×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Akutagawa, K., On spacelike hypersurfaces with constant Mean curvature in the de Sitter space, Math. Z., 196, 13-19, (1987) · Zbl 0611.53047
[2] Alías, L.J.; Brasil, A.; Colares, A.G., Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications, Proc. edinb. math. soc., 46, 465-488, (2003) · Zbl 1053.53038
[3] Alías, L.J.; Malacarne, J.M., Spacelike hypersurfaces with constant higher order Mean curvature in Minkowski space – time, J. geom. phys., 41, 359-375, (2002) · Zbl 1013.53035
[4] Alías, L.J.; Pastor, J.A., Constant Mean curvature spacelike hypersurfaces with spherical boundary in the lorentz – minkowski space, J. geom. phys., 28, 85-93, (1998) · Zbl 0945.53036
[5] Goddard, A.J., Some remarks on the existence of spacelike hypersurfaces of constant Mean curvature, Math. proc. Cambridge philos. soc., 82, 489-495, (1977) · Zbl 0386.53042
[6] Hawking, S.W.; Ellis, G.F.R., The large scale structure of spacetime, (1973), Cambridge Univ. Press Cambridge · Zbl 0265.53054
[7] López, R., Area monotonicity for spacelike surfaces with constant Mean curvature, J. geom. phys., 52, 353-363, (2004) · Zbl 1074.53051
[8] Montiel, S., An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant Mean curvature, Indiana univ. math. J., 37, 909-917, (1988) · Zbl 0677.53067
[9] Montiel, S., Complete non-compact spacelike hypersurfaces of constant Mean curvature in de Sitter spaces, J. math. soc. Japan, 55, 915-938, (2003) · Zbl 1049.53044
[10] Montiel, S., Uniqueness of spacelike hypersurfaces of constant Mean curvature in foliated spacetimes, Math. ann., 314, 529-553, (1999) · Zbl 0965.53043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.