Symplectic capacities

Abstract

In the following we introduce a special class of symplectic invariants discovered by I. Ekeland and H. Hofer in [68, 69] for subsets of ℝ²ⁿ. They were led to these invariants in their search for periodic solutions on convex energy surfaces and called them symplectic capacities. The concept of a symplectic capacity was extended to general symplectic manifolds by H. Hofer and E. Zehnder in [123]. The existence proof of these invariants is based on a variational principle; it is not intuitive, and will be postponed to the next chapter. Taking their existence for granted, the aim of this chapter is rather to deduce the rigidity of some symplectic embeddings and, in addition, the rigidity of the symplectic nature of mappings under limits in the supremum norm, which will give rise to the notion of a “symplectic homeomorphism”.