Quantitative Stratification and the Regularity of Mean Curvature Flow

Abstract

Let M be a Brakke flow of n-dimensional surfaces in R[superscript N]. The singular set S ⊂ M has a stratification S[superscript 0] ⊂ S[superscript 1] ⊂ ⋯ S, where X ∈ S[superscript j] if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata S[j over η,r] satisfying ∪[subscript η>0]∩[subscript 0<r]S[j over η,r] = S[superscript j]. Sharpening the known parabolic Hausdorff dimension bound dimS[superscript j] ≤ j, we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of S[j over η,r] satisfies Vol(T[subscript r](S[j over η,r]) ∩ B[subscript 1]) ≤ Cr[superscript N+2−j−ε]. 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[subscript r] ⊂ M the set of points with regularity scale less than r, we prove that Vol(T[subscript r](B[subscript r])) ≤ Cr[superscript n+4−k−ε]. This gives L[superscript p]-estimates for the second fundamental form for any p < n + 1 − k. In fact, the estimates are much stronger and give L[superscript 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).