Symplectic circle actions on manifolds with contact type boundary

TL;DR

This paper extends Morse-Bott techniques to symplectic manifolds with contact type boundary, showing that circle actions are Hamiltonian and analyzing the topology of such manifolds.

Contribution

It generalizes key results from closed symplectic manifolds to those with contact type boundary, focusing on circle actions and Hamiltonian properties.

Findings

Any symplectic group action on such manifolds is Hamiltonian.

The boundary of the manifold is connected under certain conditions.

Level sets of the Hamiltonian are either empty or connected after attaching cylindrical ends.

Abstract

Many of the existing results for closed Hamiltonian G-manifolds are based on the analysis of the corresponding Hamiltonian functions using Morse-Bott techniques. In general such methods fail for non-compact manifolds or for manifolds with boundary. In this article, we consider circle actions only on symplectic manifolds that have (convex) contact type boundary. In this situation we show that many of the key ideas of Morse-Bott theory still hold, allowing us to generalize several results from the closed setting. Among these, we show that in our situation any symplectic group action is always Hamiltonian, we show several results about the topology of the symplectic manifold and in particular about the connectedness of its boundary. We also show that after attaching cylindrical ends, a level set of the Hamiltonian of a circle action is either empty or connected. We concentrate mostly on circle actions, but we believe that with our methods many of the classical results can be generalized from closed symplectic manifolds to symplectic manifolds with contact type boundary.