A systolic inequality for geodesic flows on the two-sphere

Abstract

For a Riemannian metric $g$ on the two-sphere, let ${\ell }_{\min }(g)$ be the length of the shortest closed geodesic and ${\ell }_{\max }(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq π\ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.

47 pages; v2 added sharp lower bound on the length of the longest simple closed geodesic. Added appendix proving that all Zoll geodesic flows on the two-sphere are symplectically conjugate. Revised introduction

References in corpus (2)

• Open problems and questions about geodesics
• Ergodic components and topological entropy in geodesic flows of surfaces