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

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.

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.

PARABOLICITY OF SPACELIKE HYPERSURFACES IN GENERALIZED ROBERTSON–WALKER SPACETIMES: APPLICATIONS TO UNIQUENESS RESULTS

2013 ◽  
Vol 10 (08) ◽  
pp. 1360014 ◽  
Author(s):  
A. ROMERO ◽  
R. M. RUBIO ◽  
J. J. SALAMANCA
Keyword(s):  
Arbitrary Dimension ◽  
Spacelike Hypersurface ◽  
Universal Covering ◽  
New Technique ◽  
Warping Function ◽  
Complete Spacelike Hypersurfaces ◽  
Uniqueness Results ◽  
Complete Spacelike Hypersurface ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Angle

We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.

ON THE EXISTENCE OF -MAXIMAL SPACELIKE HYPERSURFACES IN CERTAIN WEIGHTED MANIFOLDS

2017 ◽  
Vol 96 (2) ◽  
pp. 317-325
Author(s):  
ARLANDSON M. S. OLIVEIRA ◽  
HENRIQUE F. DE LIMA
Keyword(s):  
Product Space ◽  
Mean Value ◽  
Sobolev Embedding ◽  
Lorentzian Manifolds ◽  
Heat Operator ◽  
Nonexistence Result ◽  
Spacelike Hypersurfaces ◽  
Mean Value Inequality ◽  
Weighted Manifolds ◽  
Lorentzian Product Space

We apply a mean-value inequality for positive subsolutions of the $f$-heat operator, obtained from a Sobolev embedding, to prove a nonexistence result concerning complete noncompact $f$-maximal spacelike hypersurfaces in a class of weighted Lorentzian manifolds. Furthermore, we establish a new Calabi–Bernstein result for complete noncompact maximal spacelike hypersurfaces in a Lorentzian product space.

scholarly journals Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker Space-Times

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Ximin Liu ◽  
Ning Zhang
Keyword(s):  
Maximum Principle ◽  
Space Time ◽  
Ambient Space ◽  
Weak Maximum Principle ◽  
Uniqueness Results ◽  
Spacelike Hypersurfaces ◽  
Generalized Maximum Principle ◽  
Special Case

Applying generalized maximum principle and weak maximum principle, we obtain several uniqueness results for spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptions. Furthermore, we also study the special case when the ambient space is static and provide some results by using Bochner’s formula.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

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