Quantitative Stratification and the Regularity of Mean Curvature Flow

Jeff Cheeger, Robert Haslhofer, Aaron Naber

Research output: Contribution to journalArticle

Abstract

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>00<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).

Original languageEnglish (US)
Pages (from-to)828-847
Number of pages20
JournalGeometric and Functional Analysis
Volume23
Issue number3
DOIs
StatePublished - Jun 2013

Fingerprint

Mean Curvature Flow
Stratification
Regularity
Estimate
Hypersurface
Symmetry
Singular Set
Second Fundamental Form
Hausdorff Dimension
Tangent line
Set of points
n-dimensional
Multiplicity

ASJC Scopus subject areas

  • Geometry and Topology
  • Analysis

Cite this

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

In: Geometric and Functional Analysis, Vol. 23, No. 3, 06.2013, p. 828-847.

Research output: Contribution to journalArticle

Cheeger, Jeff ; Haslhofer, Robert ; Naber, Aaron. / Quantitative Stratification and the Regularity of Mean Curvature Flow. In: Geometric and Functional Analysis. 2013 ; Vol. 23, No. 3. pp. 828-847.
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