On the Gauss map of complete CMC hypersurfaces in the hyperbolic space

In this paper, as suitable applications of the so-called Omori–Yau generalized maximum principle, we obtain rigidity results concerning to complete hypersurfaces with constant mean curvature in the hyperbolic space, under appropriated restrictions on their Gauss image. Furthermore, by supposing a linear dependence between support functions naturally attached to such hypersurfaces, we establish a characterization theorem.