scholarly journals ON THE STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER ORDER MEAN CURVATURE IN A DE SITTER SPACE

2014 ◽  
Vol 51 (5) ◽  
pp. 1539-1549
Author(s):  
Shicheng Zhang
Keyword(s):  
Mean Curvature ◽  
De Sitter Space ◽  
Higher Order ◽  
De Sitter ◽  
Spacelike Hypersurfaces ◽  
Higher Order Mean Curvature ◽  
The Stability

scholarly journals Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Kairen Cai ◽  
Huiqun Xu
Keyword(s):  
Positive Constant ◽  
Mean Curvature ◽  
De Sitter Space ◽  
Higher Order ◽  
De Sitter ◽  
Spacelike Hypersurfaces ◽  
Higher Order Mean Curvature ◽  
Set Up ◽  
Minkowski Formula

The authors apply the generalized Minkowski formula to set up a spherical theorem. It is shown that a compact connected hypersurface with positive constant higher-order mean curvatureHrfor some fixedr,1≤r≤n, immersed in the de Sitter spaceS1n+1must be a sphere.

Some remarks on the existence of spacelike hypersurfaces of constant mean curvature

1977 ◽  
Vol 82 (3) ◽  
pp. 489-495 ◽  
Author(s):  
A. J. Goddard
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
General Class ◽  
Constant Mean Curvature ◽  
Spacelike Hypersurface ◽  
De Sitter Space ◽  
De Sitter ◽  
Minimal Graphs ◽  
Complete Spacelike Hypersurface ◽  
Spacelike Hypersurfaces

AbstractBernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

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 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.

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.

Complete spacelike hypersurfaces in a de Sitter space

2006 ◽  
Vol 73 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Shu Shichang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
The Mean

In this paper, we characterise the n-dimensional (n ≥ 3) complete spacelike hypersurfaces Mn in a de Sitter space with constant scalar curvature and with two distinct principal curvatures. We show that if the multiplicities of such principal curvatures are greater than 1, then Mn is isometric to Hk (sinh r) × Sn−k (cosh r), 1 < k < n − 1. In particular, when Mn is the complete spacelike hypersurfaces in with the scalar curvature and the mean curvature being linearly related, we also obtain a characteristic Theorem of such hypersurfaces.

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