On static three-manifolds with positive scalar curvature

2015/03/12 by Lucas Ambrozio, Ambrozio, L. · 3 citations
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.1503.03803

openalex publication_date 2015/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein (n+1)-manifold associated to a maximal domain of a static n-manifold where the static potential is positive. There are examples where this construction inevitably produces an Einstein metric with conical singularities along a codimension-two submanifold. By proving versions of classical results for Einstein four-manifolds for the singular spaces thus obtained, we deduce some classification results for compact static three-manifolds with positive scalar curvature.

Citations

• A note on static spaces and related problems
• Normal and Integral Currents
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Cited by

• Critical metrics of the volume functional on complete manifolds
• Some rigidity results for static three-manifolds with boundary and positive scalar curvature
• Geometric inequalities for electrostatic systems with boundary

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