Rigidity and Curvature Estimates for Graphical Selfshrinkers
Abstract
Selfshrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire selfshrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \leq 6$, any smooth, complete selfshrinker $\Sigma^n\subset\mathbf{R}^{n+1}$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable selfshrinkers. A key component of this paper is the procurement of linear curvature estimates for almost stable shrinkers, and it is this step that is responsible for the restriction on $n$. Our methods also yield uniform curvature bounds for translating solitons of the mean curvature flow.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 DOI:
 10.48550/arXiv.1510.06061
 arXiv:
 arXiv:1510.06061
 Bibcode:
 2015arXiv151006061G
 Keywords:

 Mathematics  Differential Geometry;
 53C44;
 53C24
 EPrint:
 20 pages