Singular behaviour and long time behaviour of mean curvature flow

Abstract

In this thesis, we investigate two asymptotic behaviours of the mean curvature flow. The first one is the asymptotic behaviour of singularities of the mean curvature flow, and the asymptotic limit is modelled by the tangent flows. The second one is the asymptotic behaviour of the mean curvature flow as time goes to infinity. We will study several problems related to the asymptotic behaviours. The first problem is the partial regularity of the limit. The partial regularity of mean curvature flow without any curvature assumptions was first studied by Ilmanen. We will follow the idea of Ilmanen to study the partial regularity of other asymptotic limit. In particular, we introduce a generalization of Colding-Minicozzi's entropy in a closed manifold, which plays a significant role. The second problem is the genericity of the tangent flows of mean curvature flow. The generic mean curvature flow was introduced by Colding-Minicozzi. Furthermore, they introduced mean curvature flow entropy and use it to study the generic tangent flows of mean curvature flow. We study the multiplicity of the generic tangent flow. In particular, we prove that the generic compact tangent flow of mean curvature flow of surfaces has multiplicity 1. This result partially addresses the famous multiplicity 1 conjecture of Ilmanen. One key idea is defining a local version of Colding-Minicozzi's entropy. We also discuss some related results. These results include a joint work with Zhichao Wang and a joint work with Julius Baldauf.