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 $H \in L^\infty L^p_{loc}$ 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 $H \in L^\infty L^p_{loc}$. For smooth, codimension one mean curvature flows with $H \in L^\infty L^\infty_{loc}$, 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.