Quantitative Stratification and the Regularity of Mean Curvature Flow

By Jeff Cheeger, Robert Haslhofer, A. Naber. 2012

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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 Lp-estimates for the second fundamental form for any p < n + 1 − k. In fact, the estimates are much stronger and give Lp-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).

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f6b49319f1aef93a95670c6c2a093f30dac4b8c0

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Citations in the corpus

Ancient asymptotically cylindrical flows and applications (2019, unknown): “Indeed, as is explained in [CHN13] and Section 4.”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, methodology): “In fact, using quantitative stratification this can be improved to Minkowski and L-estimates, in particular in the k-convex case, see Cheeger-Haslhofer-Naber [10].”