On the rigidity of mean convex selfshrinkers
Abstract
Selfshrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. ColdingIlmanenMinicozzi showed that cylindrical selfshrinkers are rigid in a strong sense  that is, any selfshrinker 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 selfshrinkers. We also obtain a rigidity theorem in all dimensions for graphical selfshrinkers, and curvature estimates for translators of the mean curvature flow.
 Publication:

arXiv eprints
 Pub Date:
 March 2016
 DOI:
 10.48550/arXiv.1603.09435
 arXiv:
 arXiv:1603.09435
 Bibcode:
 2016arXiv160309435G
 Keywords:

 Mathematics  Differential Geometry;
 53C44;
 53C24
 EPrint:
 15 pages, comments welcome!