Rigidity of generic singularities of mean curvature flow

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, Colding and Minicozzi II (Ann. Math. 175(2):755–833, 2012) showed that the only generic are round cylinders S k ×R 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.