Bounds for tentacular Hamiltonians

TL;DR

This paper explores conditions for establishing bounds on Floer trajectories for non-compact energy levels and introduces tentacular Hamiltonians, paving the way for extending Rabinowitz Floer homology to non-compact hypersurfaces.

Contribution

It introduces tentacular Hamiltonians and analyzes conditions for $L^{ abla}$ bounds, advancing the extension of Rabinowitz Floer homology to non-compact energy hypersurfaces.

Findings

Established $L^{ abla}$ bounds for Floer trajectories under certain conditions.

Defined a new class of Hamiltonians called tentacular Hamiltonians.

Laid groundwork for defining Rabinowitz Floer homology on non-compact hypersurfaces.

Abstract

This paper represents a first step towards the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish $L^{\infty}$-bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians, which satisfy the conditions: how to define RFH for these examples will be the subject of a follow-up paper.