The singular structure and regularity of stationary varifolds

If one considers an integral varifold Im ⊆ M with bounded mean curvature, and if Sk(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 Sk(I) ≤ k. In complete generality nothing else is known about the singular sets Sk(I). In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum Sk(I) is k-rectifiable. In fact, we prove for k-a.e. point x ∈ Sk(I) that there exists a unique k-plane Vk such that every tangent cone at x is of the form V × C for some cone C. In the case of minimizing hypersurfaces In-1 ⊆ Mn 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 L7weak on I, an estimate which is sharp as |A| is not in L7 for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale rI has L7weak estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications S,rk and Sk ≡ S,k0. Roughly, x ∈ Sk ⊆ I if no ball Br(x) is -close to being k + 1-symmetric. We show that Sk is k-rectifiable and satisfies the Minkowski estimate Vol(Br(Sk)) ≤ Crn-k. The proof requires a new L2 subspace approximation theorem for integral varifolds with bounded mean curvature, and a W1,p-Reifenberg type theorem proved by the authors in [NVa].