Spacelike Hypersurfaces in Conformally Stationary Spacetimes

2020 ◽  
pp. 161-174
Author(s):  
Luis J. Alías ◽  
Antonio Caminha ◽  
F. Yure S. do Nascimento
Keyword(s):  
Spacelike Hypersurfaces ◽  
Stationary Spacetimes

scholarly journals INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

2003 ◽  
Vol 46 (2) ◽  
pp. 465-488 ◽  
Author(s):  
Luis J. Alías ◽  
Aldir Brasil ◽  
A. Gervasio Colares
Keyword(s):  
Primary 53C42 ◽  
Subject Classification ◽  
Secondary 53B30 ◽  
Lorentzian Manifolds ◽  
Conformal Field ◽  
Uniqueness Results ◽  
Spacelike Hypersurfaces ◽  
Stationary Spacetimes ◽  
Integral Formulae

AbstractIn this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99

scholarly journals r-stable spacelike hypersurfaces in conformally stationary spacetimes

2011 ◽  
Vol 34 (3) ◽  
pp. 339-351 ◽  
Author(s):  
Fernanda Camargo ◽  
Antonio Caminha ◽  
Henrique Fernandes de Lima ◽  
Marco Antonio Velásquez
Keyword(s):  
Spacelike Hypersurfaces ◽  
Stationary Spacetimes

STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER-ORDER MEAN CURVATURES LINEARLY RELATED IN CONFORMALLY STATIONARY SPACETIMES

2013 ◽  
Vol 24 (14) ◽  
pp. 1350109
Author(s):  
HENRIQUE FERNANDES DE LIMA ◽  
ANTONIO FERNANDO DE SOUSA ◽  
MARCO ANTONIO LÁZARO VELÁSQUEZ
Keyword(s):  
Sectional Curvature ◽  
Sufficient Conditions ◽  
De Sitter Space ◽  
Higher Order ◽  
First Eigenvalue ◽  
De Sitter ◽  
Higher Order Mean Curvatures ◽  
The First Eigenvalue ◽  
Spacelike Hypersurfaces ◽  
Stationary Spacetimes

In this paper, we establish the notion of (r, s)-stability concerning spacelike hypersurfaces with higher-order mean curvatures linearly related in conformally stationary spacetimes of constant sectional curvature. In this setting, we characterize (r, s)-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of an operator naturally attached to the higher-order mean curvatures. Moreover, we obtain sufficient conditions which assure the (r, s)-stability of complete spacelike hypersurfaces immersed in the de Sitter space.

scholarly journals Fermat Metrics

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1422
Author(s):  
Antonio Masiello
Keyword(s):  
Variational Methods ◽  
Morse Theory ◽  
Sagnac Effect ◽  
Variational Theory ◽  
Global Hyperbolicity ◽  
Fermat Principle ◽  
Multiple Image ◽  
Light Rays ◽  
Stationary Spacetimes

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

scholarly journals Separable geodesic action slicing in stationary spacetimes

2011 ◽  
Vol 44 (3) ◽  
pp. 603-621 ◽  
Author(s):  
Donato Bini ◽  
Andrea Geralico ◽  
Robert T. Jantzen
Keyword(s):  
Stationary Spacetimes

scholarly journals Weingarten spacelike hypersurfaces in a de Sitter space

2012 ◽  
Vol 20 (1) ◽  
pp. 387-406
Author(s):  
Junfeng Chen ◽  
Shichang Shu
Keyword(s):  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Riemannian Product ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Cylinder

Abstract We study some Weingarten spacelike hypersurfaces in a de Sitter space S1n+1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product Hk(1−coth2 ϱ)× Sn−k(1 − tanh2 ϱ), 1 < k < n − 1 in S1n+1(1), the hyperbolic cylinder H1(1 − coth2 ϱ) × Sn-1(1 − tanh2 ϱ) or spherical cylinder Hn−1(1 − coth2 ϱ) × S1(1 − tanh2 ϱ) in S1n+1 (1)

Complete spacelike hypersurfaces in a Robertson–Walker spacetime

2011 ◽  
Vol 151 (2) ◽  
pp. 271-282 ◽  
Author(s):  
ALMA L. ALBUJER ◽  
FERNANDA E. C. CAMARGO ◽  
HENRIQUE F. DE LIMA
Keyword(s):  
Maximum Principle ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Rigidity Results ◽  
Complete Spacelike Hypersurfaces ◽  
Uniqueness Results ◽  
Spacelike Hypersurfaces ◽  
Generalized Maximum Principle ◽  
Non Parametric

AbstractIn this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

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