A Finsler counterexample to the Croke conjecture for the systolic ratio on the 2-sphere

Abstract

We exhibit a Finsler metric on the 2-sphere whose systolic (Holmes-Thompson) ratio is $\frac{4\pi }{3}$. This is bigger than the conjectured maximal Riemannian systolic ratio of $2\sqrt{3}$ achieved by the Calabi-Croke metric. The construction of the Finsler metric is heavily inspired by a paper of Cossarini-Sabourau.

11 pages, 3 figures