On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces

Alex Moriani; Enrico Trebeschi

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arXiv · 2411.10352

On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces

Alex Moriani·Enrico Trebeschi

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Abstract

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.

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BibTeX RIS

Alex Moriani, Enrico Trebeschi. 2024-11-15. On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces. https://arxiv.org/abs/2411.10352

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