Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

TL;DR

This paper proves a volume estimate for minimal hypersurfaces in manifolds with nonnegative Ricci curvature, linking the hypersurface volume to the manifold's volume via a min-max approach.

Contribution

It introduces a new min-max estimate on volume width and constructs minimal hypersurfaces with volume bounds in such manifolds.

Findings

Existence of a minimal hypersurface with volume bounded by the manifold's volume.

Construction of a PL Morse function with controlled level set volume.

Establishment of a volume width estimate for manifolds with nonnegative Ricci curvature.

Abstract

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.