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

Neshan Wickramasekera (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England)

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.

Published 16 July 2019