Quantitative Stratification and the Regularity of Mean Curvature Flow
Author(s)Cheeger, Jeff; Haslhofer, Robert; Naber, Aaron Charles
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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).
Original manuscript October 29, 2012
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Geometric and Functional Analysis
Cheeger, Jeff, Robert Haslhofer, and Aaron Naber. “Quantitative Stratification and the Regularity of Mean Curvature Flow.” Geometric and Functional Analysis 23, no. 3 (June 7, 2013): 828-847.