Sharp distance comparison for curve shortening flow on the round sphere

TL;DR

This paper establishes a sharp chord-arc estimate for curve shortening flow on the round sphere, leading to a clear understanding of the long-term behavior of spherical curves, similar to the planar case.

Contribution

It extends the sharp chord-arc estimate for curve shortening flow from the plane to the round sphere, providing a new, efficient proof of curve behavior.

Findings

Curves contract to round points or converge to great circles.

Sharp chord-arc estimate holds on the sphere.

Provides a unified approach similar to planar results.

Abstract

We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting (Andrews and Bryan, Comm. Anal. Geom., 2011). As in the planar case, the sharp estimate implies control on the curvature, resulting in a direct and efficient proof that simple spherical curves either contract to round points (in finite time) or converge to great circles (in infinite time).