Quantitative Stratification and the Regularity of Mean Curvature Flow

Let $\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. The singular set $\cS\subset\cM$ has a stratification $\cS^0\subset\cS^1\subset...\cS$, where $X\in \cS^j$ if no tangent flow at $X$ has more than $j$ symmetries. Here, we define quantitative singular strata $\cS^j_{\eta,r}$ satisfying $\cup_{\eta>0}\cap_{0<r} \cS^j_{\eta,r}=\cS^j$. Sharpening the known parabolic Hausdorff dimension bound $\dim \cS^j\leq j$, we prove the effective Minkowski estimates that the volume of $r$-tubular neighborhoods of $\cS^j_{\eta,r}$ satisfies $\Vol (T_r(\cS^j_{\eta,r})\cap B_1)\leq Cr^{N+2-j-\varepsilon}$. Our primary application of this is to higher regularity of Brakke flows starting at $k$-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of $k$-convex hypersurfaces, any backwards selfsimilar limit flow with at least $k$ symmetries is in fact a static multiplicity one plane. Then, denoting by $\cB_r\subset\cM$ the set of points with regularity scale less than $r$, we prove that $\Vol(T_r(\cB_r))\leq C r^{n+4-k-\varepsilon}$. This gives $L^p$-estimates for the second fundamental form for any $p<n+1-k$. In fact, the estimates are much stronger and give $L^p$-estimates for the inverse of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (arXiv:1103.1819v3) and Cheeger and Naber (arXiv:1107.3097v1).