Mean curvature in manifolds with Ricci curvature bounded from below

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 \ Sigma is isometric to the Riemannian product Sigma x (a, b), and if Sigma separates M then the map i(*) : pi(1)(Sigma) -> pi(1)(M) induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature H >= (n - 1) root k in a manifold of Ricci curvature Ric(M) >= -(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.