## Abstract

If one considers an integral varifold I^{m} ⊆ M with bounded mean curvature, and if S^{k}(I) ≡ {x ∈ M : no tangent cone at x is k + 1-symmetric} is the standard stratification of the singular set, then it is well known that dim S^{k}(I) ≤ k. In complete generality nothing else is known about the singular sets S^{k}(I). In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum S^{k}(I) is k-rectifiable. In fact, we prove for k-a.e. point x ∈ S^{k}(I) that there exists a unique k-plane V^{k} such that every tangent cone at x is of the form V × C for some cone C. In the case of minimizing hypersurfaces I^{n-1} ⊆ M^{n} we can go further. Indeed, we can show that the singular set S(I), which is known to satisfy dim S(I) ≤ n - 8, is in fact n - 8-rectifiable with uniformly finite n - 8-measure. An effective version of this allows us to prove that the second fundamental form A has a priori estimates in L^{7}_{weak} on I, an estimate which is sharp as |A| is not in L^{7} for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale r_{I} has L^{7}_{weak} estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications S_{,r}^{k} and S^{k} ≡ S,^{k}_{0}. Roughly, x ∈ S^{k} ⊆ I if no ball B_{r}(x) is -close to being k + 1-symmetric. We show that S^{k} is k-rectifiable and satisfies the Minkowski estimate Vol(B_{r}(S^{k})) ≤ Cr^{n-k}. The proof requires a new L^{2} subspace approximation theorem for integral varifolds with bounded mean curvature, and a W^{1,p}-Reifenberg type theorem proved by the authors in [NVa].

Original language | English (US) |
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Pages (from-to) | 3305-3382 |

Number of pages | 78 |

Journal | Journal of the European Mathematical Society |

Volume | 22 |

Issue number | 10 |

DOIs | |

State | Published - Jul 19 2020 |

## Keywords

- Mean curvature
- Minimal currents
- Quantitative stratification
- Singularities
- Varifolds

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics