Dynamical convexity and elliptic periodic orbits for Reeb flows

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 (An index theory for periodic solutions of convex Hamiltonian systems, 1986) and Dell’Antonio–D’Onofrio–Ekeland (Periodic solutions of elliptic type for strongly nonlinear Hamiltonian systems, 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.