Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917351948025856 |
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| author | Sheng, Hongyi |
| author_facet | Sheng, Hongyi |
| contents | On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19169 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature Sheng, Hongyi Differential Geometry On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere. |
| title | Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2403.19169 |