A refinement of the Hofer-Zehnder theorem on the existence of closed trajectories near a hypersurface

TL;DR

This paper refines the Hofer-Zehnder theorem by demonstrating that the existence of closed characteristics near a hypersurface depends only on the finite capacity of its thickening, not on bounding a compact submanifold.

Contribution

It relaxes the conditions of the Hofer-Zehnder theorem, showing finite capacity of the hypersurface's thickening suffices for the existence of closed trajectories.

Findings

Closed characteristics exist under weaker conditions.

Finite capacity of the thickening guarantees closed trajectories.

Refinement broadens applicability of the theorem.

Abstract

The Hofer-Zehnder theorem states that almost every hypersurface in a thickening of a hypersurface $S$ in a symplectic manifold $(M,\omega)$ carries a closed characteristic provided that $S$ bounds a compact submanifold and $(M,\omega)$ has finite capacity. We show that it is enough to assume that the thickening of $S$ has finite capacity.