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Complete spacelike hypersurfaces immersed in pp-wave spacetimes

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Abstract

We study some aspects of the geometry of complete spacelike hypersurfaces immersed into a pp-wave spacetime, namely, into a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying suitable versions of the classical Hopf and Stokes theorems and a criterion of parabolicity for complete Riemannian manifolds, we obtain sufficient conditions which guarantee that a complete spacelike hypersurface is either maximal, 1-maximal or totally geodesic. As a consequence of these results, we also establish some results of nonexistence concerning such spacelike hypersurfaces. Finally, considering constant mean curvature closed spacelike hypersurfaces immersed in a pp-wave spacetime, we study a notion of stability via the first nonzero eigenvalue of the Laplacian.

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Acknowledgements

The authors would like to thank the referees for their comments and suggestions, which enabled them to reach at a considerable improvement of the original version of this paper. The first author is partially supported by CNPq, Brazil, Grant 311224/2018-0. The second author is partially supported by CNPq, Brazil, Grant 301970/2019-0.

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  1. Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58429-970, Brazil

    Marco A. L. Velásquez & Henrique F. de Lima

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  1. Marco A. L. Velásquez
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  2. Henrique F. de Lima
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Correspondence to Marco A. L. Velásquez.

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Velásquez, M.A.L., Lima, H.F.d. Complete spacelike hypersurfaces immersed in pp-wave spacetimes. Gen Relativ Gravit 52, 41 (2020). https://doi.org/10.1007/s10714-020-02692-0

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