Periodicity characterization by capacities for star-shaped domains

TL;DR

This paper characterizes Besse and Zoll Reeb flows on star-shaped domains in Euclidean space using Ekeland-Hofer capacities, completing previous spectral characterizations and providing new criteria for these flows.

Contribution

It provides a complete spectral characterization of Besse and Zoll Reeb flows on star-shaped domains using capacities, extending prior work on the standard contact sphere.

Findings

Reeb flow is Besse iff it has n coinciding capacities.

Reeb flow is Zoll iff the first n capacities coincide.

The characterization applies to star-shaped domains in Euclidean space.

Abstract

We complete the spectral characterization of Besse and Zoll Reeb flows on the standard contact sphere $S^{2n-1}$ initiated by Ginzburg-G\"urel-Mazzucchelli. Roughly speaking, it states that a Reeb flow on the boundary of any star-shaped domain in $\mathbb{R}^{2n}$ is Besse if and only if it has $n$ coinciding Ekeland-Hofer capacities, and that it is Zoll if and only if the first $n$ capacities coincide.