On the structure of Besse convex contact spheres

TL;DR

This paper investigates convex contact spheres with all Reeb orbits closed, revealing they structurally resemble contact ellipsoids with stratifications by orbit periods and spectral invariants matching action values.

Contribution

It demonstrates that such convex contact spheres have a stratification into homology spheres and their spectral invariants align with their action spectrum, resembling contact ellipsoids.

Findings

Strata are integral homology spheres.

Spectral invariants match the sequence of action values.

Convex contact spheres resemble contact ellipsoids.

Abstract

We consider convex contact spheres $Y$ all of whose Reeb orbits are closed. Any such $Y$ admits a stratification by the periods of closed Reeb orbits. We show that $Y$ "resembles" a contact ellipsoid: any stratum of $Y$ is an integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants of $Y$ coincides with the full sequence of action values, each one repeated according to its multiplicity.