Symplectic Invariants and Hamiltonian Dynamics - Hofer / Zehnder ...

ter 2
lectic capacities
efinition and application to embeddings
llowing we introduce a special class of symplectic invariants discovered by
d and H. Hofer in [68, 69] for subsets of R 2n . They were led to these inin
their search for periodic solutions on convex energy surfaces and called
plectic capacities. The concept of a symplectic capacity was extended to
ymplectic manifolds by H. Hofer and E. Zehnder in [123]. The existence
these invariants is based on a variational principle; it is not intuitive,
be postponed to the next chapter. Taking their existence for granted, the
his chapter is rather to deduce the rigidity of some symplectic embedd,
in addition, the rigidity of the symplectic nature of mappings under
the supremum norm, which will give rise to the notion of a “symplectic
orphism”.
n of symplectic capacity. We consider the class of all symplectic manifolds
ossibly with boundary and of fixed dimension 2n. A symplectic capacity is
,ω) ↦→ c(M,ω) which associates with every symplectic manifold (M,ω)
ative number or ∞, satisfying the following properties A1–A3.
otonicity: c(M,ω) ≤ c(N,τ)
xists a symplectic embedding ϕ :(M,ω) → (N,τ).
formality: c(M,αω) =|α|c(M,ω)
∈ R,α = 0.
triviality: c(B(1),ω0) =π = c(Z(1),ω0)
pen unit ball B(1) and the open symplectic cylinder Z(1) in the standard
2n ,ω0). For convenience, we recall that with the symplectic coordinates
2n ,
B(r) =
(x, y) ∈ R 2n
|x| 2 + |y| 2