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

This paper studies singularities of mean curvature flows with integral mean curvature bounds $H\in L^{\mathrm{\infty }}{L}_{loc}^p$ for some $p\in (n,\mathrm{\infty }]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$. When $p=\mathrm{\infty }$ 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 $H\in L^{\mathrm{\infty }}{L}_{loc}^p$. For smooth, codimension one mean curvature flows with $H\in L^{\mathrm{\infty }}{L}_{loc}^{\mathrm{\infty }}$, 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.