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

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

2010-12-06

arxiv.org/abs/1012.1302v1

Mathematics

Differential Geometry

10 pages, no figures

Scientific paper

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.

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