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

2011/07/26 by Mario Micallef, Micallef, Mario, Vlad Moraru +1 · 3 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1107.5346

Abstract

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.

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Cited by

• Stability and rigidity of axisymmetric marginally outer trapped two-spheres
• Rigidity of marginally outer trapped 2-spheres
• Rigidity phenomena involving scalar curvature
• Area-charge inequalities and rigidity of time-symmetric initial data sets

Related

• On Area Comparison and Rigidity Involving the Scalar Curvature
• A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundary
• Rigidity of area-minimizing hyperbolic surfaces in three-manifolds
• Rigidity of area-minimizing free boundary surfaces in mean convex\n three-manifolds
• Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature