## 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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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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DOI: https://doi.org/10.1007/s00039-013-0224-9