Preprint D36/2006
Complete Vertical Graphs with Constant Mean Curvature in Semi-Riemannian Warped Products

Henrique de Lima | Caminha, Antonio

**Keywords: **
Semi-Riemannian manifolds | Lorentz geometry | Hyperbolic space | Steady State space | Vertical graphs | Bernstein-type theorems

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.