scholarly journals Compact space-like hypersurfaces in de Sitter space

2005 ◽  
Vol 2005 (13) ◽  
pp. 2053-2069 ◽  
Author(s):  
Jinchi Lv
Keyword(s):  
Compact Space ◽  
Mean Curvature ◽  
De Sitter Space ◽  
Higher Order ◽  
De Sitter ◽  
Totally Umbilical ◽  
Higher Order Mean Curvatures ◽  
Integral Formulas

We present some integral formulas for compact space-like hypersurfaces in de Sitter space and some equivalent characterizations for totally umbilical compact space-like hypersurfaces in de Sitter space in terms of mean curvature and higher-order mean curvatures.

Strong (r, s)-stability in the de Sitter space

2015 ◽  
Vol 145 (1) ◽  
pp. 91-104
Author(s):  
Henrique F. de Lima ◽  
Antonio F. de Sousa ◽  
Marco Antonio L. Velásquez
Keyword(s):  
De Sitter Space ◽  
Higher Order ◽  
Closed Space ◽  
De Sitter ◽  
Round Sphere ◽  
Totally Umbilical ◽  
Higher Order Mean Curvatures ◽  
Geometric Conditions ◽  
Stable Space

In this paper, we establish the notion of strong (r, s)-stability concerning closed space-like hypersurfaces immersed with higher-order mean curvatures linearly related in the de Sitter space . In this setting, we prove that totally umbilical round spheres of are strongly (r, s)-stable. Afterwards, we obtain sufficient geometric conditions that guarantee that a closed strongly (r, s)-stable space-like hypersurface in must be a totally umbilical round sphere.

scholarly journals On the umbilicity of linear Weingarten spacelike submanifolds immersed in the de Sitter space

2020 ◽  
pp. 1-12
Author(s):  
Weiller F. C. Barboza ◽  
Eudes L. de Lima ◽  
Henrique F. de Lima ◽  
Marco Antonio L. Velásquez
Keyword(s):  
Mean Curvature ◽  
Curvature Vector ◽  
De Sitter Space ◽  
Index Formula ◽  
Curvature Function ◽  
De Sitter ◽  
Totally Umbilical ◽  
Spacelike Submanifold ◽  
Totally Umbilical Submanifolds ◽  
Spacelike Submanifolds

We investigate the umbilicity of [Formula: see text]-dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field in the de Sitter space [Formula: see text] of index [Formula: see text]. We recall that a spacelike submanifold is said to be linear Weingarten when its mean curvature function [Formula: see text] and its normalized scalar curvature [Formula: see text] satisfy a linear relation of the type [Formula: see text], for some constants [Formula: see text]. Under suitable constraints on the values of [Formula: see text] and [Formula: see text], we apply a generalized maximum principle for a modified Cheng–Yau operator [Formula: see text] in order to show that such a spacelike submanifold must be either totally umbilical or isometric to a product [Formula: see text], where the factors [Formula: see text] are totally umbilical submanifolds of [Formula: see text] which are mutually perpendicular along their intersections. Moreover, we also study the case in which these spacelike submanifolds are [Formula: see text]-parabolic.

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.

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.

On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

2017 ◽  
Vol 148 (1) ◽  
pp. 199-210 ◽  
Author(s):  
Oscar Palmas ◽  
Francisco J. Palomo ◽  
Alfonso Romero
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Upper Bound ◽  
Mean Curvature ◽  
Light Cone ◽  
Minkowski Spacetime ◽  
Totally Umbilical ◽  
Total Mean Curvature ◽  
Lower Estimation ◽  
Compact Submanifold

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

Sign in / Sign up

Export Citation Format

Share Document