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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Abstract

If one considers an integral varifold with bounded mean curvature, and if no tangent cone at is -symmetric} is the standard stratification of the singular set, then it is well known that . In complete generality nothing else is known about the singular sets . In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum is -rectifiable. In fact, we prove for -a.e. point that there exists a unique -plane such that every tangent cone at is of the form for some cone .

In the case of minimizing hypersurfaces we can go further. Indeed, we can show that the singular set , which is known to satisfy , is in fact rectifiable with uniformly finite measure. An effective version of this allows us to prove that the second fundamental form has a priori estimates in on , an estimate which is sharp as is not in for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale has -estimates.

The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications and . Roughly, if no ball is -close to being -symmetric. We show that is -rectifiable and satisfies the Minkowski estimate . The proof requires a new -subspace approximation theorem for integral varifolds with bounded mean curvature, and a -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