Mean curvature in manifolds with Ricci curvature bounded from below

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 $i_{\ast }:{\pi }_1(\Sigma )\to {\pi }_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\ge (n-1)\sqrt{k}$ in a manifold of Ricci curvature Ric${}_M\ge -(n-1)k$, $k>0$, and for a free boundary minimal hypersurface in an $n$-dimensional manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact $(n-1)$-dimensional manifold $N$ with the number of generators of ${\pi }_1(N)<n-1$ cannot be minimally embedded in the flat torus $T^n$.