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