On isolated singularities and generic regularity of min-max CMC hypersurfaces

TL;DR

This paper proves that min-max constant mean curvature hypersurfaces in positively curved manifolds have isolated singularities that are area-minimising, and demonstrates the existence of smooth CMC hypersurfaces in generic 8D manifolds.

Contribution

It establishes regularity and isolated singularity properties of min-max CMC hypersurfaces and constructs smooth CMC hypersurfaces in generic high-dimensional manifolds.

Findings

Isolated singularities are area-minimising.

Min-max CMC hypersurfaces have regularity properties.

Existence of smooth CMC hypersurfaces in generic 8D manifolds.

Abstract

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure of Bellettini-Wickramasekera (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real $\lambda$ we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature $\lambda$; the minimal case ($\lambda$ = 0) of this result was obtained in work by Chodosh-Liokumovich-Spolaor.