Embeddedness of Min-Max CMC Hypersurfaces in Manifolds with Positive Ricci Curvature

TL;DR

This paper proves that in manifolds with positive Ricci curvature, the min-max scheme produces embedded constant mean curvature hypersurfaces without certain singularities, confirming their existence and regularity.

Contribution

It establishes the embeddedness and regularity of min-max constant mean curvature hypersurfaces in positively curved manifolds, extending previous results by ruling out certain singularities.

Findings

Existence of embedded, closed λ-CMC hypersurfaces with Morse index 1

No even-multiplicity minimal hypersurfaces in the min-max interface

Absence of quasi-embedded points in the min-max hypersurface

Abstract

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $\lambda$, produces an embedded hypersurface of constant mean curvature $\lambda$ ($\lambda$-CMC). More precisely, we prove that the interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the first author and N. Wickramasekera). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed $\lambda$-CMC hypersurfaces (with Morse index $1$) for any prescribed non-zero constant $\lambda$, with the expected singular set when the ambient dimension is $8$ or higher.