Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime |
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| Authors: | Henrique Fernandes de Lima Marco Antonio Lázaro Velásquez |
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| Affiliation: | 1. Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba, Brazil;2. +55 83 2101 1510+55 83 2101 1030 |
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| Abstract: | We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so‐called generalized Robertson‐Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett 29 . |
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| Keywords: | Timelike conformal vector field conformally stationary spacetimes Einstein spacetimes generalized Robertson‐Walker spacetimes timelike convergence condition complete spacelike hypersurfaces higher order mean curvatures Primary: 53C42 Secondary: 53B30 53C50 53Z05 83C99 |
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