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Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes. (English) Zbl 0908.53034
In Section 3, the relation between spacelike hypersurfaces and spacelike graphs in generalized Robertson-Walker spacetimes is given, and the conditions under which all these hypersurfaces are compact are studied. Section 4 contains two new integral formulas for compact spacelike hypersurfaces. In Section 5, a new technique to solve the problem: “when is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice?” is introduced. Finally, the natural conditions \(f'' \leq 0\) and \(\text{Ric} > (n - 1) (ff'' - f'{}^2)\) are imposed to solve a Bernstein-type problem.
Reviewer: P.Stavre (Craiova)

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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[1] Bartnik, R. (1984).Commun. Math. Phys. 94, 155. · Zbl 0548.53054 · doi:10.1007/BF01209300
[2] Brill, D., and Flaherty, F. (1976).Commun. Math. Phys. 50, 157. · Zbl 0337.53051 · doi:10.1007/BF01617993
[3] do Carmo, M. P. (1992).Riemannian Geometry (Birkhäuser, Boston). · Zbl 0752.53001
[4] Cheng, S. Y., and Yau, S. T. (1976).Ann. of Math. 104, 407. · Zbl 0352.53021 · doi:10.2307/1970963
[5] Choquet-Bruhat, Y. (1975)Comptes Rend. Acad. Sci. (Paris) Serie A 281, 577.
[6] Choquet-Bruhat, Y. (1976).Ann. Scuola Norm. Sup. Pisa 3, 361.
[7] Hawking, S. W., and Ellis, G. F. R. (1973).The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge). · Zbl 0265.53054
[8] Marsden, J. E., and Tipler, F. J. (1980)Phys. Rep. 66, 109. · doi:10.1016/0370-1573(80)90154-4
[9] Nishikawa, S. (1984).Nagoya Math. J. 95, 117.
[10] O’Neill, B. (1983).Semi-Riemannian geometry with applications to relativity (Academic Press, New York).
[11] Romero, A., and Sánchez, M. (1994).Geom. Dedicata 50, to appear.
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