A New Capacity for Symplectic Manifolds

Abstract

This chapter discusses the axioms for a symplectic capacity. At present, very little is known about the nature of a symplectic map. The axioms for a symplectic capacity are useful for a more systematic study of the symplectic embedding problem; they led to a new rigidity result. The axioms such as monotonicity, conformality, local nontriviality, and nontriviality do not determine a capacity function uniquely. There are many ways to construct different capacity functions. The capacity of every symplectic manifold is positive or ∞. Every capacity singles out the subgroup of homeomorphisms of R2n preserving the capacity. The elements of this distinguished group of homeomorphisms have the additional property that they are symplectic or anti-symplectic in case they are differentiable. The associated pseudogroup can be used to define a topological symplectic manifold.