### 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) |
---|---|

Pages (from-to) | 828-847 |

Number of pages | 20 |

Journal | Geometric and Functional Analysis |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2013 |

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### ASJC Scopus subject areas

- Geometry and Topology
- Analysis

### Cite this

*Geometric and Functional Analysis*,

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

**Quantitative Stratification and the Regularity of Mean Curvature Flow.** / Cheeger, Jeff; Haslhofer, Robert; Naber, Aaron.

Research output: Contribution to journal › Article

*Geometric and Functional Analysis*, vol. 23, no. 3, pp. 828-847. https://doi.org/10.1007/s00039-013-0224-9

}

TY - JOUR

T1 - Quantitative Stratification and the Regularity of Mean Curvature Flow

AU - Cheeger, Jeff

AU - Haslhofer, Robert

AU - Naber, Aaron

PY - 2013/6

Y1 - 2013/6

N2 - 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 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).

AB - 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 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).

UR - http://www.scopus.com/inward/record.url?scp=84878566214&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878566214&partnerID=8YFLogxK

U2 - 10.1007/s00039-013-0224-9

DO - 10.1007/s00039-013-0224-9

M3 - Article

AN - SCOPUS:84878566214

VL - 23

SP - 828

EP - 847

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 3

ER -