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Spacelike hypersurfaces with constant mean curvature in the steady state space. (English) Zbl 1162.53043

The paper studies complete space-like hypersurfaces with constant mean curvature $H$ in the de Sitter space.

The first result is that if the surface lies between two concentric horospheres, then $H=1$. The second theorem assumes that the surface lies between a horosphere of center $a$ and the light-like hyperplane orthogonal to $a$ in the ambient Minkowski space (part of de Sitter space delimited by this hyperplane is the so-called steady state space). If, moreover the hypersurface has a future pointing mean curvature vector then a bound for $H$ is given.

If the dimension of de Sitter space is 3, the conditions of both theorems imply that the surface is actually totally umbilical.

The first theorem is then extended to hypersurfaces embedded in particular Lorentzian manifolds. They are a generalization of those obtained by quotient of steady state space by isometries acting cocompactly on horospheres (“de Sitter cusps”).

There exist many results about space-like hypersurfaces with constant mean curvature in de Sitter space, see for example [S. Montiel, Constant mean curvature space-like hypersurfaces in de Sitter spaces, Suh, Young Jin (ed.) et al., Proceedings of the 9th international workshop on differential geometry, Taegu, Korea, November 12–13, 2004. Taegu: Kyungpook National University. 17-30 (2004; Zbl 1075.53071)]

##### MSC:
 53C42 Immersions (differential geometry) 53C50 Lorentz manifolds, manifolds with indefinite metrics