The singular structure and regularity of stationary varifolds

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 ⊆ 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 L⁷_weak 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 r_I has L⁷_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,^k₀. 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 L² subspace approximation theorem for integral varifolds with bounded mean curvature, and a W^1,p-Reifenberg type theorem proved by the authors in [NVa].