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Complete spacelike hypersurfaces with constant mean curvature in \(-\mathbb R\times \mathbb H^n\). (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)].

53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI
[1] Albujer, A.L., New examples of entire maximal graphs in \(\mathbb{H}^2 \times \mathbb{R}_1\), Differential geom. appl., 26, 456-462, (2008) · Zbl 1147.53047
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[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
[8] Omori, H., Isometric immersions of Riemannian manifolds, J. math. soc. Japan, 19, 205-214, (1967) · Zbl 0154.21501
[9] O’Neill, B., Semi-Riemannian geometry with applications to relativity, (1983), Academic Press London · Zbl 0531.53051
[10] Yau, S.T., Harmonic functions on complete Riemannian manifolds, Comm. pure appl. math., 28, 201-228, (1975) · Zbl 0291.31002
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