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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> H\in L^{\infty}L^{p}_{\mathrm{loc}} for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> p\in(n,\infty] . For such flows, any backward tangent flow is given by the flow of a stationary cone 𝐂. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⊆</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:math> U\subseteq\mathbb{R}^{N} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> H\in L^{\infty}L^{p}_{\mathrm{loc}} . For smooth, codimension-one mean curvature flows with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> 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.