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

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 $$\Omega \mapsto \mathrm{Area}(\partial \Omega )-{\int }_{\Omega }h+f(\mathrm{Vol}(\Omega )).$$

Added positive Ricci curvature case