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

TL;DR

This paper proves that certain hypersurfaces in nonnegatively curved manifolds influence the fundamental group's structure, showing that such manifolds with specific boundary conditions are simply connected and have a single end.

Contribution

It establishes a new link between the geometry of hypersurfaces and the topology of manifolds with nonnegative Ricci curvature, using Lorentzian techniques.

Findings

Fundamental group homomorphism is surjective under given conditions.

Manifolds with these hypersurfaces have only one end if asymptotically flat.

Such manifolds are simply connected in dimensions greater than 3.

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.