# The singular structure and regularity of stationary varifolds

### Aaron Naber

Northwestern University, Evanston, USA### Daniele Valtorta

Universität Zürich, Switzerland

## 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 $dimS_{k}≤k$. 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 $x∈S_{k}$ 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 $dimS(I)≤n−8$, 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 $L_{weak}$ 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 $L_{weak}$-estimates.

The above results are in fact just applications of a new class of estimates we prove on the *quantitative* stratifications $S_{ϵ,r}$ and $S_{ϵ}≡S_{ϵ,0}$. Roughly, $x∈S_{ϵ}⊆I$ if no ball $B_{r}(x)$ is $ϵ$-close to being $k+1$-symmetric. We show that $S_{ϵ}$ is $k$-rectifiable and satisfies the Minkowski estimate $Vol(B_{r}S_{ϵ})≤C_{ϵ}r_{n−k}$. 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].

## Cite this article

Aaron Naber, Daniele Valtorta, The singular structure and regularity of stationary varifolds. J. Eur. Math. Soc. 22 (2020), no. 10, pp. 3305–3382

DOI 10.4171/JEMS/987