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Complete spacelike hypersurfaces with constant mean curvature in $-\Bbb R\times \Bbb H^n$. (English) Zbl 1193.53124
By applying the Omori-Yau generalized maximum principle for complete Riemannian manifolds, the authors prove the following Bernstein-type results: Theorem 1.1. Let $\psi:\Sigma^n\to-\mathbb{R}\times\mathbb{H}^n$ be a complete space-like hypersurface with constant mean curvature $H$. If the height function $h$ of $\Sigma^n$ satisfies, for some constant $0<\alpha<1$, $\displaystyle|\nabla h|^2\leq\frac{n\alpha}{n-1}H^2$, then $\Sigma^n$ is a slice. Theorem 1.2. Let $\psi:\Sigma^n\to-\mathbb{R}\times\mathbb{H}^n$ be a complete space-like hypersurface with constant mean curvature $H$, and the 2-mean curvature $H_2$ bounded from below. If the height function $h$ of $\Sigma^n$ satisfies, for some constant $0<\alpha<1$, $\displaystyle |\nabla h|^2\leq\frac{\alpha}{n-1}|A|^2$, where $|A|^2$ denotes the squared norm of the shape operator $A$, then $\Sigma^n$ is a slice. The first author has given some examples of complete and non-complete entire maximal graphs in $-\mathbb{R}\times\mathbb{H}^2$ which are not slices [see, Differ. Geom. Appl. 26, No. 4, 456--462 (2008; Zbl 1147.53047)].

53C40Global submanifolds (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI
[1] Albujer, A. L.: New examples of entire maximal graphs in H2$\times R$1, Differential geom. Appl. 26, 456-462 (2008) · Zbl 1147.53047 · doi:10.1016/j.difgeo.2007.11.035
[2] Albujer, A. L.; Alías, L. J.: Calabi -- Bernstein results for maximal surfaces in Lorentzian product spaces, J. geom. Phys. 59, 620-631 (2009) · Zbl 1173.53025 · doi:10.1016/j.geomphys.2009.01.008
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[4] Alías, L. J.; Jr., A. Brasil; 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 · doi:10.1017/S0013091502000500 · http://journals.cambridge.org/bin/bladerunner?REQUNIQ=1087480707&REQSESS=3723236&118000REQEVENT=&REQINT1=163427&REQAUTH=0
[5] Alías, L. J.; Colares, A. G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson -- Walker spacetimes, Math. proc. Cambridge philos. Soc. 143, 703-729 (2007) · Zbl 1131.53035 · doi:10.1017/S0305004107000576
[6] Alías, L. J.; Romero, A.; Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson -- Walker spacetimes, Gen. relativity gravitation 27, 71-84 (1995) · Zbl 0908.53034 · doi:10.1007/BF02105675
[7] Caminha, A.; De Lima, H. F.: Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. belg. Math. soc. 16, 91-105 (2009) · Zbl 1160.53362 · euclid:bbms/1235574194
[8] Omori, H.: Isometric immersions of Riemannian manifolds, J. math. Soc. Japan 19, 205-214 (1967) · Zbl 0154.21501 · doi:10.2969/jmsj/01920205
[9] O’neill, B.: Semi-Riemannian geometry with applications to relativity, (1983)
[10] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Comm. pure appl. Math. 28, 201-228 (1975) · Zbl 0291.31002 · doi:10.1002/cpa.3160280203