COMPLETE CMC HYPERSURFACES IN THE HYPERBOLIC SPACE WITH PRESCRIBED GAUSS MAPPING.

Our aim in this paper is to show that a complete hypersurface x : Mn → ℍn+1 immersed with constant mean curvature into the hyperbolic space ℍn+1 is totally umbilical provided that its Gauss mapping ν has some suitable behavior. In this setting, our first result requires that the image ν(M) lies in a totally umbilical spacelike hypersurface of the de Sitter space Sn+1 1 , while in our second one we suppose that Mn has scalar curvature bounded from below and that ν(M) is contained in the closure of a domain enclosed by a totally umbilical spacelike hypersurface of Sn+1 1 determined by some vector a of the Minkowski space Ln+2, with the tangential component of a with respect to Mn having Lebesgue integrable norm. [ABSTRACT FROM AUTHOR]