Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory

TL;DR

This paper proves that under generic or positive Ricci curvature conditions, closed manifolds of dimension 3 to 5 contain infinitely many distinct constant mean curvature hypersurfaces that enclose half the volume, using advanced min-max theory.

Contribution

It develops a new Almgren-Pitts type min-max theory for non-local functionals and applies it to establish the existence of infinitely many constant mean curvature hypersurfaces.

Findings

Existence of infinitely many constant mean curvature hypersurfaces in certain manifolds.

Development of a min-max theory for non-local geometric functionals.

Application to manifolds with positive Ricci curvature or generic metrics.

Abstract

Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant mean curvature hypersurfaces, each enclosing half the volume of $M$. As an essential part of the proof, we develop an Almgren-Pitts type min-max theory for certain non-local functionals of the general form $$\Omega \mapsto \operatorname{Area}(\partial \Omega) - \int_\Omega h + f(\operatorname{Vol}(\Omega)).$$