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Quantitative Stratification and the Regularity of Mean Curvature Flow

Abstract

Let \({\mathcal{M}}\) be a Brakke flow of n-dimensional surfaces in \({\mathbb{R}^N}\). The singular set \({\mathcal{S} \subset \mathcal{M}}\) has a stratification \({\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}\), where \({X \in \mathcal{S}^j}\) if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata \({\mathcal{S}^j_{\eta, r}}\) satisfying \({\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}\). Sharpening the known parabolic Hausdorff dimension bound \({{\rm dim} \mathcal{S}^j \leq j}\), we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of \({\mathcal{S}^j_{\eta, r}}\) satisfies \({{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}\). 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 \({\mathcal{B}_r \subset \mathcal{M}}\) the set of points with regularity scale less than r, we prove that \({{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}\). 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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Correspondence to Jeff Cheeger.

Additional 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.

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Cheeger, J., Haslhofer, R. & Naber, A. Quantitative Stratification and the Regularity of Mean Curvature Flow. Geom. Funct. Anal. 23, 828–847 (2013). https://doi.org/10.1007/s00039-013-0224-9

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Keywords

  • Fundamental Form
  • Regularity Theory
  • Tubular Neighborhood
  • Elliptic Regularization
  • Monotonicity Formula