![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 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. 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 vol-
ume 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 sup-
ported 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.
GlobalpropertiesdistinguishingHamiltonianorsymplecticdiffeomorphismsfrom
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.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-18-320.jpg)
![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 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 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 C0-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 is C1-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 Gro-
mov and Eliashberg. In the early 1970s, Gromov proved the following alternative.
Gromov’sAlternative. The group of symplectomorphisms of R2n is either C0-closed
in the group of all diffeomorphisms (hardness), or its C0-closure is the group of volume
preserving diffeomorphisms (softness).
Notice that “symplectic” is a C1-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.
C0-stability for symplectomorphisms. The group of symplectomorphisms of R2n
is C0-closed in the group of all diffeomorphisms.
It follows that volume preserving diffeomorphisms of R2n cannot be C0-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 map ϕ : U → R2n de-
fined 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
preserve the volume form 0 = 1
n! ωn
0 and are thus immersions.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-19-320.jpg)
![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, r2/2[. Then the map
R2
→]0, r2
/2[×R, (x, y) →
f (x),
y
f (x)
, (1.2.1)
is a symplectomorphism.
Step 2. The map
]0, r2
/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 em-
beddings 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 x2
+ y2
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
symplectically embeds into V via a Hamiltonian diffeomorphism of R2 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 Hamilto-
nian 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.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-20-320.jpg)
![1.2 Symplectic embedding obstructions 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 inAppendixA. 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 inAppendix 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
B2n
πr2
=
6
(x, y) ∈ R2n
.n
i=1
x2
i + y2
i r2
7
and the open 2n-dimensional symplectic cylinder
Z2n
(π) = (x, y) ∈ R2n
| x2
1 + y2
1 1 .
While the ball B2n(a) has finite volume for each a, the symplectic cylinder Z2n(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 B2n(a) symplectically embeds into the
cylinder Z2n(π) if and only if a ≤ π.
Remarks. 1. Proposition 1 shows that for n ≥ 2 the whole R2n embeds into Z2n(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)](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-21-320.jpg)
![8 1 Introduction
of R2n embeds the ball of radius −1 into Z2n(π).
2. The “symplectic cylinder” Z2n(π) in the Nonsqueezing Theorem cannot be
replaced by the “Lagrangian cylinder”
(x, y) ∈ R2n
| x2
1 + x2
2 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
C0-Stability Theorem at once. In [20], Ekeland and Hofer observed that the C0-
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
of B2n(a) into Z2n(π) for a π is the symplectic area
,
S ω0 of a holomorphic curve
S in B2n(a) passing through the centre, which is at least a π.
Using Gromov’s compactness theorem for pseudo-holomorphic discs with La-
grangian boundary conditions, Sikorav found another amazing Nonsqueezing Theo-
rem. Let S1 be the unit circle in R2(x, y), and consider the torus T n = S1 × · · · × S1
in R2n. Sikorav proved in [79] that there does not exist a symplectomorphism of R2n
which maps T n into Z2n(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 HedgehogTheorem. For n ≥ 2, no starshaped domain in R2n containing
the torus T n symplectically embeds into the cylinder Z2n(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
λU = {λz | z ∈ U}.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-22-320.jpg)
![1.2 Symplectic embedding obstructions 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
B2n(a) with a π and a starshaped domain containing the “round” torus T n, the set
V was the “long and thin” cylinder Z2n(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
E(a1, . . . , an) =
6
(z1, . . . , zn) ∈ Cn
.n
i=1
π|zi|2
ai
1
7
.
Here, | · | denotes the Euclidean norm in R2. Notice that E(a, . . . , a) = B2n(a).
Since a permutation of the symplectic coordinate planes is a (linear) symplectic map,
we may assume a1 ≤ a2 ≤ · · · ≤ an. If, for instance, a1 = · · · = an−1 and an is
much larger than a1, then E(a1, . . . , an) is indeed “long and thin”. With these choices
for U and V , Problem UV specializes to
Problem EB. What is the smallest ball B2n(A) into which E(a1, . . . , an) symplecti-
cally embeds?
Of course, the inclusion symplectically embeds E(a1, . . . , an) into B2n(A) if
A ≥ an. The following rigidity result shows that one cannot do better if the ellipsoid
is still “quite round”.
Theorem 1. Assume an ≤ 2a1. Then the ellipsoid E(a1, . . . , an) does not symplecti-
cally embed into the ball B2n(A) if A an.
In the case n = 2, Theorem 1 was proved in [26] as an application of symplectic
homology. Our proof is simpler and works in all dimensions. It uses the n’th Ekeland–
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 domains U and V in R2n a
symplectic capacity c associates numbers c(U) and c(V ) in [0, ∞] in such a way that](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-23-320.jpg)
![1.3 Symplectic embedding constructions 11
is called the action spectrum of U. In [20], Ekeland and Hofer associated with U
a number c1(U) defined via critical point theory applied to the classical action func-
tional of Hamiltonian dynamics, and in this way obtained a symplectic capacity c1
satisfying c1(U) ∈ (U). If U is convex, then c1(U) is the smallest number in (U).
Therefore, c1 (E(a1, . . . , an)) = a1. From this, the Nonsqueezing Theorem follows
at once, and also the Symplectic Hedgehog Theorem can be proved by using the sym-
plectic capacity c1, see [79] and [83]. But again, c1 is useless for Problem EB. In [21],
then, Ekeland and Hofer repeated their construction from [20] in an S1-equivariant
setting and used S1-equivariant cohomology to obtain a whole family c1 ≤ c2 ≤ · · ·
of symplectic capacities satisfying cj (U) ∈ (U). Besides c1, these capacities are
not normalized, and for an ellipsoid E = E(a1, . . . , an) they are given by
{c1(E) ≤ c2(E) ≤ · · · } = {kai | k = 1, 2, . . . ; i = 1, . . . , n} .
For an ellipsoid E(a1, . . . , an) with an ≤ 2a1 and for a ball B2n(A) we therefore find
cn (E(a1, . . . , an)) = an and cn
B2n
(A)
= A,
so that Theorem 1 follows in view of the monotonicity of cn. In Chapter 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, andthatthemethodsusedintheirproofsarequitedifferent. Theobstructions
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-called h-principle for symplectic embeddings
of codimension at least 2 and via the symplectic blow-up operation. While the former](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-25-320.jpg)
![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 assumption an ≤ 2a1 in Theorem 1 cannot be omitted.
Symplectic embeddings via the h-principle. For any a 0 there exists an 0
such that the 2n-dimensional ellipsoid E(, . . . , , a) symplectically embeds into
B2n(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 B2(a) into B2n(π). According to [32, p. 335] or [24, Theorem 12.1.1],
arbitrarily C0-close to ϕ0 there exists a symplectic embedding ϕ1 : B2(a) → B2n(π),
meaning that ϕ∗
1 ω0 = ω0. Using the Symplectic Neighbourhood Theorem, we find
0 such that ϕ1 extends to a symplectic embedding of B2n−2() × B2(a) and in
particular to a symplectic embedding of E(, . . . , , a). 2
Since the C0-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
(x, y) →
Ax,
AT
−1
y
(1.3.1)
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 Propo-
sition 1 there are plenty of such embeddings. An example are the “inverse symplectic
polar coordinates”
(x, y) →
√
2x cos y,
√
2x sin y
(1.3.2)
embedding
0, a/2π
× ]0, 2π[ into B2(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](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-26-320.jpg)
![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 defi-
nition (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 Hamiltonian vector field of the Hamiltonian function (u, v, x, y) → −f (u)x is
0, f (u)x, 0, f (u)
, so that its time-1-map
(u, v, x, y) →
u, v + f
(u)x, x, y + f (u)
(1.3.3)
fixesthehalfspace{u ≤ 0}andliftsthespace{u ≥ 1}by1inthey-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 (1.3.4)
we find “true” lifts (called elevators in the US, I guess). 3
At first glance, these elementary symplectic embeddings look useless for Prob-
lem EB. Indeed, none of them embeds the ellipsoid E(a1, . . . , an) into a ball B2n(A)
with A an. 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. Given λ 0
the ellipsoid λE(a1, a2) symplectically embeds into the ball λB4(A) if and only if
E(a1, a2) symplectically embeds into B4(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
→ B4
(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](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-27-320.jpg)
![14 1 Introduction
a product of area preserving embeddings to view an ellipsoid as a Lagrangian prod-
uct × 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)/2πZ2 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’sWrapping 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 NonsqueezingTheorem 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(B4(a2)), so that E does not fit into a ball B4(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 B4(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 B2n(an − δ) for every δ ∈
0, an
2 − 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 B2n(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
B6(A) if A 2a, but we do not know the answer to
Question 1. Does the ellipsoid E(a, 2a, 3a) symplectically embed into B6(A) for
some A 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](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-28-320.jpg)
![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 case n = 2. A refinement of their sym-
plectic folding construction in dimension 4 will prove Theorem 2 in all dimensions.
Thesymplecticfoldingconstructionisacompositionofproductsofareapreserving
embeddings and a lift. Viewing an ellipsoid E(a1, a2) as fibred over the larger disc
B2(a2), this construction first separates the smaller fibres from the larger ones by a
suitable area preserving embedding of B2(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
preservingembedding. AnideaoftheconstructioncanbeobtainedfromFigure3.12on
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) → B4(A) are encoded in the
“characteristic function” χEB on [π, ∞[ defined by
χEB(a) = inf A | E(π, a) symplectically embeds into B4
(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) ≥ 2π. This information is vacuous if a ≥ 4π, since the
volume condition Vol
E(π, a)
≤ Vol
B4(χEB(a))
translates to χEB(a) ≥
√
πa.
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 B4
(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+
fEB(2π + ) − 2π
≤
3
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+
χEB(2π + ) − 2π
3
7
?](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-29-320.jpg)
![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)−
√
πa between wEB and the volume condition is bounded
by (3 −
√
3)π. We shall also prove that fEB(a) −
√
πa is bounded. It follows that
χEB(a) −
√
πa is bounded. We in particular have
lim
a→∞
Vol (E(π, a))
Vol
B4 (χEB(a))
= 1. (1.3.5)
This means that the embedding obstructions encountered for small a more and more
disappear as a → ∞.
2π
2π
3π
4π
4π
5π
6π
6π 8π 12π 15π 20π 24π
a
A
fEB(a)
wEB(a) χEB(a) ?
A = a A = a
2 + π
A =
√
πa
c2
Figure 1.1. What is known about the embedding problem E(π, a) → B4(A).
Denote by D(a) = B2(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).](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-30-320.jpg)
![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 ϕ∗
U ω0 = ω.
Therefore, a symplectic manifold is a smooth 2n-dimensional manifold admitting
an atlas {(U, ϕU )} such that all coordinate changes
ϕV ϕ−1
U : 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
where as before 0 = 1
n! ωn
0 and = 1
n! ωn. 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)
where we set Vol (M, ω) = 1
n!
,
M ωn. For volume preserving embeddings, this obvi-
ous 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](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-32-320.jpg)
![1.3 Symplectic embedding constructions 19
the product (M, ω) = (S2 × S2, σ ⊕ kσ), where σ is an area form on the 2-sphere S2
of area
,
S2 σ = π and where k ≥ 1. Then the ball B4(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 2n-dimensional symplectic manifold
(M, ω) of finite volume Vol (M, ω) = 1
n!
,
M ωn, and define for each a ≥ π the
numbers
pE
a (M, ω) = sup
λ
Vol
λE (π, . . . , π, a)
Vol (M, ω)
,
pP
a (M, ω) = sup
λ
Vol
λP (π, . . . , π, a)
Vol (M, ω)
,
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
a→∞
pE
a (M, ω) = 1 and lim
a→∞
pP
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 inTheorem 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 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 embeddings ϕ of the ball B2n(a) into the symplectic cylinder
Z2n(π) whose image ϕ(B2n(a)) ⊂ Z2n(π) has a specific property. By Gromov’s
Nonsqueezing Theorem there does not exist a symplectic embedding of B2n(a) into
Z2n(π) if a π. So fix a ∈ ]0, π]. The simply connected hull T̂ 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 µ̂(T ) = µ(T̂ ). For the unit circle](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-33-320.jpg)
![20 1 Introduction
S1 in R2 we have µ(S1) = 0 π = µ̂(S1). As is well-known, the Nonsqueezing
Theorem is equivalent to each of the identities
a = inf
ϕ
µ
p ϕ
B2n
(a)
,
a = inf
ϕ
µ̂
p ϕ
B2n
(a)
,
where ϕ varies over all symplectomorphisms of R2n which embed B2n(a) into Z2n(π)
and where p: Z2n(π) → B2(π) is the projection, see [22] and Appendix C.2. Fol-
lowing McDuff, [59], we consider sections of the image ϕ(B2n(a)) instead of its
projection, and define
ζ(a) := inf
ϕ
sup
x
µ
p ϕ(B2n
(a)) ∩ Dx
,
ζ̂(a) := inf
ϕ
sup
x
µ̂
p ϕ(B2n
(a)) ∩ Dx
,
where ϕ again varies over all symplectomorphisms of R2n which embed B2n(a) into
Z2n(π), and where Dx ⊂ Z2n(π) denotes the disc
Dx = B2
(π) × {x}, x ∈ R2n−2
.
Clearly,
ζ(a) ≤ ζ̂(a) ≤ a.
It is also well-known that the Nonsqueezing Theorem is equivalent to the identity
ζ̂(π) = π. (1.3.6)
Indeed, the Nonsqueezing Theorem implies that for every symplectomorphism ϕ
of R2n which embeds B2n(π) into Z2n(π) there exists an x in R2n−2 such that
ϕ(B2n(π)) ∩ Dx contains the unit circle S1 × {x}. On her search for symplectic
rigidity phenomena beyond the Nonsqueezing Theorem, McDuff 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 ζ̂(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.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-34-320.jpg)
![1.3 Symplectic embedding constructions 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 B2n(a) symplectically embeds into (M, ω)?
Equivalently, one studies the k’th symplectic packing number
pk(M, ω) = sup
a
k Vol
B2n(a)
Vol (M, ω)
where the supremum is taken over all those a for which
1k
i=1 B2n(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(B2n(π), ω0) ≤ k
2n 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
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 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 maximalsymplecticpackingswereconstructedbyKarshon[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 S2 × S2 of 2-spheres of equal area by 5 balls can be described by
Figure 1.2.
Thesesymplecticpackingsaresimplyobtainedviaproductsα1×α2 ofsuitablearea
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 ln balls for each l ∈ N
in a most simple way.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-35-320.jpg)
![22 1 Introduction
Figure 1.2. Maximal symplectic packings of the 4-ball by 5 or 6 balls and of S2 ×S2 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 of k for 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
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 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 the C∞-category, i.e., all manifolds and dif-
feomorphisms are assumed to be C∞-smooth, and so are all symplectic forms and
maps.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-36-320.jpg)
![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.
WedenotebyO(n)thesetofboundeddomainsinR2n diffeomorphictoaball. Each
U ∈ O(n) is endowed with the standard symplectic structure ω0 =
+n
i=1 dxi ∧ dyi.
Orienting U by the volume form 0 = 1
n! ωn
0 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},](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-37-320.jpg)
![24 2 Proof of Theorem 1
and we set U = C2n(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
and meets LXω0 = dιXω0 = ω0. E.g., the radial vector field X = 1
2
∂
∂r on R2n
meets LXω0 = 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 B2n(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 C0-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)](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-38-320.jpg)
![2.2 Rigidity for ellipsoids 25
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) = λ2c(S) for all λ ∈ R {0}.
A3. Nontriviality: 0 c(B2n(π)) and c(Z2n(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(δ2δ1a) 2 E(δ1a
) 2 E(a).
Now the monotonicity property (A1) of the capacities and the set of relations in (2.2.3)
immediately imply that a = a. This completes the proof of Proposition 2.2.1. 2](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-39-320.jpg)
![26 2 Proof of Theorem 1
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 ≤ a
i 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 char-
acteristic 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
H(z) =
+n
i=1
π|zi|2
ai
. Using definition (1.1.3) we find
XH (z) = −2πJ
1
a1
z1, . . . , 1
an
zn
where J =
√
−1 ∈ C = R2(x, y) is the standard complex structure. The char-
acteristic on ∂E(a) through z = z(0) can therefore by parametrized as z(t) =
(z1(t), . . . , zn(t)), where
zi(t) = e−2πJt/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−2πJt/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(a1
) ∪ · · · ∪ ∂E(ad
)
where a1∪· · ·∪ad 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 ,
where dλ = ω0. Denoting by a(γ ) the smallest subset of a such that γ ⊂ ∂E (a(γ )),
and choosing λ =
+n
i=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 sym-
plectomorphism 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
/
ϕ(γ )
λ =
/
ϕ(D)
ω0 =
/
D
ϕ∗
ω0 =
/
D
ω0 =
/
γ
λ,](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-40-320.jpg)
![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 discus-
sion, 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 di-
vide 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
induction, and the identitivity of 2 is proved in an elementary way. 3
Recall that 2 does not imply 1 in general. However, a suitable pinching condi-
tion guarantees that “linear” and “non linear” coincide:
Theorem 2.2.4. Let κ ∈
b
2 , b
. Then the following statements are equivalent:
(i) B2n(κ) 1 E(a) 1 E(a) 1 B2n(b),
(ii) B2n(κ) 2 E(a) 2 E(a) 2 B2n(b),
(iii) B2n(κ) 3 E(a) 3 E(a) 3 B2n(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,
B2n
(κ) 2 E(a) 2 B2n
(b).](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-41-320.jpg)
![28 2 Proof of Theorem 1
Hence, by the monotonicity of the first Ekeland–Hofer capacity c1 we obtain
κ ≤ a1 ≤ b, (2.2.7)
and by the monotonicity of cn
κ ≤ cn(E(a)) ≤ b. (2.2.8)
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)) = a
i for i = 1, . . . , n, and from E(a) 2 E(a) we conclude ai ≤ a
i.
(iii) ⇒ (i) now follows by a similar reasoning as in the proof of the identitivity of
3. Indeed, starting from
B2n
(κ) 3 E(a) 3 E(a
) 3 B2n
(b),
the implication (2.2.2) shows that for any δ1, δ2, δ3 ∈ ]0, 1[
B2n
(δ3δ2δ1κ) 2 E(δ2δ1a) 2 E(δ1a
) 2 B2n
(b).
Choosing δ1, δ2, δ3 so large that δ3δ2δ1κ b/2 we can apply the already proved
implication to see
B2n
(δ3δ2δ1κ) 1 E(δ2δ1a) 1 E(δ1a) 1 B2n
(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 Theo-
rem 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 B2n(A) for some A an. Then
an 2a1.
Proof. Arguing by contradiction we assume E(a1, . . . , an) 3 B2n(A) for some
A an and an ≤ 2a1. A volume comparison shows a1 A. Hence, a1 ∈
A
2 , A
.
Therefore, B2n(a1) 3 E(a1, . . . , an) 3 B2n(A), Theorem 2.2.4 and the equiva-
lence (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](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-42-320.jpg)
![2.3 Rigidity for polydiscs ? 29
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)) = ja1, 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 πr2 such that the poly-
disc P2n(π, . . . , π, πr2) embeds into the cube C2n(A) = P2n(A, . . . , A) by a linear
symplectomorphism.
Proof. It is enough to prove the lemma for n = 2. Consider the linear symplecto-
morphism given by
(z1, z2) → (z
1, z
2) =
1
√
2
(z1 + z2, z1 − z2).
For (z1, z2) ∈ P(π, πr2) and i = 1, 2 we have
z
i
2
≤
1
2
|z1|2
+ |z2|2
+ 2 |z1| |z2|
1
2
+
r2
2
+ r. (2.3.2)
The right hand side of (2.3.2) is strictly smaller than r2 provided that r 1 +
√
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.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-43-320.jpg)
![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 ellipsoid E(a1, . . . , an) is defined by
E(a1, . . . , an) =
6
(z1, . . . , zn) ∈ Cn
.n
i=1
π|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. Assume a 2π. Then E2n(π, . . . , π, a) symplectically embeds into
B2n
a
2 + π +
for every 0.
ThesymplecticfoldingconstructionofLalondeandMcDuffconsidersa4-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 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
: E(a, π) → B4 a
2
+ π +
satisfying
π|(z1, z2)|2
a
2
+ +
π2|z1|2
a
+ π|z2|2
for all (z1, z2) ∈ E(a, π).](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-45-320.jpg)
![3.1 Reformulation of Theorem 2 33
a family of concentric circles and LV is a family of rectangles with smooth corners
and width larger than a positive constant, then no bijection from U to V mapping
loops to loops is continuous at the origin. 3
Proof of Lemma 3.1.5. Denote the concentric circle of radius r by C(r). We may
assume that LU = {C(r)}, 0 r R. Let β be the diffeomorphism parametrizing
(V {p}, LV ). After reparametrizing the r-variable by a diffeomorphism of ]0, R[
which is the identity near 0 we may assume that β maps the loop C(r) of radius r
to the loop L(r) in LV which encloses the domain V (r) of area πr2. We denote the
Jacobian of β at reiϕ by β(reiϕ). Since β is a translation near the origin and U is
connected, det β(reiϕ) 0. By our choice of β,
πr2
= |V (r)| =
/
D(πr2)
det β
=
/ r
0
ρ dρ
/ 2π
0
det β
(ρeiϕ
) dϕ.
Differentiating in r we obtain
2π =
/ 2π
0
det β
(reiϕ
) dϕ. (3.1.2)
Define the smooth function h: ]0, R[ ×R → R as the unique solution of the initial
value problem
d
dt h(r, t) = 1/ det β
(reih(r,t)
), t ∈ R
h(r, t) = 0, t = 0
(3.1.3)
depending on the parameter r. We claim that
h(r, t + 2π) = h(r, t) + 2π. (3.1.4)
It then follows, since the function h is strictly increasing in the variable t, that for
every r fixed the map h(r, ·): R → R induces a diffeomorphism of the circle R/2πZ.
In order to prove the claim (3.1.4) we denote by t0(r) 0 the unique solution of
h(r, t0(r)) = 2π. Substituting ϕ = h(r, t) into formula (3.1.2) we obtain, using
det β(reih(r,t)) · d
dt h(r, t) = 1, that
2π =
/ t0(r)
0
dt = t0(r).
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)
)](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-47-320.jpg)
![34 3 Proof of Theorem 2
we obtain for the determinant of the Jacobian
1
r
·
∂h
∂ϕ
(r, ϕ) · r · det β
(reih(r,ϕ)
) = 1,
where we again have used (3.1.3). Finally, α is 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. 2
Consider a bounded domain U ⊂ C and a continuous function f : U → R0.
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
E(a, b) = F
D(a), f (z1) = b
1 −
π|z1|2
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
a
b
u
fibre capacity
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”.](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-48-320.jpg)
![3.1 Reformulation of Theorem 2 35
Lemma 3.1.8. Assume 0. Then
(i) E(a, b) symplectically embeds into T (a + , b + ),
(ii) T 4(a) symplectically embeds into B4(a + ).
Proof. (i) Set = a2/(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.
1
v
u
a
1
2
4a
4
2
3
4
L0
L1
2
L1
Figure 3.2. Constructing the embedding α.
In an “optimal world” we would choose the loops L̂u, 0 u a, in the image
R(a) as the boundaries of the rectangles with corners (0, 0), (0, 1), (u, 0), (u, 1). If
the family L̂ = L̂u induced a map α̂, we would then have u
α̂(z1)
≤ π |z1|2
for
all (z1, z2) ∈ R(a). The non admissible family L̂ 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 , 1
2
. 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
4 , 1 −
4a
, and let L1 be the boundary of R(a). We define a family of loops Ls
by linearly interpolating between L0 and L1, i.e., Ls is the boundary of the rectangle
with corners
(1 − s)
4
, (1 − s)
4a
and
us, 1 −
4a
+
4a
s
, s ∈ [0, 1],](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-49-320.jpg)
![36 3 Proof of Theorem 2
where us = 3
4 + s
a − 3
4
. Since us a, the area enclosed by Ls is estimated
from below by
us −
4
1 − 2
4a
us −
3
4
. (3.1.6)
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
Ls more and more rectangular as s → 1, we can arrange that the set
1
0s1 Ls is the
domain bounded by L0 and L1. Moreover, by choosing all smooth corners rectangular
enough, we can arrange that the area enclosed by Ls and Ls is less than /4. In view
of (3.1.6), the area enclosed by Ls is then at least us −. Complete the families {L(r)}
and {Ls} to an admissible family L of loops in R(a) and let α : D(a) → R(a) be the
map defined by L. Fix (z1, z2) ∈ D(a). If α(z1) lies on a loop in L {Ls}0s1,
then u (α(z1)) 3
4 ≤ π |z1|2
+ , and so the required estimate (3.1.5) is satisfied.
If α(z1) ∈ Ls for some s ∈ ]0, 1[, then the area enclosed by Ls is π |z1|2
, and so
π |z1|2
+ us ≥ u (α(z1)), whence (3.1.5) is again satisfied. This completes the
construction of a symplectomorphism α : D(a) → R(a) satisfying (3.1.5). In the
sequel, we will illustrate a map like α by a picture like in Figure 3.3.
To continue the proof of (i) we shall show that (α(z1), z2) ∈ T (a + , b + )
for every (z1, z2) ∈ E(a, b), so that the symplectic map α × id embeds E(a, b) into
T (a+, b+). Take (z1, z2) ∈ E(a, b). Then, using the definition (3.1.1) of E(a, b),
the estimate (3.1.5) and the definition of we find
π|z2|2
b
1 −
π|z1|2
a
≤ b
1 −
u (α(z1))
a
+
a
b
1 −
u (α(z1))
a +
+ b
a
= b
1 −
u (α(z1))
a +
+ −
a +
a +
≤ b
1 −
u (α(z1))
a +
+ −
a +
u (α(z1))
= (b + )
1 −
u(α(z1))
a +
.
It follows that
(α(z1), z2) ∈ T (a + , b + ) = F
R(a + ), (b + )
1 −
u
a +
as claimed.
In order to prove (ii) we shall construct an area preserving diffeomorphism ω from
a rectangular neighbourhood of R(a) having smooth corners and area a+ to D(a+)
such that
π|ω(z1)|2
≤ u + for all z1 = (u, v) ∈ R(a). (3.1.7)](https://image.slidesharecdn.com/25561864-250514164529-4cf892ca/85/Embedding-Problems-In-Symplectic-Geometry-Felix-Schlenk-50-320.jpg)
![40 3 Proof of Theorem 2
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
P1 = U ∩
6
u ≤
a
2
+ δ
7
,
P2 = U ∩
6
u ≥
a + π
2
+ 11δ
7
,
L = U (P1 ∪ P2).
v
u
u
1
P1 P2
L
U
δ
a
2 + δ a+π
2 + 11δ a + π
2 + 12δ
f
π
π
2
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, a
2
,
(ii) h(w) 0 for w ∈
a
2 , a
2 + δ2
,
(iii) h
a
2 + δ2
= δ,
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Embedding Problems In Symplectic Geometry Felix Schlenk
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- 5. 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
- 6. 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.) 27 Algebra in the Stone-Čech Compactification, N. Hindman, D. Strauss 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, Péter 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. Chu- barikov, Anatoly A. Karatsuba
- 7. Embedding Problems in Symplectic Geometry by Felix Schlenk ≥ Walter de Gruyter · Berlin · New York
- 8. Author Felix Schlenk Mathematisches Institut Universität Leipzig Augustusplatz 10/11 04109 Leipzig Germany e-mail: schlenk@math.uni-leipzig.de 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 앪 앝 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 516.316⫺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.
- 11. 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 B2n(r) of radius r into the infinite cylinder B2(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 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-
- 12. 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 teşekkür 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
- 13. 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 2 Proof of Theorem 1 23 2.1 Comparison of the relations ≤i 23 2.2 Rigidity for ellipsoids 24 2.3 Rigidity for polydiscs ? 28 3 Proof 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 6 Proof of Theorem 3 107 6.1 Proof of lima→∞ pP a (M, ω) = 1 107 6.2 Proof of lima→∞ pE a (M, ω) = 1 123 6.3 Asymptotic embedding invariants 147
- 14. x Contents 7 Symplectic wrapping 149 7.1 The wrapping construction 149 7.2 Folding versus wrapping 157 8 Proof of Theorem 4 162 8.1 A more general statement 162 8.2 A further motivation for Problem ζ 165 8.3 Proof by symplectic folding 168 8.4 Proof by symplectic lifting 177 9 Packing symplectic manifolds by hand 188 9.1 Motivations for the symplectic packing problem 189 9.2 The packing numbers of the 4-ball and CP2 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 invariants cB and cC 224 D Computer programs 235 E Some other symplectic embedding problems 238 References 241 Index 247
- 15. 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”, “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 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, ẍ = 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 ẍ(t) = ∇U(x(t), t). Introducing the auxiliary variables y = ẋ, this second order system becomes the first order system of twice as many equations ẋ(t) = y(t), ẏ(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 sys- tem (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 2 − U(x, t), (1.1.2)
- 16. 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 ẋ(t) = ∂H ∂y (x, y, t), ẏ(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 ω0 = n . i=1 dxi ∧ dyi (1.1.4) on R2n. It is called the standard symplectic form on R2n. It’s n’th exterior product is ωn 0 = 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 factor (−1) n(n−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 now that XH = ∂H ∂y , −∂H ∂x , so that the Hamiltonian system (1.1.3) takes the compact form ż(t) = XH (z(t), t). (1.1.6) 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 {ϕt H } defined by d dt ϕt H (z) = XH ϕt H (z), t , ϕ0 H (z) = z, z ∈ R2n , (1.1.7) is called the Hamiltonian flow of H. Each diffeomorphism ϕt H is called a Hamiltonian diffeomorphism. More generally, any time-t-map ϕt H 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 consider- ations. As the following facts show, there are important pay-offs, however. Fact 1. If H is time-independent, then H is preserved by the flow ϕt H .
- 17. 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), d dt H ϕt H = dH ϕt H d dt ϕt H = dH ϕt H XH ϕt H = ω0 XH , XH ϕt H , which vanishes because ω0 is skew-symmetric. 2 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 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. Therefore, d dt ϕt H ∗ ω0 = ϕt H ∗ LXH ω0 = 0 for all t, so that ϕt H ∗ ω0 = ω0. 2 A diffeomorphism ϕ of R2n is called symplectic diffeomorphism or symplecto- morphism if it preserves the symplectic form ω0, 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 (R2n, 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 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 0. Proof. ϕ∗ 0 = ϕ∗ 1 n! ωn 0 = 1 n! (ϕ∗ω0)n = 1 n! ωn 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.
- 18. 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 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. 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 vol- ume 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 sup- ported 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. GlobalpropertiesdistinguishingHamiltonianorsymplecticdiffeomorphismsfrom 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.
- 19. 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 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 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 C0-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 is C1-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 Gro- mov and Eliashberg. In the early 1970s, Gromov proved the following alternative. Gromov’sAlternative. The group of symplectomorphisms of R2n is either C0-closed in the group of all diffeomorphisms (hardness), or its C0-closure is the group of volume preserving diffeomorphisms (softness). Notice that “symplectic” is a C1-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. C0-stability for symplectomorphisms. The group of symplectomorphisms of R2n is C0-closed in the group of all diffeomorphisms. It follows that volume preserving diffeomorphisms of R2n cannot be C0-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 map ϕ : U → R2n de- fined 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 preserve the volume form 0 = 1 n! ωn 0 and are thus immersions.
- 20. 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, r2/2[. Then the map R2 →]0, r2 /2[×R, (x, y) → f (x), y f (x) , (1.2.1) is a symplectomorphism. Step 2. The map ]0, r2 /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 em- beddings 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 x2 + y2 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 symplectically embeds into V via a Hamiltonian diffeomorphism of R2 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 Hamilto- nian 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.
- 21. 1.2 Symplectic embedding obstructions 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 inAppendixA. 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 inAppendix 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 B2n πr2 = 6 (x, y) ∈ R2n .n i=1 x2 i + y2 i r2 7 and the open 2n-dimensional symplectic cylinder Z2n (π) = (x, y) ∈ R2n | x2 1 + y2 1 1 . While the ball B2n(a) has finite volume for each a, the symplectic cylinder Z2n(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 B2n(a) symplectically embeds into the cylinder Z2n(π) if and only if a ≤ π. Remarks. 1. Proposition 1 shows that for n ≥ 2 the whole R2n embeds into Z2n(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)
- 22. 8 1 Introduction of R2n embeds the ball of radius −1 into Z2n(π). 2. The “symplectic cylinder” Z2n(π) in the Nonsqueezing Theorem cannot be replaced by the “Lagrangian cylinder” (x, y) ∈ R2n | x2 1 + x2 2 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 C0-Stability Theorem at once. In [20], Ekeland and Hofer observed that the C0- 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 of B2n(a) into Z2n(π) for a π is the symplectic area , S ω0 of a holomorphic curve S in B2n(a) passing through the centre, which is at least a π. Using Gromov’s compactness theorem for pseudo-holomorphic discs with La- grangian boundary conditions, Sikorav found another amazing Nonsqueezing Theo- rem. Let S1 be the unit circle in R2(x, y), and consider the torus T n = S1 × · · · × S1 in R2n. Sikorav proved in [79] that there does not exist a symplectomorphism of R2n which maps T n into Z2n(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 HedgehogTheorem. For n ≥ 2, no starshaped domain in R2n containing the torus T n symplectically embeds into the cylinder Z2n(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 λU = {λz | z ∈ U}.
- 23. 1.2 Symplectic embedding obstructions 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 B2n(a) with a π and a starshaped domain containing the “round” torus T n, the set V was the “long and thin” cylinder Z2n(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 E(a1, . . . , an) = 6 (z1, . . . , zn) ∈ Cn .n i=1 π|zi|2 ai 1 7 . Here, | · | denotes the Euclidean norm in R2. Notice that E(a, . . . , a) = B2n(a). Since a permutation of the symplectic coordinate planes is a (linear) symplectic map, we may assume a1 ≤ a2 ≤ · · · ≤ an. If, for instance, a1 = · · · = an−1 and an is much larger than a1, then E(a1, . . . , an) is indeed “long and thin”. With these choices for U and V , Problem UV specializes to Problem EB. What is the smallest ball B2n(A) into which E(a1, . . . , an) symplecti- cally embeds? Of course, the inclusion symplectically embeds E(a1, . . . , an) into B2n(A) if A ≥ an. The following rigidity result shows that one cannot do better if the ellipsoid is still “quite round”. Theorem 1. Assume an ≤ 2a1. Then the ellipsoid E(a1, . . . , an) does not symplecti- cally embed into the ball B2n(A) if A an. In the case n = 2, Theorem 1 was proved in [26] as an application of symplectic homology. Our proof is simpler and works in all dimensions. It uses the n’th Ekeland– 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 domains U and V in R2n a symplectic capacity c associates numbers c(U) and c(V ) in [0, ∞] in such a way that
- 24. 10 1 Introduction A1. Monotonicity: c(U) ≤ c(V ) if U symplectically embeds into V . A2. Conformality: c(λU) = λ2c(U) for all λ ∈ R {0}. A3. Nontriviality: 0 c B2n(π) and c Z2n(π) ∞. A symplectic capacity c is normalized if A3. Normalization: c B2n(π) = c Z2n(π) = 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, B2n (a1) ⊂ E(a1, . . . , an) ⊂ Z2n (a1) := B2 (a1) × R2n−2 , so that c (E(a1, . . . , an)) = a1 for any normalized symplectic capacity. Given a sym- plectic embedding E(a1, . . . , an) → B2n(A), such a symplectic capacity therefore only yields a1 ≤ A, an information 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 domain U ⊂ R2n with smooth boundary ∂U. A closed characteristic on ∂U is an embedded circle in ∂U tangent to the characteristic line bundle LU = {(x, ξ) ∈ T ∂U | ω0(ξ, η) = 0 for all η ∈ Tx ∂U} . If ∂U is represented as a regular energy surface {x ∈ R2n | H(x) = const} of a smooth function H on R2n, then the Hamiltonian vector field XH restricted to ∂U is a section of LU in view of its definition (1.1.5), and so the traces of the periodic orbits of XH on ∂U are the closed characteristics on ∂U. The action A (γ ) of a closed characteristic γ on ∂U is defined as A (γ ) = / γ λ , where λ is any primitive of ω0. In view of Stokes’ Theorem, A(γ ) is the symplectic area , D ω0 of a closed disc D ⊂ R2n with boundary γ . The set (U) = kA (γ ) | k = 1, 2, . . . ; γ is a closed characteristic on ∂U
- 25. 1.3 Symplectic embedding constructions 11 is called the action spectrum of U. In [20], Ekeland and Hofer associated with U a number c1(U) defined via critical point theory applied to the classical action func- tional of Hamiltonian dynamics, and in this way obtained a symplectic capacity c1 satisfying c1(U) ∈ (U). If U is convex, then c1(U) is the smallest number in (U). Therefore, c1 (E(a1, . . . , an)) = a1. From this, the Nonsqueezing Theorem follows at once, and also the Symplectic Hedgehog Theorem can be proved by using the sym- plectic capacity c1, see [79] and [83]. But again, c1 is useless for Problem EB. In [21], then, Ekeland and Hofer repeated their construction from [20] in an S1-equivariant setting and used S1-equivariant cohomology to obtain a whole family c1 ≤ c2 ≤ · · · of symplectic capacities satisfying cj (U) ∈ (U). Besides c1, these capacities are not normalized, and for an ellipsoid E = E(a1, . . . , an) they are given by {c1(E) ≤ c2(E) ≤ · · · } = {kai | k = 1, 2, . . . ; i = 1, . . . , n} . For an ellipsoid E(a1, . . . , an) with an ≤ 2a1 and for a ball B2n(A) we therefore find cn (E(a1, . . . , an)) = an and cn B2n (A) = A, so that Theorem 1 follows in view of the monotonicity of cn. In Chapter 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, andthatthemethodsusedintheirproofsarequitedifferent. Theobstructions 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-called h-principle for symplectic embeddings of codimension at least 2 and via the symplectic blow-up operation. While the former
- 26. 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 assumption an ≤ 2a1 in Theorem 1 cannot be omitted. Symplectic embeddings via the h-principle. For any a 0 there exists an 0 such that the 2n-dimensional ellipsoid E(, . . . , , a) symplectically embeds into B2n(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 B2(a) into B2n(π). According to [32, p. 335] or [24, Theorem 12.1.1], arbitrarily C0-close to ϕ0 there exists a symplectic embedding ϕ1 : B2(a) → B2n(π), meaning that ϕ∗ 1 ω0 = ω0. Using the Symplectic Neighbourhood Theorem, we find 0 such that ϕ1 extends to a symplectic embedding of B2n−2() × B2(a) and in particular to a symplectic embedding of E(, . . . , , a). 2 Since the C0-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 (x, y) → Ax, AT −1 y (1.3.1) 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 Propo- sition 1 there are plenty of such embeddings. An example are the “inverse symplectic polar coordinates” (x, y) → √ 2x cos y, √ 2x sin y (1.3.2) embedding 0, a/2π × ]0, 2π[ into B2(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
- 27. 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 defi- nition (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 Hamiltonian vector field of the Hamiltonian function (u, v, x, y) → −f (u)x is 0, f (u)x, 0, f (u) , so that its time-1-map (u, v, x, y) → u, v + f (u)x, x, y + f (u) (1.3.3) fixesthehalfspace{u ≤ 0}andliftsthespace{u ≥ 1}by1inthey-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 (1.3.4) we find “true” lifts (called elevators in the US, I guess). 3 At first glance, these elementary symplectic embeddings look useless for Prob- lem EB. Indeed, none of them embeds the ellipsoid E(a1, . . . , an) into a ball B2n(A) with A an. 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. Given λ 0 the ellipsoid λE(a1, a2) symplectically embeds into the ball λB4(A) if and only if E(a1, a2) symplectically embeds into B4(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 → B4 (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
- 28. 14 1 Introduction a product of area preserving embeddings to view an ellipsoid as a Lagrangian prod- uct × 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)/2πZ2 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’sWrapping 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 NonsqueezingTheorem 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(B4(a2)), so that E does not fit into a ball B4(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 B4(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 B2n(an − δ) for every δ ∈ 0, an 2 − 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 B2n(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 B6(A) if A 2a, but we do not know the answer to Question 1. Does the ellipsoid E(a, 2a, 3a) symplectically embed into B6(A) for some A 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
- 29. 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 case n = 2. A refinement of their sym- plectic folding construction in dimension 4 will prove Theorem 2 in all dimensions. Thesymplecticfoldingconstructionisacompositionofproductsofareapreserving embeddings and a lift. Viewing an ellipsoid E(a1, a2) as fibred over the larger disc B2(a2), this construction first separates the smaller fibres from the larger ones by a suitable area preserving embedding of B2(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 preservingembedding. AnideaoftheconstructioncanbeobtainedfromFigure3.12on 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) → B4(A) are encoded in the “characteristic function” χEB on [π, ∞[ defined by χEB(a) = inf A | E(π, a) symplectically embeds into B4 (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) ≥ 2π. This information is vacuous if a ≥ 4π, since the volume condition Vol E(π, a) ≤ Vol B4(χEB(a)) translates to χEB(a) ≥ √ πa. 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 B4 (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+ fEB(2π + ) − 2π ≤ 3 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+ χEB(2π + ) − 2π 3 7 ?
- 30. 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)− √ πa between wEB and the volume condition is bounded by (3 − √ 3)π. We shall also prove that fEB(a) − √ πa is bounded. It follows that χEB(a) − √ πa is bounded. We in particular have lim a→∞ Vol (E(π, a)) Vol B4 (χEB(a)) = 1. (1.3.5) This means that the embedding obstructions encountered for small a more and more disappear as a → ∞. 2π 2π 3π 4π 4π 5π 6π 6π 8π 12π 15π 20π 24π a A fEB(a) wEB(a) χEB(a) ? A = a A = a 2 + π A = √ πa c2 Figure 1.1. What is known about the embedding problem E(π, a) → B4(A). Denote by D(a) = B2(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).
- 31. 1.3 Symplectic embedding constructions 17 The “n-cube” P 2n(a, . . . , a) will be denoted by C2n(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) C2n(π) is the convex hull of T n, we could equally well have taken polydiscs and cubes. Problem PC. What is the smallest cube C2n(A) into which P(a1, . . . , an) symplec- tically 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 ellip- soids 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 that = 1 n! ωn is a volume form on M, so that M is orientable. The non-degeneracy together with the closedness of ω imply that (M, ω) is locally isomorphic to (R2n, ω0) with ω0 as in (1.1.4):
- 32. 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 ϕ∗ U ω0 = ω. Therefore, a symplectic manifold is a smooth 2n-dimensional manifold admitting an atlas {(U, ϕU )} such that all coordinate changes ϕV ϕ−1 U : 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 where as before 0 = 1 n! ωn 0 and = 1 n! ωn. 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) where we set Vol (M, ω) = 1 n! , M ωn. For volume preserving embeddings, this obvi- ous 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
- 33. 1.3 Symplectic embedding constructions 19 the product (M, ω) = (S2 × S2, σ ⊕ kσ), where σ is an area form on the 2-sphere S2 of area , S2 σ = π and where k ≥ 1. Then the ball B4(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 2n-dimensional symplectic manifold (M, ω) of finite volume Vol (M, ω) = 1 n! , M ωn, and define for each a ≥ π the numbers pE a (M, ω) = sup λ Vol λE (π, . . . , π, a) Vol (M, ω) , pP a (M, ω) = sup λ Vol λP (π, . . . , π, a) Vol (M, ω) , 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 a→∞ pE a (M, ω) = 1 and lim a→∞ pP 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 inTheorem 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 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 embeddings ϕ of the ball B2n(a) into the symplectic cylinder Z2n(π) whose image ϕ(B2n(a)) ⊂ Z2n(π) has a specific property. By Gromov’s Nonsqueezing Theorem there does not exist a symplectic embedding of B2n(a) into Z2n(π) if a π. So fix a ∈ ]0, π]. The simply connected hull T̂ 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 µ̂(T ) = µ(T̂ ). For the unit circle
- 34. 20 1 Introduction S1 in R2 we have µ(S1) = 0 π = µ̂(S1). As is well-known, the Nonsqueezing Theorem is equivalent to each of the identities a = inf ϕ µ p ϕ B2n (a) , a = inf ϕ µ̂ p ϕ B2n (a) , where ϕ varies over all symplectomorphisms of R2n which embed B2n(a) into Z2n(π) and where p: Z2n(π) → B2(π) is the projection, see [22] and Appendix C.2. Fol- lowing McDuff, [59], we consider sections of the image ϕ(B2n(a)) instead of its projection, and define ζ(a) := inf ϕ sup x µ p ϕ(B2n (a)) ∩ Dx , ζ̂(a) := inf ϕ sup x µ̂ p ϕ(B2n (a)) ∩ Dx , where ϕ again varies over all symplectomorphisms of R2n which embed B2n(a) into Z2n(π), and where Dx ⊂ Z2n(π) denotes the disc Dx = B2 (π) × {x}, x ∈ R2n−2 . Clearly, ζ(a) ≤ ζ̂(a) ≤ a. It is also well-known that the Nonsqueezing Theorem is equivalent to the identity ζ̂(π) = π. (1.3.6) Indeed, the Nonsqueezing Theorem implies that for every symplectomorphism ϕ of R2n which embeds B2n(π) into Z2n(π) there exists an x in R2n−2 such that ϕ(B2n(π)) ∩ Dx contains the unit circle S1 × {x}. On her search for symplectic rigidity phenomena beyond the Nonsqueezing Theorem, McDuff 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 ζ̂(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.
- 35. 1.3 Symplectic embedding constructions 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 B2n(a) symplectically embeds into (M, ω)? Equivalently, one studies the k’th symplectic packing number pk(M, ω) = sup a k Vol B2n(a) Vol (M, ω) where the supremum is taken over all those a for which 1k i=1 B2n(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(B2n(π), ω0) ≤ k 2n 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 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 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 maximalsymplecticpackingswereconstructedbyKarshon[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 S2 × S2 of 2-spheres of equal area by 5 balls can be described by Figure 1.2. Thesesymplecticpackingsaresimplyobtainedviaproductsα1×α2 ofsuitablearea 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 ln balls for each l ∈ N in a most simple way.
- 36. 22 1 Introduction Figure 1.2. Maximal symplectic packings of the 4-ball by 5 or 6 balls and of S2 ×S2 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 of k for 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 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 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 the C∞-category, i.e., all manifolds and dif- feomorphisms are assumed to be C∞-smooth, and so are all symplectic forms and maps.
- 37. 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. WedenotebyO(n)thesetofboundeddomainsinR2n diffeomorphictoaball. Each U ∈ O(n) is endowed with the standard symplectic structure ω0 = +n i=1 dxi ∧ dyi. Orienting U by the volume form 0 = 1 n! ωn 0 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},
- 38. 24 2 Proof of Theorem 1 and we set U = C2n(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 and meets LXω0 = dιXω0 = ω0. E.g., the radial vector field X = 1 2 ∂ ∂r on R2n meets LXω0 = 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 B2n(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 C0-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)
- 39. 2.2 Rigidity for ellipsoids 25 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) = λ2c(S) for all λ ∈ R {0}. A3. Nontriviality: 0 c(B2n(π)) and c(Z2n(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(δ2δ1a) 2 E(δ1a ) 2 E(a). Now the monotonicity property (A1) of the capacities and the set of relations in (2.2.3) immediately imply that a = a. This completes the proof of Proposition 2.2.1. 2
- 40. 26 2 Proof of Theorem 1 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 ≤ a i 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 char- acteristic 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 H(z) = +n i=1 π|zi|2 ai . Using definition (1.1.3) we find XH (z) = −2πJ 1 a1 z1, . . . , 1 an zn where J = √ −1 ∈ C = R2(x, y) is the standard complex structure. The char- acteristic on ∂E(a) through z = z(0) can therefore by parametrized as z(t) = (z1(t), . . . , zn(t)), where zi(t) = e−2πJt/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−2πJt/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(a1 ) ∪ · · · ∪ ∂E(ad ) where a1∪· · ·∪ad 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 , where dλ = ω0. Denoting by a(γ ) the smallest subset of a such that γ ⊂ ∂E (a(γ )), and choosing λ = +n i=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 sym- plectomorphism 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 / ϕ(γ ) λ = / ϕ(D) ω0 = / D ϕ∗ ω0 = / D ω0 = / γ λ,
- 41. 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 discus- sion, 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 di- vide 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 induction, and the identitivity of 2 is proved in an elementary way. 3 Recall that 2 does not imply 1 in general. However, a suitable pinching condi- tion guarantees that “linear” and “non linear” coincide: Theorem 2.2.4. Let κ ∈ b 2 , b . Then the following statements are equivalent: (i) B2n(κ) 1 E(a) 1 E(a) 1 B2n(b), (ii) B2n(κ) 2 E(a) 2 E(a) 2 B2n(b), (iii) B2n(κ) 3 E(a) 3 E(a) 3 B2n(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, B2n (κ) 2 E(a) 2 B2n (b).
- 42. 28 2 Proof of Theorem 1 Hence, by the monotonicity of the first Ekeland–Hofer capacity c1 we obtain κ ≤ a1 ≤ b, (2.2.7) and by the monotonicity of cn κ ≤ cn(E(a)) ≤ b. (2.2.8) 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)) = a i for i = 1, . . . , n, and from E(a) 2 E(a) we conclude ai ≤ a i. (iii) ⇒ (i) now follows by a similar reasoning as in the proof of the identitivity of 3. Indeed, starting from B2n (κ) 3 E(a) 3 E(a ) 3 B2n (b), the implication (2.2.2) shows that for any δ1, δ2, δ3 ∈ ]0, 1[ B2n (δ3δ2δ1κ) 2 E(δ2δ1a) 2 E(δ1a ) 2 B2n (b). Choosing δ1, δ2, δ3 so large that δ3δ2δ1κ b/2 we can apply the already proved implication to see B2n (δ3δ2δ1κ) 1 E(δ2δ1a) 1 E(δ1a) 1 B2n (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 Theo- rem 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 B2n(A) for some A an. Then an 2a1. Proof. Arguing by contradiction we assume E(a1, . . . , an) 3 B2n(A) for some A an and an ≤ 2a1. A volume comparison shows a1 A. Hence, a1 ∈ A 2 , A . Therefore, B2n(a1) 3 E(a1, . . . , an) 3 B2n(A), Theorem 2.2.4 and the equiva- lence (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
- 43. 2.3 Rigidity for polydiscs ? 29 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)) = ja1, 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 πr2 such that the poly- disc P2n(π, . . . , π, πr2) embeds into the cube C2n(A) = P2n(A, . . . , A) by a linear symplectomorphism. Proof. It is enough to prove the lemma for n = 2. Consider the linear symplecto- morphism given by (z1, z2) → (z 1, z 2) = 1 √ 2 (z1 + z2, z1 − z2). For (z1, z2) ∈ P(π, πr2) and i = 1, 2 we have z i 2 ≤ 1 2 |z1|2 + |z2|2 + 2 |z1| |z2| 1 2 + r2 2 + r. (2.3.2) The right hand side of (2.3.2) is strictly smaller than r2 provided that r 1 + √ 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.
- 44. 30 2 Proof of Theorem 1 Question 2.3.2. Assume that there exist symplectic embeddings P(a1, a2, a3) → P(a1, a 2, a 3) and P(a1, a 2, a 3) → P(a1, a2, a3). Is it then true that a2 = a 2 and a3 = a 3? We also 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.
- 45. 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 ellipsoid E(a1, . . . , an) is defined by E(a1, . . . , an) = 6 (z1, . . . , zn) ∈ Cn .n i=1 π|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. Assume a 2π. Then E2n(π, . . . , π, a) symplectically embeds into B2n a 2 + π + for every 0. ThesymplecticfoldingconstructionofLalondeandMcDuffconsidersa4-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 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 : E(a, π) → B4 a 2 + π + satisfying π|(z1, z2)|2 a 2 + + π2|z1|2 a + π|z2|2 for all (z1, z2) ∈ E(a, π).
- 46. 32 3 Proof of Theorem 2 We recall 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 Proposition 3.1.2. Then the composition of the permutation E2n(π, . . . , π, a) → E2n(a, π, . . . , π) with the restriction of × id2n−4 to E2n(a, π, . . . , π) embeds E2n(π, . . . , π, a) into B2n a 2 + π + . Proof. Let z = (z1, . . . , zn) ∈ E2n(a, π, . . . , π). By Proposition 3.1.2 and the definition (3.1.1) of the ellipsoid, π | × id2n−4(z)|2 = π |(z1, z2)|2 + n . i=3 |zi|2 a 2 + + π2|z1|2 a + π n . i=2 |zi|2 = a 2 + + π π|z1|2 a + n . i=2 π|zi|2 π a 2 + + π, as claimed. 2 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 . We now make sure that we can describe such a map by prescribing it on an exhausting and nested family of embedded loops. Recall that D(a) denotes the open disc of area a centred at the origin, and that |U| denotes the area of a domain U ⊂ R2. Definition 3.1.4. A family L of loops in a simply connected domain U ⊂ R2 is called admissible if there is a diffeomorphism β : D(|U|) {0} → U {p} for some point p ∈ U such that (i) concentric circles are mapped to elements of L, (ii) in a neighbourhood of the origin β is a translation. Lemma 3.1.5. Let U and V be bounded and simply connected domains in R2 of equal area and let LU and LV be admissible families of loops in U and V , respectively. Then there is a symplectomorphism between U and V mapping 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, if LU is
- 47. 3.1 Reformulation of Theorem 2 33 a family of concentric circles and LV is a family of rectangles with smooth corners and width larger than a positive constant, then no bijection from U to V mapping loops to loops is continuous at the origin. 3 Proof of Lemma 3.1.5. Denote the concentric circle of radius r by C(r). We may assume that LU = {C(r)}, 0 r R. Let β be the diffeomorphism parametrizing (V {p}, LV ). After reparametrizing the r-variable by a diffeomorphism of ]0, R[ which is the identity near 0 we may assume that β maps the loop C(r) of radius r to the loop L(r) in LV which encloses the domain V (r) of area πr2. We denote the Jacobian of β at reiϕ by β(reiϕ). Since β is a translation near the origin and U is connected, det β(reiϕ) 0. By our choice of β, πr2 = |V (r)| = / D(πr2) det β = / r 0 ρ dρ / 2π 0 det β (ρeiϕ ) dϕ. Differentiating in r we obtain 2π = / 2π 0 det β (reiϕ ) dϕ. (3.1.2) Define the smooth function h: ]0, R[ ×R → R as the unique solution of the initial value problem d dt h(r, t) = 1/ det β (reih(r,t) ), t ∈ R h(r, t) = 0, t = 0 (3.1.3) depending on the parameter r. We claim that h(r, t + 2π) = h(r, t) + 2π. (3.1.4) It then follows, since the function h is strictly increasing in the variable t, that for every r fixed the map h(r, ·): R → R induces a diffeomorphism of the circle R/2πZ. In order to prove the claim (3.1.4) we denote by t0(r) 0 the unique solution of h(r, t0(r)) = 2π. Substituting ϕ = h(r, t) into formula (3.1.2) we obtain, using det β(reih(r,t)) · d dt h(r, t) = 1, that 2π = / t0(r) 0 dt = t0(r). 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) )
- 48. 34 3 Proof of Theorem 2 we obtain for the determinant of the Jacobian 1 r · ∂h ∂ϕ (r, ϕ) · r · det β (reih(r,ϕ) ) = 1, where we again have used (3.1.3). Finally, α is 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. 2 Consider a bounded domain U ⊂ C and a continuous function f : U → R0. 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 E(a, b) = F D(a), f (z1) = b 1 − π|z1|2 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 a b u fibre capacity 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”.
- 49. 3.1 Reformulation of Theorem 2 35 Lemma 3.1.8. Assume 0. Then (i) E(a, b) symplectically embeds into T (a + , b + ), (ii) T 4(a) symplectically embeds into B4(a + ). Proof. (i) Set = a2/(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. 1 v u a 1 2 4a 4 2 3 4 L0 L1 2 L1 Figure 3.2. Constructing the embedding α. In an “optimal world” we would choose the loops L̂u, 0 u a, in the image R(a) as the boundaries of the rectangles with corners (0, 0), (0, 1), (u, 0), (u, 1). If the family L̂ = L̂u induced a map α̂, we would then have u α̂(z1) ≤ π |z1|2 for all (z1, z2) ∈ R(a). The non admissible family L̂ 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 , 1 2 . 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 4 , 1 − 4a , and let L1 be the boundary of R(a). We define a family of loops Ls by linearly interpolating between L0 and L1, i.e., Ls is the boundary of the rectangle with corners (1 − s) 4 , (1 − s) 4a and us, 1 − 4a + 4a s , s ∈ [0, 1],
- 50. 36 3 Proof of Theorem 2 where us = 3 4 + s a − 3 4 . Since us a, the area enclosed by Ls is estimated from below by us − 4 1 − 2 4a us − 3 4 . (3.1.6) 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 Ls more and more rectangular as s → 1, we can arrange that the set 1 0s1 Ls is the domain bounded by L0 and L1. Moreover, by choosing all smooth corners rectangular enough, we can arrange that the area enclosed by Ls and Ls is less than /4. In view of (3.1.6), the area enclosed by Ls is then at least us −. Complete the families {L(r)} and {Ls} to an admissible family L of loops in R(a) and let α : D(a) → R(a) be the map defined by L. Fix (z1, z2) ∈ D(a). If α(z1) lies on a loop in L {Ls}0s1, then u (α(z1)) 3 4 ≤ π |z1|2 + , and so the required estimate (3.1.5) is satisfied. If α(z1) ∈ Ls for some s ∈ ]0, 1[, then the area enclosed by Ls is π |z1|2 , and so π |z1|2 + us ≥ u (α(z1)), whence (3.1.5) is again satisfied. This completes the construction of a symplectomorphism α : D(a) → R(a) satisfying (3.1.5). In the sequel, we will illustrate a map like α by a picture like in Figure 3.3. To continue the proof of (i) we shall show that (α(z1), z2) ∈ T (a + , b + ) for every (z1, z2) ∈ E(a, b), so that the symplectic map α × id embeds E(a, b) into T (a+, b+). Take (z1, z2) ∈ E(a, b). Then, using the definition (3.1.1) of E(a, b), the estimate (3.1.5) and the definition of we find π|z2|2 b 1 − π|z1|2 a ≤ b 1 − u (α(z1)) a + a b 1 − u (α(z1)) a + + b a = b 1 − u (α(z1)) a + + − a + a + ≤ b 1 − u (α(z1)) a + + − a + u (α(z1)) = (b + ) 1 − u(α(z1)) a + . It follows that (α(z1), z2) ∈ T (a + , b + ) = F R(a + ), (b + ) 1 − u a + as claimed. In order to prove (ii) we shall construct an area preserving diffeomorphism ω from a rectangular neighbourhood of R(a) having smooth corners and area a+ to D(a+) such that π|ω(z1)|2 ≤ u + for all z1 = (u, v) ∈ R(a). (3.1.7)
- 51. 3.1 Reformulation of Theorem 2 37 Such a map ω can again be obtained with the help of Lemma 3.1.5. In an “optimal world” we would choose the loops L̂u in the domain R(a) as before. This time, we perturb this non admissible family to an admissible family L of loops as illustrated in Figure 3.3. If the smooth corners of all those loops in L which enclose an area greater than /2 lie outside R(a) and if the upper, left and lower edges of all these loops are close enough, then the induced map ω will satisfy (3.1.7). D(a) D(a + ) z1 z1 α ω 1 1 v v u u a a Figure 3.3. The first and the last base deformation. Restricting ω to R(a) we obtain a symplectic embedding ω × id: T 4(a) → R4. For (z1, z2) ∈ T 4(a) we have π |z2|2 a (1 − u/a), where z1 = (u, v) ∈ R(a). In view of (3.1.7) we conclude that π |ω(z1)|2 + |z2|2 u + + a 1 − u a = u + + a − u = a + , and so (ω × id)(z1, z2) ∈ B4(a + ) for all (z1, z2) ∈ T 4(a). 2 Lemma 3.1.8 allows us to reformulate Proposition 3.1.2 as follows.
- 52. 38 3 Proof of Theorem 2 Proposition 3.1.9. Assume a 2π. Given 0, there exists a symplectic embed- ding : T (a, π) → T 4 a 2 + π + , (z1, z2) → (z 1, z 2), z1 = (u, v) and z 1 = (u, v), satisfying u + π|z 2|2 a 2 + + πu a + π|z2|2 for all (u, v, z2) ∈ T (a, π). (3.1.8) Postponingtheproof, wefirstshowthatProposition3.1.9impliesProposition3.1.2. Corollary 3.1.10. Assume the statement of Proposition 3.1.9 holds true. Then there exists a symplectic embedding : E(a, π) → B4 a 2 + π + satisfying π|(z1, z2)|2 a 2 + + π2|z1|2 a + π|z2|2 for all (z1, z2) ∈ E(a, π). (3.1.9) Proof. Let 0 be so small that ca + 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 ω × id: T 4 ca 2 + π + → B4 ca 2 + π + 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 : T (ca + , π) → T 4 ca 2 + π + , (z1, z2) → (1(z1, z2), 2(z1, z2)), satisfying u (1(α(z1), z2))+π |2 (α(z1), z2)|2 ca 2 + + πu(α(z1)) ca + +π |z2|2 (3.1.12)
- 53. 3.2 The folding construction 39 for all (u(α(z1)), v, z2) ∈ T (ca + , π). Set ˆ = (ω × id) (α × id). Then ˆ symplectically embeds E(ca, cπ) into B4 ca 2 + π + 2 . Moreover, if (z1, z2) ∈ E(ca, cπ), then π ˆ (z1, z2) 2 = π |ω (1(α(z1), z2))|2 + π |2(α(z1), z2)|2 (3.1.11) ≤ u(1(α(z1), z2)) + + π |2(α(z1), z2)|2 (3.1.12) ca 2 + 2 + πu(α(z1)) ca + + π|z2|2 (3.1.10) ≤ ca 2 + 2 + π2|z1|2 ca + + π ca + a()2 caπ + a + π + π|z2|2 ca 2 + 3 + π2|z1|2 ca + π|z2|2 where in the last step we again used ca + 2π. Now choose 0 so small that π+3 c π + . We denote the dilatation by √ c in R4 also by √ c, and define : E(a, π) → R4 by = √ c −1 ˆ √ c. Then symplectically embeds E(a, π) into B4 a 2 + π+2 c ⊂ B4 a 2 +π + , and since π |z1|2 a for all (z1, z2) ∈ E(a, π) and by the choice of , π |(z1, z2)|2 = π c ˆ √ c z1, √ c z2 2 1 c ca 2 + 3 + π2|z1|2 a + πc|z2|2 = a 2 + 3 c + 1 c π2|z1|2 a + π|z2|2 a 2 + + π2|z1|2 a + π|z2|2 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.
- 54. 40 3 Proof of Theorem 2 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 P1 = U ∩ 6 u ≤ a 2 + δ 7 , P2 = U ∩ 6 u ≥ a + π 2 + 11δ 7 , L = U (P1 ∪ P2). v u u 1 P1 P2 L U δ a 2 + δ a+π 2 + 11δ a + π 2 + 12δ f π π 2 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, a 2 , (ii) h(w) 0 for w ∈ a 2 , a 2 + δ2 , (iii) h a 2 + δ2 = δ, (iv) h(w) = h(a − w) for all w ∈ [0, a + δ].
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