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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)].

##### MSC:
 53C40 Global submanifolds 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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##### References:
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