TY - JOUR

T1 - Quantitative Stratification and the Regularity of Mean Curvature Flow

AU - Cheeger, Jeff

AU - Haslhofer, Robert

AU - Naber, Aaron

N1 - Funding Information:
Jeff Cheeger was partially supported by NSF grant DMS1005552. Robert Haslhofer was partially supported by the Swiss National Science Foundation. Aaron Naber was partially supported by NSF postdoctoral grant 0903137.

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).

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U2 - 10.1007/s00039-013-0224-9

DO - 10.1007/s00039-013-0224-9

M3 - Article

AN - SCOPUS:84878566214

SN - 1016-443X

VL - 23

SP - 828

EP - 847

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 3

ER -