Rabinowitz Floer homology for tentacular Hamiltonians

TL;DR

This paper generalizes Rabinowitz Floer homology to certain non-compact hypersurfaces called strongly tentacular Hamiltonians, providing a new tool for studying closed characteristics and the Weinstein conjecture.

Contribution

It introduces a framework for Rabinowitz Floer homology on non-compact hypersurfaces, specifically for strongly tentacular Hamiltonians, under suitable compactness conditions.

Findings

Defined Rabinowitz Floer homology for strongly tentacular Hamiltonians.

Enabled proofs of the Weinstein conjecture in new non-compact settings.

Provided a method to establish existence of closed characteristics.

Abstract

This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces being the level sets of specific Hamiltonians: strongly tentacular Hamiltonians, for which the compactness conditions are satisfied, thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics.