Uniqueness of compact tangent flows in Mean Curvature Flow

TL;DR

This paper proves the uniqueness of tangent flows in mean curvature flow when the tangent flow is a smooth, embedded self-similar shrinker, and analyzes stability and convergence near such shrinkers using the Simon-Lojasiewicz inequality.

Contribution

It establishes the uniqueness of tangent flows under specific conditions and demonstrates stability and convergence properties near self-similar shrinkers in mean curvature flow.

Findings

Uniqueness of tangent flows for certain self-similar shrinkers.

Stability of solutions close to a given self-similar shrinker.

Convergence of solutions to possibly different self-similarly shrinking solutions.

Abstract

We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings. Furthermore, given such a closed, multiplicity-one, smoothly embedded self-similarly shrinker $\Sigma$, we show that any solution of the rescaled flow, which is sufficiently close to $\Sigma$, with Gaussian density ratios greater or equal to that of $\Sigma$, stays for all time close to $\Sigma$ and converges to a possibly different self-similarly shrinking solution $\Sigma'$. The central point in the argument is a direct application of the Simon-{\L}ojasiewicz inequality to Huisken's monotone Gaussian integral for Mean Curvature Flow.