Some rigidity results for static three-manifolds with boundary and positive scalar curvature
Vladimir Medvedev
TL;DR
The paper addresses rigidity of compact static 3-manifolds with boundary under positive scalar curvature and a Ricci bound, showing that the orientable quotient $Nar_{-1,1}(\mathbb S^{2})/A$ is the unique example (up to equivalence) with connected boundary when $V^{-1}(0)$ is connected and disjoint from the boundary. It introduces Robin static triples and employs a Robin-boundary framework, a nonexistence/structure lemma for stable minimal surfaces, and a Pohozaev-type integral identity to derive strong rigidity conclusions. The main contributions are twofold: (i) a Ricci bound classification proving equivalence to $Nar_{-1,1}(\mathbb S^{2})/A$ under $|\mathring{\mathrm{Ric}}_g|^2\le 6$; (ii) a Cruz–Nunes-type hemispherical rigidity giving area bounds and, in equality cases, identification with the standard hemisphere with $V$ aligned to coordinate functions. These results advance the understanding of rigidity in static vacua with boundary and connect to Nariai and Schwarzschild–de Sitter geometries, while providing tools (Robin static triples and Pohozaev identities) for further analysis.
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 $Nar_{-1,1}(\mathbb S^2)$ 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.