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

2014 ◽  
Vol 287 (11-12) ◽  
pp. 1223-1240 ◽  
Author(s):  
Henrique Fernandes de Lima ◽  
Marco Antonio Lázaro Velásquez
Keyword(s):  
Higher Order ◽  
Higher Order Mean Curvatures ◽  
Complete Spacelike Hypersurfaces ◽  
Stationary Spacetime ◽  
Spacelike Hypersurfaces

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.

Uniqueness of Spacelike Hypersurfaces in a GRW Spacetime via Higher Order Mean Curvatures

2016 ◽  
Vol 48 (1) ◽  
pp. 45-61
Author(s):  
Cícero P. Aquino ◽  
Jogli G. Araújo ◽  
Márcio Batista ◽  
Henrique F. de Lima
Keyword(s):  
Higher Order ◽  
Higher Order Mean Curvatures ◽  
Spacelike Hypersurfaces

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 Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

2007 ◽  
Vol 143 (3) ◽  
pp. 703-729 ◽  
Author(s):  
LUIS J. ALÍAS ◽  
A. GERVASIO COLARES
Keyword(s):  
Mean Curvature ◽  
Differential Operators ◽  
Spacelike Hypersurface ◽  
Convergence Condition ◽  
Higher Order ◽  
Spacelike Hypersurfaces ◽  
Newton Transformations ◽  
Higher Order Mean Curvature ◽  
Associated Differential Operators

AbstractIn this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

scholarly journals UNIQUENESS OF COMPLETE HYPERSURFACES WITH BOUNDED HIGHER ORDER MEAN CURVATURES IN SEMI-RIEMANNIAN WARPED PRODUCTS

2011 ◽  
Vol 54 (1) ◽  
pp. 201-212 ◽  
Author(s):  
C. P. AQUINO ◽  
H. F. DE LIMA
Keyword(s):  
Steady State ◽  
Maximum Principle ◽  
Higher Order ◽  
Hyperbolic Type ◽  
Warped Products ◽  
Higher Order Mean Curvatures ◽  
Uniqueness Results ◽  
State Type ◽  
Type Spaces ◽  
Complete Hypersurfaces

AbstractIn this paper, we deal with complete hypersurfaces immersed with bounded higher order mean curvatures in steady state-type spacetimes and in hyperbolic-type spaces. By applying a generalised maximum principle for the Yau's square operator [11], we obtain uniqueness results in each of these ambient spaces.

scholarly journals Erratum to “Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes”. Math. Proc. Camb. Phil. Soc. (2012), 152, 365–383.

2013 ◽  
Vol 155 (2) ◽  
pp. 375-377
Author(s):  
LUIS J. ALÍAS ◽  
DEBORA IMPERA ◽  
MARCO RIGOLI
Keyword(s):  
Mean Curvature ◽  
Higher Order ◽  
Spacelike Hypersurfaces ◽  
Higher Order Mean Curvature ◽  
Formula Group ◽  
Image Position ◽  
Correct Expression

The proof of Corollary 4⋅3 in our paper [1] is not correct because there is a mistake in the expression given for ∥X* ∧ Y*∥2 on page 374. In fact, the correct expression for this term is \begin{eqnarray*} \norm{X^*\wedge Y^*}^2 & = & \norm{X^*}^2\norm{Y^*}^2-\pair{X^*,Y^*}^2\\ {} & = & 1+\pair{X,T}^2+\pair{Y,T}^2\geq 1, \end{eqnarray*} and then the inequality (4⋅9) is no longer true. Observe that all the previous reasoning before the wrong expression for ∥X* ∧ Y*∥2 is correct.

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