(z
1))
ca+≥
τ
+π|z 2|
2
(3.1.12)

3.2 The folding construction 39
for all(u(α(z
1)), v, z2)∈T(ca+≥
τ
,π). Setˆ˚=(ω×id)ˇﬁˇ(α×id). Then
ˆ˚symplectically embedsE(ca,cπ)intoB
4
α
ca
2
+π+2≥
τ
β
. Moreover, if(z
1,z2)∈
E(ca,cπ), then
π

ˆ˚(z
1,z2)

2
=π|ω(ﬁ 1(α(z1), z2))|
2
+π|ﬁ 2(α(z1), z2)|
2
(3.1.11)
≤u(ﬁ 1(α(z1), z2))+≥
τ
+π|ﬁ 2(α(z1), z2)|
2
(3.1.12)
<
ca
2
+2≥
τ
+
πu(α(z
1))
ca+≥
τ
+π|z 2|
2
(3.1.10)
≤
ca
2
+2≥
τ
+
π
2
|z1|
2
ca+≥
τ
+
π
ca+≥
τ
a(≥
τ
)
2
caπ+a≥
τ
+≥
τ
+π|z 2|
2
<
ca
2
+3≥
τ
+
π
2
|z1|
2
ca
+π|z
2|
2
where in the last step we again usedca+≥
τ
>2π.Nowchoose≥
τ
>0sosmall
that
π+3≥
τ
c
<π+≥.Wedenote the dilatation by
√
cinR
4
also by
√
c, and deﬁne
˚:E(a,π)→R
4
by˚=
α√
c
β
−1
ˇˆ˚ˇ
√
c. Then˚symplectically embedsE(a,π)
intoB
4
α
a
2
+
π+2≥
τ
c
β
⊂B
4
α
a2
+π+≥
β
, and sinceπ|z 1|
2
<afor all(z 1,z2)∈E(a,π)
and by the choice of≥
τ
,
π|˚(z
1,z2)|
2
=
π
c
ˆ˚
α√
cz1,
√
cz2
β
2
<
1
c
ρ
ca
2
+3≥
τ
+
π
2
|z1|
2
a
+πc|z
2|
2
σ
=
a
2
+
3≥
τ
c
+
1
c
π
2
|z1|
2
a
+π|z
2|
2
<
a
2
+≥+
π
2
|z1|
2
a
+π|z
2|
2
for all(z 1,z2)∈E(a,π). This proves the required estimate (3.1.9), and so the proof
of Corollary 3.1.10 is complete. α
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 embeddingﬁas 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
byz
1=(u, v)andz 2=(x, y), respectively.

40 3 Proof of Theorem 2
Step 1.Following [49, Lemma 2.1] we ﬁrst separate the “low” regions overR(a)
from the “high” ones. We may do this using Lemma 3.1.5. We prefer, however, to
give an explicit construction.
Letδ>0besmall. SetF=F(U, f ), where Uandfare described in Figure 3.4,
and write
P
1=U∩
6
u≤
a
2
+δ
7
,
P
2=U∩
6
u≥
a+π
2
+11δ
7
,
L=U\(P
1∪P2).
v
u
u
1
P
1 P2
L
U
δ
a
2
+δ
a+π
2
+11δa +
π
2
+12δ
f
π
π
2
Figure 3.4. Separating the low ﬁbres from the large ﬁbres.
Hence,Uis the disjoint union
U=P
1
2
L
2
P
2.
Choose a smooth functionh:[0,a+δ]→]0,1]as in Figure 3.5, i.e.
(i)h(w)=1 forw∈
γ
0,
a
2
δ
,
(ii)h
τ
(w) <0 forw∈
δ
a
2
,
a
2
+δ
2
γ
,
(iii)h
α
a
2
+δ
2
β
=δ,
(iv)h(w)=h(a−w)for allw∈[0,a+δ].

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Embedding Problems In Symplectic Geometry Felix Schlenk

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