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Bernstein-type theorems in semi-Riemannian warped products. (English) Zbl 1223.53045
If \(M^n\) is an \(n\)-dimensional Riemannian manifold, the Lorentzian warped product \(-\mathbb{R}\times_{e^t}M^n\) is called a steady-state type space while the Riemannian warped product \(\mathbb{R}\times_{e^t}M^n\) is known as a hyperbolic type space. For \(M^n=\mathbb{R}^n\), the classical steady state and the hyperbolic space are obtained, respectively. Assume that \(M^n\) is a complete and connected Riemannian manifold. For a given \(t_0\in\mathbb{R}\), a slice \(\Sigma_{t_0}=\{t_0\}\times M^n\) is a complete, connected (space-like) hypersurface with mean curvature \(H=1\) (after choosing a suitable orientation).
In this paper, the authors study complete hypersurfaces, in both steady-state type spaces and in hyperbolic type spaces, with bounded mean curvature and with the gradient of the height function satisfying suitable conditions. By using a technique of S. T. Yau, they obtain Bernstein-type results for this class of immersions.
For instance, the authors prove that, if \(\Sigma\) is a complete connected space-like hypersurface of a steady-state type space-time, contained in a slab bounded by two slices, with bounded mean curvature \(H\geq 1\), and such that the gradient of its height function has integrable norm, then \(\Sigma\) must be a slice. Analogously, in the Riemannian setting the authors prove that, if \(\Sigma\) is a complete, connected hypersurface of a hyperbolic type space-time contained in a slab, with bounded mean curvature \(0<H\leq1\), and satisfying the condition that the gradient of the height function has integrable norm, then \(\Sigma\) has to be a slice.
The last part of the paper is devoted to extend, under similar assumptions, the previous results to the case in which the conditions on the mean curvature are substituted by the following conditions on the \(r\)-th mean curvatures of the hypersurfaces: \(0<H_r\leq H_{r+1}\) in the Lorentzian case; and \(H_r\geq H_{r+1}>0\) in the Riemannian case.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53Z05 Applications of differential geometry to physics
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