Complete spacelike hypersurfaces in a Robertson–Walker spacetime

Albujer, Alma L.; Camargo, Fabiola; Lima, Henrique F. de

doi:10.1017/s0305004111000351

Cited by 27 publications

(25 citation statements)

References 22 publications

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“…In [3], the first and third authors, jointly with A. L. Albujer, proved the next result when the ambient space is an RW spacetime obeying NCC. Here, we use Proposition 3·1 to extend [3, theorem 3·3] for some positive constants α and β, then n is a slice.…”

Section: Bernstein-type Theorems In Grw Spacetimesmentioning

confidence: 87%

Generalized maximum principles and the rigidity of complete spacelike hypersurfaces

Camargo¹,

Caminha²,

Lima³

et al. 2012

Math. Proc. Camb. Phil. Soc.

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Link to this article: http://journals.cambridge.org/abstract_S0305004112000369 How to cite this article: FERNANDA CAMARGO, ANTONIO CAMINHA, HENRIQUE DE LIMA and ULISSES PARENTE (2012). Generalized maximum principles and the rigidity of complete spacelike hypersurfaces. AbstractIn this paper, we apply several forms of generalized maximum principles to the study of the uniqueness of complete, non-compact spacelike hypersurfaces immersed in a class of Lorentzian warped products obeying a suitable convergence condition.

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Section: Bernstein-type Theorems In Grw Spacetimesmentioning

confidence: 87%

Generalized maximum principles and the rigidity of complete spacelike hypersurfaces

Camargo¹,

Caminha²,

Lima³

et al. 2012

Math. Proc. Camb. Phil. Soc.

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show abstract

“…However, it can be proved that if P n is complete and |Du| 2 P ≤ 2 (u) − c for certain positive constant c > 0, then Σ n (u) is complete. Although a particular case of this claim is proven in [4,Theorem 4.1] and the general proof is analogous, we will expose it here for the sake of completeness. Proposition 1.…”

Section: Entire Vertical Graphs In a Spatially Weighted Grw Spacetimementioning

confidence: 77%

“…The first results in this context were obtained by Alías, Romero and Sánchez in [9] for the case of compact spacelike hypersurfaces with constant mean curvature. From them on many authors have obtained unicity results not only for compact, but for complete (non-necessarily compact) spacelike hypersurfaces with constant mean curvature in a GRW satisfying certain energy conditions (see, for instance, [4,8,12,13,15,28,29,30]).…”

Section: Introductionmentioning

confidence: 99%

Rigidity of complete spacelike hypersurfaces in spatially weighted generalized Robertson–Walker spacetimes

Albujer

Lima

Oliveira

et al. 2017

Differential Geometry and its Applications

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Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f -mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient space. In this setting, we also obtain new Calabi-Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.2010 Mathematics Subject Classification. Primary 53C42, 53A07; Secondary 35P15.

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“…Remark 2. 2 We point out that spacelike hypersurfaces having some constant higher order mean curvature in a conformally stationary spacetime are critical points of certain Jacobi functionals for volume-preserving variations (see, for instance, [9], [16], [19] and [20] for more details).…”

Section: Preliminariesmentioning

confidence: 99%

Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime

Lima

Velásquez

2014

Mathematische Nachrichten

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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 .

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Complete spacelike hypersurfaces in a Robertson–Walker spacetime

Cited by 27 publications

References 22 publications

Generalized maximum principles and the rigidity of complete spacelike hypersurfaces

Generalized maximum principles and the rigidity of complete spacelike hypersurfaces

Rigidity of complete spacelike hypersurfaces in spatially weighted generalized Robertson–Walker spacetimes

Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime

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