Harnack estimate for mean curvature flow on the sphere

TL;DR

This paper establishes a differential Harnack inequality for convex solutions to mean curvature flow on the sphere and classifies ancient solutions using reflection techniques.

Contribution

It introduces a Harnack inequality for weakly convex hypersurfaces evolving on the sphere and classifies ancient solutions via reflection methods.

Findings

Proved a differential Harnack inequality for convex solutions.

Classified convex ancient solutions on the sphere.

Applied Aleksandrov reflection to achieve classification.

Abstract

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the mean curvature flow on the sphere.