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Uniqueness and nonexistence of complete spacelike hypersurfaces, Calabi–Bernstein type results and applications to Einstein–de Sitter and steady state type spacetimes

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Abstract

We investigate the geometry of complete spacelike hypersurfaces (immersed) in a generalized Robertson-Walker spacetime \(-I\times _fM^{n}\). Under suitable constraints on the sectional curvature of the Riemannian fiber \(M^n\), on the warping function f and on the future mean curvature (that is, the mean curvature function with respect to the future-pointing Gauss map of the spacelike hypersurface), we are able to prove that such a spacelike hypersurface must be a slice \(\{t\}\times M^{n}\) of the ambient spacetime. Nonexistence and Calabi–Bernstein type results concerning entire spacelike graphs constructed over the Riemannian fiber \(M^n\) are also obtained, as well as applications to the Einstein–de Sitter and steady state type spacetimes are given. Our approach is based on the so-called Omori–Yau’s generalized maximum principle and on certain integrability properties due to Yau.

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Acknowledgements

The authors would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments which improved the paper. The second author is partially supported by CNPq, Brazil, Grant 301970/2019-0. The third author is partially supported by CAPES, Brazil.

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Correspondence to Henrique F. de Lima.

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Araújo, J.G., de Lima, H.F. & Gomes, W.F. Uniqueness and nonexistence of complete spacelike hypersurfaces, Calabi–Bernstein type results and applications to Einstein–de Sitter and steady state type spacetimes. Rev Mat Complut 34, 653–673 (2021). https://doi.org/10.1007/s13163-020-00375-7

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  • DOI: https://doi.org/10.1007/s13163-020-00375-7

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