Abstract
Our aim in this paper is to study the uniqueness of complete spacelike hypersurfaces immersed in the anti-de Sitter space \({{\mathbb {H}}}_1^{n+1}\), through the behavior of their higher order mean curvatures. This is done by applying a suitable maximum principle concerning smooth vector fields whose norm is Lebesgue integrable on a complete Riemannian manifold. We also infer the nullity of complete r-maximal spacelike hypersurfaces and, in particular, we establish a nonexistence result concerning complete 1-maximal spacelike hypersurfaces in \({{\mathbb {H}}}_1^{n+1}\).
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Acknowledgements
The first author is partially supported by CAPES, Brazil. The second and third authors are partially supported by CNPq, Brazil, grants 301970/2019-0 and 311224/2018-0, respectively.
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Barboza, W.F.C., de Lima, H.F. & Velásquez, M.A.L. Uniqueness and nullity of complete spacelike hypersurfaces immersed in the anti-de Sitter space. Ann Univ Ferrara 69, 95–109 (2023). https://doi.org/10.1007/s11565-022-00403-y
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DOI: https://doi.org/10.1007/s11565-022-00403-y
Keywords
- Anti-de Sitter space
- Complete spacelike hypersurfaces
- Totally umbilical spacelike hypersurfaces
- Higher order mean curvatures
- r-maximal hypersurfaces
- Index of nullity