Uniqueness and nonexistence of complete spacelike hypersurfaces, Calabi–Bernstein type results and applications to Einstein–de Sitter and steady state type spacetimes

2020 ◽  
Author(s):  
Jogli G. Araújo ◽  
Henrique F. de Lima ◽  
Wallace F. Gomes
Keyword(s):  
Steady State ◽  
De Sitter ◽  
Bernstein Type ◽  
Complete Spacelike Hypersurfaces ◽  
State Type ◽  
Spacelike Hypersurfaces

New characterizations of spacelike hyperplanes in the steady state space

2020 ◽  
Vol 126 (1) ◽  
pp. 61-72
Author(s):  
Cícero P. Aquino ◽  
Halyson I. Baltazar ◽  
Henrique F. De Lima
Keyword(s):  
Steady State ◽  
State Space ◽  
Riemannian Manifolds ◽  
De Sitter ◽  
Main Hypothesis ◽  
Spacelike Hypersurfaces ◽  
Open Region ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle ◽  
Hyperbolic Cylinders

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb {S}^{n+1}_{1}$ which is known as the steady state space $\mathcal {H}^{n+1}$. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of $\mathcal {H}^{n+1}$. Furthermore, through the analysis of the hyperbolic cylinders of $\mathcal {H}^{n+1}$, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.

On the geometry of complete spacelike hypersurfaces in the anti-de Sitter space

2014 ◽  
Vol 174 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Cícero P. Aquino ◽  
Henrique F. de Lima ◽  
Marco Antonio L. Velásquez
Keyword(s):  
De Sitter Space ◽  
De Sitter ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces

Spacelike hypersurfaces with constant rth mean curvature in steady state type spacetimes

2014 ◽  
Vol 106 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Cícero P. Aquino ◽  
Henrique F. de Lima ◽  
Fábio R. dos Santos ◽  
Marco Antonio L. Velásquez
Keyword(s):  
Steady State ◽  
Mean Curvature ◽  
State Type ◽  
Spacelike Hypersurfaces

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.

scholarly journals Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-Riemannian warped product

2015 ◽  
Vol 23 (2) ◽  
pp. 259-277
Author(s):  
Yaning Wang ◽  
Ximin Liu
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Curvature Tensor ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Bernstein Type ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
Comparison Inequality

Abstract In this paper, by supposing a natural comparison inequality on the positive r-th mean curvatures of the hypersurface, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces immersed in a semi-Riemannian warped product of constant sectional curvature. Generalizing the above results, under a restriction on the sectional curvature or the Ricci curvature tensor of the fiber of a warped product, we also prove some new rigidity theorems in semi-Riemannian warped products. Our main results extend some recent Bernstein-type theorems proved in [12, 13, 14].

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