  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.
