On spacelike hypersurfaces with constant mean curvature in the de Sitter space

1987 ◽  
Vol 196 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
De Sitter Space ◽  
De Sitter ◽  
Spacelike Hypersurfaces

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 Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

2009 ◽  
Vol 2009 ◽  
pp. 1-12
Author(s):  
Henrique F. de Lima ◽  
Joseilson R. de Lima
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Spacelike Hypersurface ◽  
De Sitter Space ◽  
De Sitter ◽  
Height Estimate ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Domains

We obtain a height estimate concerning to a compact spacelike hypersurfaceΣnimmersed with constant mean curvatureHin the anti-de Sitter spaceℍ1n+1, when its boundary∂Σis contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic spaceℍn. Our estimate depends only on the value ofHand on the geometry of∂Σ.As applications of our estimate, we obtain a characterization of hyperbolic domains ofℍ1n+1and nonexistence results in connection with such types of hypersurfaces.

scholarly journals Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.

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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