Let M be a Brakke flow of n-dimensional surfaces in ℝN. The singular set S ⊂ M has a stratification S0 ⊂ S1...S, where X ∈ Sj if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata Sjηr satisfying ∪η>0 ∩0<r Sjη,r= Sj. Sharpening the known parabolic Hausdorff dimension bound of dim Sj ≤ j, we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of Sjηr satisfies Vol(Tr(Sjηr)∩B1) ≤ CrN+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 Br ⊂ M the set of points with regularity scale less than r, we prove that Vol(Tr(Br)) ≤ Crn+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).
ASJC Scopus subject areas
- Geometry and Topology