Rigidity of generic singularities of mean curvature flow

TL;DR

This paper proves a strong rigidity property of round cylinders as the only generic singularities in mean curvature flow, showing that any close shrinker must be a round cylinder, with implications for understanding singularities in geometric flows.

Contribution

It establishes the first general rigidity theorem for singularities in a nonlinear geometric flow, demonstrating that round cylinders are uniquely characterized among shrinkers.

Findings

Round cylinders are the only generic singularities in MCF.

Any shrinker close to a round cylinder on a large set must be a round cylinder.

Results hold in all dimensions without requiring smoothness.

Abstract

Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, [CM1] showed that the only generic are round cylinders $\SS^k\times \RR^{n-k}$. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder. To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows. Our results hold in all dimensions and do not require any a priori smoothness.