### Abstract

Let M be a Brakke flow of n-dimensional surfaces in ℝ^{N}. The singular set S ⊂ M has a stratification S^{0} ⊂ S^{1}...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})∩B_{1}) ≤ 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).

Original language | English (US) |
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Pages (from-to) | 828-847 |

Number of pages | 20 |

Journal | Geometric and Functional Analysis |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2013 |

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

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## Cite this

*Geometric and Functional Analysis*,

*23*(3), 828-847. https://doi.org/10.1007/s00039-013-0224-9