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 \(\mathcal{H}^{n+1}\subset \mathbb{S}^{n+1}\).


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
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