Some rigidity results for static three-manifolds with boundary and positive scalar curvature

2025/10/08 by В. И. Медведев, Vladimir Medvedev, Medvedev, Vladimir
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2510.07296

openalex publication_date 2025/10/08 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

This paper studies three-dimensional compact static manifolds with boundary and positive scalar curvature. We prove that, under a suitable bound on the Ricci curvature, the orientable quotient of the Nariai static manifold with boundary Nar_-1,1(\mathbb S²) is the only such manifold with connected boundary, provided that the zero-level set of the potential is connected and does not intersect the boundary. We also establish a rigidity theorem for the upper hemisphere with the standard static potential, in the spirit of Cruz and Nunes.

Citations

• Static manifolds with boundary: Their geometry and some uniqueness theorems
• Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature
• Localized Deformation of the Scalar Curvature and the Mean Curvature
• On static manifolds satisfying an overdetermined Robin type condition on the boundary
• Rigidity of non-compact static domains in hyperbolic space via positive mass theorems
• A local rigidity theorem for minimal two-spheres in an electrovacuum spacetime
• Critical metrics and curvature of metrics with unit volume or unit area of the boundary
• Deformation of the scalar curvature and the mean curvature
• On static three-manifolds with positive scalar curvature
• The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass
• Rigidity of area-minimizing two-spheres in three-manifolds
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• On the Fundamental Group of a Compact Minimal Submanifold

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