On Bernstein-Type Theorems in Semi-Riemannian Warped Products

Abstract

Complete spacelike hypersurfaces immersed in semi-Riemannian warped products are investigated. By using a technique according to Yau (1976) and a reasonable restriction on the mean curvature of the hypersurfaces, we obtain some new Bernstein-type theorems which extend some known results proved by .

Space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products and generalized Calabi–Bernstein-type problems

“…is weaker than the natural comparison hypothesis H 2 f 2 (τ )/f 2 (τ ) commonly used in several works (see [3,10,19,20,30,33,35]). The latter inequality can be interpreted geometrically as follows: the mean curvature of the space-like hypersurface, at any point, is bounded, in absolute value, by the mean curvature of the space-like slice at that point.…”

“…A more general approach to this problem on Riemannian surfaces with finite total curvature is given in [33]. In [35], Wang and Liu studied the parametric version of a similar problem for arbitrary dimension, making use of a technique provided by Yau [36]. A non-parametric approach for this type of problem when the mean curvature is constant is given by Aquino and de Lima in [10].…”

Space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products and generalized Calabi–Bernstein-type problems

“…f 2 (τ ) commonly used in several works (see [30], [33], [35], [10], [3], [19], [26]). This last inequality can be geometrically interpreted as follows: the mean curvature of the spacelike hypersurface, at any point, is bounded, in absolute value, by the mean curvature of the spacelike slice at that point.…”

“…A more general approach to this problem on Riemannian surfaces with finite total curvature is given in [33]. In [35], Wang and Liu studied the parametric version of a similar problem for arbitrary dimension making use of a technique provided by Yau [36]. A non-parametric approach for this type of problem is given in [10] by Aquino and de Lima, when the mean curvature is constant.…”

Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-Riemannian warped product

“…It is worth to point out that the Laplacian of integral of the warping function was studied by the present authors in [20,21] to obtain some uniqueness results. Throughout this paper, we denote by L(Σ n ) the space of Lebesgue integrable functions on spacelike hypersurface Σ n .…”