Mean curvature in manifolds with Ricci curvature bounded from below

TL;DR

This paper investigates the geometric and topological properties of minimal hypersurfaces in manifolds with nonnegative Ricci curvature, revealing a dichotomy in their structure and implications for fundamental groups.

Contribution

It establishes a dichotomy for minimal hypersurfaces in nonnegative Ricci curvature manifolds and proves surjectivity of induced fundamental group maps under certain curvature conditions.

Findings

If the hypersurface does not separate, it is totally geodesic and splits the manifold.

If it separates, the induced fundamental group map is surjective.

A manifold with fewer generators than its dimension cannot embed minimally in a flat torus.

Abstract

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $\Sigma$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $\Sigma$ does not separate $M$ then $\Sigma$ is totally geodesic and $M\setminus\Sigma$ is isometric to the Riemannian product $\Sigma\times(a,b)$, and if $\Sigma$ separates $M$ then the map $i_*:\pi_1(\Sigma)\rightarrow \pi_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\geq(n-1)\sqrt{k}$ in a manifold of Ricci curvature $Ric_M\geq-(n-1)k,k>0$, and for a free boundary minimal hypersurface in a manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact $n$-dimensional manifold $N$ with the number of generators of $\pi_1(N)<n$ cannot be minimally embedded in the flat torus $T^{n+1}$.