Quantitative Stratification and the Regularity of Mean Curvature Flow
arXiv:1207.3619
Abstract
Let $\cM$ be a Brakke flow of -dimensional surfaces in . 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 has more than symmetries. Here, we define quantitative singular strata $\cS^j_{η,r}$ satisfying $\cup_{η>0}\cap_{0<r} \cS^j_{η,r}=\cS^j$. Sharpening the known parabolic Hausdorff dimension bound $\dim \cS^j\leq j$, we prove the effective Minkowski estimates that the volume of -tubular neighborhoods of $\cS^j_{η,r}$ satisfies $\Vol (T_r(\cS^j_{η,r})\cap B_1)\leq Cr^{N+2-j-\varepsilon}$. Our primary application of this is to higher regularity of Brakke flows starting at -convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of -convex hypersurfaces, any backwards selfsimilar limit flow with at least 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 , we prove that $\Vol(T_r(\cB_r))\leq C r^{n+4-k-\varepsilon}$. This gives -estimates for the second fundamental form for any . In fact, the estimates are much stronger and give -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).