Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces |
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Authors: | A Gervasio Colares Henrique F de Lima |
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Institution: | (1) Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, 60455-760, Brazil;(2) Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58109-970, Brazil |
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Abstract: | In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ
n
immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space , and whose boundary is contained into a slice {t} × M
n
. By considering the hyperbolic caps of the Lorentz–Minkowski space , 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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Keywords: | Lorentzian products spaces Space-like hypersurfaces Higher order mean curvatures Height estimates |
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