On symplectic manifolds with some contact properties

Abstract

We show in this article that a symplectic manifold bounded by the standard contact sphere is, under some additional hypotheses, a ball. This gives a tool for the recognition of nonstandard structures on spheres, and we show here that exotic contact structures on spheres of dimension > 3 do exist. These examples are of a completely different nature than Bennequin's nonstandard contact structure on S 3 (see The new nonstandard structures on high-dimensional spheres are s-fillable (see 3 below); i.e., they bound symplectic manifolds, while Bennequin's nonstandard structure is overtwisted (see In the other paper (see