JournalsjemsVol. 22, No. 10pp. 3305–3382

The singular structure and regularity of stationary varifolds

  • Aaron Naber

    Northwestern University, Evanston, USA
  • Daniele Valtorta

    Universität Zürich, Switzerland
The singular structure and regularity of stationary varifolds cover

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If one considers an integral varifold ImMI^m\subseteq M with bounded mean curvature, and if Sk(I){xM:S^k(I)\equiv\{x\in M: no tangent cone at xx is k+1k+1-symmetric} is the standard stratification of the singular set, then it is well known that dimSkk\mathrm {dim} S^k\leq k. In complete generality nothing else is known about the singular sets Sk(I)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 Sk(I)S^k(I) is kk-rectifiable. In fact, we prove for kk-a.e. point xSkx\in S^k that there exists a unique kk-plane VkV^k such that every tangent cone at xx is of the form V×CV\times C for some cone CC.

In the case of minimizing hypersurfaces In1MnI^{n-1}\subseteq M^n we can go further. Indeed, we can show that the singular set S(I)S(I), which is known to satisfy dimS(I)n8\mathrm {dim} S(I)\leq n-8, is in fact n8n-8 rectifiable with uniformly finite n8n-8 measure. An effective version of this allows us to prove that the second fundamental form AA has a priori estimates in Lweak7L^7_{\mathrm {weak}} on II, an estimate which is sharp as A|A| is not in L7L^7 for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale rIr_I has Lweak7L^7_{weak}-estimates.

The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications Sϵ,rkS^k_{\epsilon,r} and SϵkSϵ,0kS^k_{\epsilon}\equiv S^k_{\epsilon,0}. Roughly, xSϵkIx\in S^k_{\epsilon}\subseteq I if no ball Br(x)B_r(x) is ϵ\epsilon-close to being k+1k+1-symmetric. We show that SϵkS^k_\epsilon is kk-rectifiable and satisfies the Minkowski estimate Vol(BrSϵk)Cϵrnk\mathrm {Vol}(B_r\,S_\epsilon^k)\leq C_\epsilon r^{n-k}. The proof requires a new L2L^2-subspace approximation theorem for integral varifolds with bounded mean curvature, and a W1,pW^{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