Symplectically hyperbolic manifolds

TL;DR

This paper explores symplectically hyperbolic manifolds, establishing their properties, construction methods, and relations to bounded cohomology, isoperimetric inequalities, and fundamental group characteristics.

Contribution

It provides new characterizations of symplectically hyperbolic manifolds, including their equivalence to isoperimetric inequalities and the non-amenability of their fundamental groups.

Findings

Bounded cohomology class implies hyperbolicity

Hyperbolicity linked to isoperimetric inequality

Fundamental group is non-amenable

Abstract

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology class then it is hyperbolic. * The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality. * The fundamental group of symplectically hyperbolic manifold is non-amenable. We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations, discuss the dependenc of the symplectic hyperbolicity on the fundamental group and discuss some properties of the group of symplectic diffeomorphisms of a symplectically hyperbolic manifold.