Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

TL;DR

This paper derives sharp, universal upper bounds on the shortest closed geodesic length for complete punctured spheres with finite area, extending results to Finsler metrics and many punctures.

Contribution

It provides the first sharp curvature-free bounds for punctured spheres with three or four ends and extends these bounds to Finsler metrics and spheres with many punctures.

Findings

Sharp upper bounds expressed in terms of area.

Extremal metrics modeled on Calabi-Croke and tetrahedral spheres.

Optimal bounds extended to Finsler metrics and large numbers of punctures.

Abstract

We establish sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. These sharp curvature-free upper bounds are expressed in terms of the area of the punctured sphere. In both cases, we describe the extremal metrics, which are modeled on the Calabi-Croke sphere or the tetrahedral sphere. We also extend these optimal inequalities for reversible and non-necessarily reversible Finsler metrics. In this setting, we obtain optimal bounds for spheres with a larger number of punctures. Finally, we present a roughly asymptotically optimal upper bound on the length of the shortest closed geodesic for spheres/surfaces with a large number of punctures in terms of the area.