Convex solutions to the mean curvature flow

• 成果类型: Article
• 署名作者: Wang, Xu-Jia
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.3.1
• 发表日期: 2011
• 页码: 1185-1239
• 关键词: hessian equations; dirichlet problem; SINGULARITIES; SURFACES; sets; hypersurfaces

摘要

In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rn+1. An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2 <= k <= n, namely whether its level set is a sphere or cylinder Sk-1 x Rn-k. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is k-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space Rn+1 is a shrinking sphere or cylinder.