Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

TL;DR

This paper extends classical mean curvature flow results by relaxing convexity assumptions, proving a generalized Harnack inequality for mean convex solutions, and characterizing ancient solutions with curvature bounds.

Contribution

It introduces a generalized differential Harnack inequality for mean convex solutions with curvature bounds and characterizes ancient solutions without convexity assumptions.

Findings

Generalized Harnack inequality for mean curvature flow

Characterizations of shrinking spheres in ancient solutions

Relaxation of convexity assumptions in classical results

Abstract

The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean convex solutions to mean curvature flow with a lower bound on $\frac{\lambda_1}{H}$ where $\lambda_1$ is the smallest principal curvature. Then, we use classical maximum principle to provide several characterizations of family of shrinking spheres for closed, mean convex ancient solution to mean curvature flow with a lower bound on $\frac{\lambda_1 + .. + \lambda_k}{H}$ for some $1 \leq k \leq d-1$, where $\lambda_1 \leq \lambda_2 \leq .. \leq \lambda_d$ are the principal curvatures.