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

TL;DR

This paper constructs a Finsler metric on the 2-sphere with a systolic ratio exceeding the maximum known for Riemannian metrics, providing a counterexample to the Croke conjecture.

Contribution

It presents the first explicit Finsler metric on the 2-sphere surpassing the Riemannian systolic ratio conjectured maximum, challenging existing conjectures.

Findings

Finsler metric with systolic ratio of 4π/3 on the 2-sphere

Exceeds the Riemannian maximum systolic ratio of 2√3

Inspired by Cossarini-Sabourau's work

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.