Self-shrinkers and translating solitons of mean curvature flow

Abstract

We study singularity models of mean curvature flow ("MCF") and their generalizations. In the first part, we focus on rigidity and curvature estimates for self-shrinkers. We give a rigidity theorem proving that any self-shrinker which is graphical in a large ball must be a hyperplane. This result gives a stronger version of the Bernstein type theorem for shrinkers proved by Ecker-Huisken. One key ingredient is a curvature estimate for almost stable shrinkers. By proving curvature estimates for mean convex shrinkers, we show that any shrinker which is mean convex in a large ball must be a round cylinder. This generalizes a result by Colding-Ilmanen-Minicozzi : no curvature bound assumption is needed. This part is joint work with Jonathan Zhu. In the second part, we consider [lambda]-hypersurfaces which can be thought of as a generalization of shrinkers. We first give various gap and rigidity theorems. We then establish the Bernstein type theorem for [lambda]-hypersurfaces and classify [lambda]-curves. In the last part, we study translating solitons of MCF from four aspects: volume growth, entropy, stability, and curvature estimates. First, we show that every properly immersed translator has at least linear volume growth. Second, using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. Third, we estimate the spectrum of the stability operator L for translators and give a rigidity result of L-stable translators. Finally, we provide curvature estimates for L-stable translators, graphical translators and translators with small entropy.