Quantitative Stratification and the Regularity of Mean Curvature Flow

Abstract

Let \({\mathcal{M}}\) be a Brakke flow of n-dimensional surfaces in \({\mathbb{R}^N}\). The singular set \({\mathcal{S} \subset \mathcal{M}}\) has a stratification \({\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}\), where \({X \in \mathcal{S}^j}\) if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata \({\mathcal{S}^j_{\eta, r}}\) satisfying \({\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}\). Sharpening the known parabolic Hausdorff dimension bound \({{\rm dim} \mathcal{S}^j \leq j}\), we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of \({\mathcal{S}^j_{\eta, r}}\) satisfies \({{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}\). 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 \({\mathcal{B}_r \subset \mathcal{M}}\) the set of points with regularity scale less than r, we prove that \({{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}\). 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 reciprocal 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 (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).