On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

Abstract

This paper studies singularities of mean curvature flows with integral mean curvature bounds $H\in L^{\infty }{L}_{loc}^p$ for some $p\in (n,\infty ]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$. When $p=\infty$ and $\mathbf{C}$ is a regular cone, we prove that the tangent flow is unique. These results hold for general integral Brakke flows of arbitrary codimension in an open subset $U\subseteq {\mathbb{R}}^N$ with $H\in L^{\infty }{L}_{loc}^p$. For smooth, codimension one mean curvature flows with $H\in L^{\infty }{L}_{loc}^{\infty }$, we also show that, at points where a tangent flow is given by an area-minimizing Simons cone, there is an accompanying limit flow given by a smooth Hardt-Simon minimal surface.

33 pages, comments welcome