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 $\dim S^k\le k$. 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 $\dim S(I)\le n-8$, 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 $L_{\mathrm{weak}}^7$ 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 $L_{weak}^7$-estimates.

The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications $S_{ϵ,r}^k$ and $S_{ϵ}^k\equiv S_{ϵ,0}^k$. Roughly, $x\in S_{ϵ}^k\subseteq I$ if no ball $B_r(x)$ is $ϵ$-close to being $k+1$-symmetric. We show that $S_{ϵ}^k$ is $k$-rectifiable and satisfies the Minkowski estimate $\mathrm{Vol}(B_r{S}_{ϵ}^k)\le C_{ϵ}{r}^{n-k}$. 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].