Complete Vertical Graphs with Constant Mean Curvature in Semi-Riemannian Warped Products

Abstract

In this paper we study complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. We first derive suitable formulas for the Laplacians of the height function and of a support-like function naturally attached to the graph; then, under appropriate restrictions on the values of the mean curvature and the growth of the height function, we obtain necessary conditions for the existence of such a graph. In the two-dimensional case we apply this analytical framework to state and prove Bernstein-type results in each of these ambient spaces. This paper deals with complete non-compact constant mean curvature graphs over a horosphere of the Hyperbolic space, as well as over horizontal hyperplanes (slices) in the Steady State space. In connection with our work, L. Al´oas and M. Dajczer (cf. (2)) studied properly immersed complete surfaces of the 3−dimensional Hyperbolic space contained between two horospheres, obtaining a Bernstein-type result for the case of constant mean curvature between −1 and 1. In de Sitter space, K. Akutagawa (cf. (5)) proved that complete spacelike hypersurfaces having constant mean curvature in a specific interval of the real line are totally umbilical. Also for de Sitter space, among other interesting results S. Montiel (cf. (14)) proves that, under an appropriate restriction on their Hyperbolic Gauss map, complete spacelike Hypersurfaces of constant mean curvature greater than or equal to 1 must actually have mean curvature 1. For the Lorentz case, our motivation to restrict attention to the Steady State space comes from the fact that there exists a natural duality between the Gauss maps of Riemannian hypersurfaces of this space and those of the Hyperbolic space, provided we model these as hyperquadrics of the Lorentz-Minkowski space (cf. section 5). Besides, in physical context the Steady State space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesics normal to horizontal hyperplanes; these, in turn, serve as the initial data for the Cauchy problem associated to those equations (cf. (8), chapter 5).