On the classification of ancient solutions to curvature flows on the sphere

TL;DR

This paper classifies convex, quasi-ancient solutions to curvature flows on the sphere, showing they are either stationary or shrinking geodesic spheres under certain conditions, using geometric maximum principles and reflection techniques.

Contribution

It introduces the concept of quasi-ancient solutions on the sphere and provides a classification result without assuming homogeneity or convexity/concavity of the flow speed.

Findings

Convex, quasi-ancient solutions with bounded mean curvature are either stationary or shrinking spheres.

The classification applies broadly without restrictions on the flow speed's homogeneity or convexity.

The methods include maximum principle, sphere rigidity, and Alexandrov reflection.

Abstract

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Alexandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.