4Symplectic embeddings
III Symplectic Geometry
4 Symplectic embeddings
We end with a tiny chapter on symplectic embeddings, as promised in the course
description.
Definition
(Symplectic embedding)
.
A symplectic embedding is an embedding
ϕ
:
M
1
→ M
2
such that
ϕ
∗
ω
2
=
ω
1
. The notation we use is (
M
1
, ω
1
)
s
→
(
M
2
, ω
2
).
A natural question to ask is, if we have two symplectic manifolds, is there a
symplectic embedding between them?
For concreteness, take (
C
n
, ω
0
)
∼
=
(
R
2n
, ω
0
), and consider the subsets
B
2n
(
r
)
and
Z
2n
(
R
) =
B
2
(
R
)
×R
2n−2
(where the product is one of symplectic manifolds).
If
r ≤ R
, then there is a natural inclusion of
B
2n
(
r
) into
Z
2n
(
R
)? If we only
ask for volume-preserving embeddings, then we can always embed
B
2n
(
r
) into
Z
2n
(
R
), since
Z
2n
(
R
) has infinite volume. It turns out, if we require the
embedding to be symplectic, we have
Theorem
(Non-squeezing theorem, Gromov, 1985)
.
There is an embedding
B
2
(n) → Z
2n
(R) iff r < R.
When studying symplectic embeddings, it is natural to consider the following:
Definition
(Symplectic capacity)
.
A symplectic capacity is a function
c
from
the set of 2n-dimensional manifolds to [0, ∞] such that
(i) Monotonicity: if (M
1
, ω
1
) → (M
2
, ω
2
), then c(M
1
, ω
1
) ≤ c(M
2
, ω
2
).
(ii) Conformality: c(M, λω) = λc(M, ω).
(iii) Non-triviality: c(B
2n
(1), ω
0
) > 0 and c(Z
2n
(1), ω
0
) < ∞.
If we only have (i) and (ii), this is called a generalized capacity.
Note that the volume is a generalized capacity, but not a symplectic capacity.
Proposition.
The existence of a symplectic capacity is equivalent to Gromov’s
non-squeezing theorem.
Proof.
The
⇒
direction is clear by monotonicity and conformality. Conversely,
if we know Gromov’s non-squeezing theorem, we can define the Gromov width
W
G
(M, ω) = sup{πr
2
| (B
2n
(r), ω
0
) → (M, ω)}.
This clearly satisfies (i) and (ii), and (iii) follows from Gromov non-squeezing.
Note that Darboux’s theorem says there is always an embedding of
B
2n
(
r
) into
any symplectic manifold as long as r is small enough.