Dynamical convexity and elliptic periodic orbits for Reeb flows

TL;DR

This paper proves that convex hypersurfaces in n have elliptic closed orbits under Reeb flows, generalizing previous results using contact homology and dynamical convexity, with applications to geodesic and magnetic flows.

Contribution

It introduces a generalized approach using contact homology and dynamical convexity to establish the existence of elliptic orbits for broader classes of convex hypersurfaces.

Findings

Existence of elliptic closed orbits for convex hypersurfaces in n

Application to geodesic and magnetic flows under pinching conditions

Extension of previous results using contact homology and dynamical convexity

Abstract

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satisfying suitable pinching conditions and for antipodal invariant convex hypersurfaces respectively. In this work we present a generalization of these results using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact forms on $S^3$. Applications include geodesic flows under pinching conditions, magnetic flows and toric contact manifolds.