An Alternative for Constant Mean Curvature Hypersurfaces

TL;DR

This paper investigates the existence of closed hypersurfaces with constant mean curvature in certain manifolds, showing either infinitely many with mean curvature c or infinitely many with less than c enclosing half the volume.

Contribution

It establishes a dichotomy for the existence of constant mean curvature hypersurfaces in generic manifolds of dimension 3 to 7.

Findings

Either infinitely many hypersurfaces with mean curvature c exist.

Or infinitely many with mean curvature less than c enclose half the volume.

Results apply to generic Riemannian metrics on manifolds of dimension 3 to 7.

Abstract

Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$ equipped with a generic Riemannian metric $g$. Let $c$ be a positive number. We show that, either there exist infinitely many distinct closed hypersurfaces with constant mean curvature equal to $c$, or there exist infinitely many distinct closed hypersurfaces with constant mean curvature less than $c$ but enclosing half the volume of $M$.