Splitting of 3-Manifolds and Rigidity of Area-Minimising Surfaces

Details Splitting of 3-Manifolds and Rigidity of Area-Minimising Surfaces Splitting of 3-Manifolds and Rigidity of Area-Minimising Surfaces

2011-07-26

arxiv.org/abs/1107.5346v3

Mathematics

Differential Geometry

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Scientific paper

In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.

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