The singular structure and regularity of stationary varifolds

Abstract

If one considers an integral varifold I^m\subseteq M with bounded mean curvature, and if S^k(I)\equiv\{x\in M: no tangent cone at x is k+1 -symmetric} is the standard stratification of the singular set, then it is well known that \mathrm {dim} S^k\leq 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\in S^k that there exists a unique k -plane V^k such that every tangent cone at x is of the form V\times C for some cone C . In the case of minimizing hypersurfaces I^{n-1}\subseteq M^n we can go further. Indeed, we can show that the singular set S(I) , which is known to satisfy \mathrm {dim} S(I)\leq 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_{\mathrm {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^k_{\epsilon,r} and S^k_{\epsilon}\equiv S^k_{\epsilon,0} . Roughly, x\in S^k_{\epsilon}\subseteq I if no ball B_r(x) is \epsilon -close to being k+1 -symmetric. We show that S^k_\epsilon is k -rectifiable and satisfies the Minkowski estimate \mathrm {Vol}(B_r\,S_\epsilon^k)\leq C_\epsilon r^{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].