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Complete spacelike hypersurfaces with constant mean curvature in $-ℝ×{ℍ}^{n}$. (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 $\psi :{{\Sigma }}^{n}\to -ℝ×{ℍ}^{n}$ be a complete space-like hypersurface with constant mean curvature $H$. If the height function $h$ of ${{\Sigma }}^{n}$ satisfies, for some constant $0<\alpha <1$, ${|\nabla h|}^{2}\le \frac{n\alpha }{n-1}{H}^{2}$, then ${{\Sigma }}^{n}$ is a slice.

Theorem 1.2. Let $\psi :{{\Sigma }}^{n}\to -ℝ×{ℍ}^{n}$ 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 $0<\alpha <1$, ${|\nabla h|}^{2}\le \frac{\alpha }{n-1}{|A|}^{2}$, 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 $-ℝ×{ℍ}^{2}$ which are not slices [see, Differ. Geom. Appl. 26, No. 4, 456–462 (2008; Zbl 1147.53047)].

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