Symplectic Capacities in Manifolds

Abstract

Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set. 1. Preliminaries. For an introduction to symplectic capacities, non-smooth Hamiltonian systems and characteristic differential inclusions we refer to a previous talk given at the Banach Center in October 93 [K93]. The aim of this note is to show that some calculations of symplectic capacities can be simplified through embeddings of non-smooth convex sets . No approximations by families of Hamiltonian functions are needed. We show that definitions of capacities of convex sets in the symplectic model space ( R, ω ) suffice to define and to calculate in some cases symplectic capacities for subsets in any symplectic manifolds. Moreover, some applications of the product formula for convex sets derived in [K90] are given. To define the setting, let us consider the standard symplectic linear space V :=