Quantitative Stratification and the Regularity of Mean Curvature Flow

Let M be a Brakke flow of n-dimensional surfaces in ℝ^N. The singular set S ⊂ M has a stratification S⁰ ⊂ S¹...S, where X ∈ S^j if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata S^j_ηr satisfying ∪_η>0 ∩_0<r S^j_η,r= S^j. Sharpening the known parabolic Hausdorff dimension bound of dim S^j ≤ j, we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of S^j_ηr satisfies Vol(T_r(S^j_ηr)∩B₁) ≤ Cr^N+2-j-∈. 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 B_r ⊂ M the set of points with regularity scale less than r, we prove that Vol(T_r(B_r)) ≤ Cr^n+4-k-∈. 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).