Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic

TL;DR

This paper provides an alternative proof that the Calabi-Croke sphere is a local extremum for the shortest closed geodesic length among certain Riemannian and Finsler spheres, avoiding the use of uniformization.

Contribution

It offers a new proof of a known extremality result and extends it to Finsler metrics, broadening the scope of the original theorem.

Findings

Calabi-Croke sphere is a local extremum for shortest closed geodesic length.

The proof does not rely on uniformization theorem.

The extremality result is extended to Finsler metrics.

Abstract

Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theorem, which does not make use of the uniformization theorem, and extend the result to Finsler metrics.