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# Current Developments in Mathematics

## Volume 2017

### Regularity and compactness for stable codimension $1$ CMC varifolds

Pages: 87 – 174

DOI: https://dx.doi.org/10.4310/CDM.2017.v2017.n1.a3

#### Author

#### Abstract

According to the Allard regularity theory, the set of singular points (i.e. non $C^{1,\alpha}$-embedded points) of an integral $n$-varifold with generalized mean curvature locally in $L^p$ for some $p \gt n$ is a nowhere dense (closed) subset of the support of the varifold. A well-known codimension $1$ example due to Brakke shows that not much can be said about the Hausdorff measure of the singular set; it need not have zero $n$-dimensional measure. We survey recent work that shows nonetheless, that in codimension $1$, all is well whenever those parts of the varifold that are regular (in certain specific ways) satisfy further hypotheses, namely: (a) that the orientable portions of the $C^{1,\alpha}$ embedded part and the $C^2$ immersed part are respectively stationary (or equivalently CMC) and stable with respect to the area functional for volume preserving deformations, and (b) that there is appropriate control on two types of singularities—called classical and touching singularities—that are formed by $C^{1,\alpha}$ embedded pieces of the varifold coming together in a regular fashion. This work builds on and extends the recent codimension $1$ theory for zero mean curvature stable varifolds with no classical singularities, and the earlier fundamental curvature estimates of Schoen–Simon–Yau and of Schoen–Simon. We include a brief discussion of these previous works and their role in (different approaches to) the existence theory for minimal hypersurfaces in compact Riemannian manifolds. The main focus of the survey is on the novel aspects of the CMC regularity and compactness theory and the associated curvature estimates.

This article is an expanded version of the themes covered in the author’s lectures at the 2017 Current Developments in Mathematics conference at Harvard University. The author thanks the organizers—David Jerison, Mark Kisin, Paul Seidel, Richard Stanley, Horng-Tzer Yau, and Shing-Tung Yau—for their invitation to deliver the lectures and for their hospitality during the event.

Published 16 July 2019