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

2010/12/06 by I. P. Costa e Silva, Silva, I. P. Costa e
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1012.1302

openalex publication_date 2010/12/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• A remark on the rigidity case of the positive energy theorem
• On the proof of the positive mass conjecture in general relativity
• The mass of an asymptotically flat manifold
• A new proof of the positive energy theorem

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• On the topology of manifolds with nonnegative Ricci curvature and linear volume growth
• The Boundary between Compact and Noncompact Complete Riemann Manifolds
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