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Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

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Abstract

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σn immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space \({-\mathbb{R} \times M^n}\) , and whose boundary is contained into a slice {t} × M n. By considering the hyperbolic caps of the Lorentz–Minkowski space \({\mathbb{L}^{n+1}}\) , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, 60455-760, Brazil

    A. Gervasio Colares

  2. Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58109-970, Brazil

    Henrique F. de Lima

Authors
  1. A. Gervasio Colares
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  2. Henrique F. de Lima
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Correspondence to Henrique F. de Lima.

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Colares, A.G., de Lima, H.F. Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces. Gen Relativ Gravit 40, 2131–2147 (2008). https://doi.org/10.1007/s10714-008-0621-9

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  • DOI: https://doi.org/10.1007/s10714-008-0621-9

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