  Self-shrinkers model singularities of the mean curvature flow; they are
defined as the special solutions that contract homothetically under the flow.
Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a
strong sense - that is, any self-shrinker that is mean convex with uniformly
bounded curvature on a large, but compact, set must be a round cylinder. Using
this result, Colding and Minicozzi were able to establish uniqueness of blowups
at cylindrical singularities, and provide a detailed description of the
singular set of generic mean curvature flows.
  In this paper, we show that the bounded curvature assumption is unnecessary
for the rigidity of the cylinder if either n is at most 6, or if the mean
curvature is bounded below by a positive constant. These results follow from
curvature estimates that we prove for strictly mean convex self-shrinkers. We
also obtain a rigidity theorem in all dimensions for graphical self-shrinkers,
and curvature estimates for translators of the mean curvature flow.
