Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

TL;DR

This paper classifies all closed, embedded ancient solutions to the curve shortening flow on the sphere, showing they are only equators or shrinking circles, and establishes key inequalities and symmetry properties to prove this.

Contribution

It proves the classification of ancient solutions on the sphere, introducing a Harnack inequality and symmetry preservation techniques.

Findings

Ancient solutions are only equators or shrinking circles.

Solutions converge to an equator backwards in time.

The methods include a Harnack inequality and Aleksandrov reflection.

Abstract

We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain the result, we first prove a Harnack inequality for the curve shortening flow on the sphere. Then an application of the Gauss-Bonnet, easily allows us to obtain curvature bounds for ancient solutions leading to backwards smooth convergence to an equator. To complete the proof, we use an Aleksandrov reflection argument to show that maximal symmetry is preserved under the flow.