Spheres of small diameter with long sweep-outs

TL;DR

This paper demonstrates that there is no universal upper limit on the lengths of curves in sweep-outs of Riemannian 2-spheres, impacting the understanding of geodesic existence.

Contribution

It proves the non-existence of a universal diameter bound for sweep-out curves on Riemannian 2-spheres, challenging assumptions about geodesic length estimates.

Findings

No universal diameter bound exists for sweep-out curves.

A bound would imply simple proofs for short geodesics.

Results affect the study of geodesic existence on small-diameter spheres.

Abstract

We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small diameter.