The length of the shortest closed geodesic on positively curved 2-spheres

TL;DR

This paper establishes an upper bound on the length of the shortest closed geodesic on positively curved 2-spheres, improving bounds through a new isoperimetric inequality for pinched curvature.

Contribution

It introduces a novel isoperimetric inequality for 2-spheres with pinched curvature, leading to improved bounds on geodesic lengths.

Findings

Shortest closed geodesic length ≤ 3 × diameter for non-negative curvature

New isoperimetric inequality for 2-spheres with pinched curvature

Enhanced bounds in the pinched curvature setting

Abstract

We show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. We prove a new isoperimetric inequality for 2-spheres with pinched curvature; this allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting.