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Complete spacelike hypersurfaces with constant mean curvature in -× 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 ψ:Σ n -× n be a complete space-like hypersurface with constant mean curvature H. If the height function h of Σ n satisfies, for some constant 0<α<1, |h| 2 nα n-1H 2 , then Σ n is a slice.

Theorem 1.2. Let ψ:Σ n -× 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 Σ n satisfies, for some constant 0<α<1, |h| 2 α n-1|A| 2 , where |A| 2 denotes the squared norm of the shape operator A, then Σ n is a slice.

The first author has given some examples of complete and non-complete entire maximal graphs in -× 2 which are not slices [see, Differ. Geom. Appl. 26, No. 4, 456–462 (2008; Zbl 1147.53047)].

MSC:
53C40Global submanifolds (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics
References:
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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 · doi:euclid:bbms/1235574194
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