Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space.(English)Zbl 1111.53049

Let $$\mathbb{L}^{n+2}$$ denote the $$(n+2)$$-dimensional Lorentz-Minkowski space$$(n\geq2)$$. The $$(n+1)$$-dimensional de Sitter space $$\mathbb{S}^{n+1}$$ is a hyperquadric of $$\mathbb{L}^{n+2}$$, which is given as $$\mathbb{S}^{n+1}=\{p\in\mathbb{L}^{n+2} :\langle p, p\rangle =1\}$$, where $$\langle,\rangle$$ denotes the Lorentzian metric. Let $$a\in\mathbb{L}^{n+2}$$ be a non-zero null vector in the past half of the null cone (with vertex in the origin), that is $$\langle a,a\rangle=0$$ and $$\langle a,e_{n+2}\rangle>0$$, where $$e_{n+2}=(0,\ldots,0,1)$$. Then the open region of the de Sitter space $$\mathbb{S}^{n+1}$$ given by $$\mathcal{H}^{n+1}=\{x\in\mathbb{S}^{n+1} :\langle x,a\rangle >0\}$$ is the so-called steady state space.
Compact space-like hypersurfaces in $$(n+1)$$-dimensional Minkowski space with spherical boundary and constant mean or scalar curvature have been classified by L. J. Alias and J. A. Pastor in [J. Geom. Phys. 28, No. 1–2, 85–93 (1998; Zbl 0945.53036)]. Their results are extended to the case of constant higher order mean curvature by L. J. Alias and J. M. Malacarne in [J. Geom. Phys. 41, No. 4, 359–375 (2002; Zbl 1013.53035)].
In the paper under review, the author gives Minkowski-type formulae for compact space-like immersed hypersurfaces with boundary having some constant higher order mean curvature in de Sitter space $$\mathbb{S}^{n+1}$$, then he studies them to establish a relation between the mean curvature and the geometry of the boundary, when it is a geodesic sphere contained in a horizontal hyperplane of the steady state space $$\mathcal{H}^{n+1}\subset \mathbb{S}^{n+1}$$.

MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Citations:

Zbl 0945.53036; Zbl 1013.53035
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References:

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