On the spectral characterization of Besse and Zoll Reeb flows

TL;DR

This paper characterizes Besse contact forms on convex spheres and tangent bundles using spectral invariants, and provides conditions for Besse properties in symplectic hypersurfaces via capacities.

Contribution

It introduces spectral invariant criteria for identifying Besse contact forms and links symplectic capacities to the Besse property in specific hypersurfaces.

Findings

Spectral invariants characterize Besse contact forms on convex spheres.

Ekeland-Hofer capacities provide sufficient conditions for Besse properties.

Results connect spectral geometry with Reeb flow periodicity.

Abstract

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic Euclidean spaces, we give a sufficient condition for the Besse property via the Ekeland-Hofer capacities.