On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912687023194112 |
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| author | Moriani, Alex Trebeschi, Enrico |
| author_facet | Moriani, Alex Trebeschi, Enrico |
| contents | We study in this article the curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. We show that the scalar curvature of these submanifolds is nonpositive in every signature. This gives, together with a result of Ishihara, a sharp bound on the scalar curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces of every signature. We show that achieving the bound at a point is equivalent to achieving it identically, and explicitely describe the submanifolds achieving the bound. When the codimension is equal to 1, we deduce a sharp upper bound on the Ricci curvature of complete maximal hypersurfaces in Anti-de Sitter spaces, and characterize the hypersurfaces achieving it. Finally, we discuss the link between scalar curvature and Gromov-hyperbolicity for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces Moriani, Alex Trebeschi, Enrico Differential Geometry We study in this article the curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. We show that the scalar curvature of these submanifolds is nonpositive in every signature. This gives, together with a result of Ishihara, a sharp bound on the scalar curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces of every signature. We show that achieving the bound at a point is equivalent to achieving it identically, and explicitely describe the submanifolds achieving the bound. When the codimension is equal to 1, we deduce a sharp upper bound on the Ricci curvature of complete maximal hypersurfaces in Anti-de Sitter spaces, and characterize the hypersurfaces achieving it. Finally, we discuss the link between scalar curvature and Gromov-hyperbolicity for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. |
| title | On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2411.10352 |