Compact Mean Convex Hypersurfaces and the Fundamental Group of Manifolds with Nonnegative Ricci Curvature

Abstract

We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism between the fundamental groups of S and N induced by the inclusion is surjective, provided only that N - S has two connected components, one of which has noncompact closure and trivial homotopy relative to S. The idea of the proof is to view N as a spacelike hypersurface in a suitable Lorentz manifold and then apply a recent version of certain classic results by Gannon and Lee on the topology of spacetimes. As an application, we show that if N is asymptotically flat, then it has only one end, and N is simply connected for n larger than 3.