Complete CMC hypersurfaces in the hyperbolic space with prescribed Gauss mapping

Abstract:

Our aim in this paper is to show that a complete hypersurface $x:M^{n}\to \mathbb {H}^{n+1}$ immersed with constant mean curvature into the hyperbolic space $\mathbb {H}^{n+1}$ is totally umbilical provided that its Gauss mapping $\nu$ has some suitable behavior. In this setting, our first result requires that the image $\nu (M)$ lies in a totally umbilical spacelike hypersurface of the de Sitter space $\mathbb S_1^{n+1}$, while in our second one we suppose that $M^n$ has scalar curvature bounded from below and that $\nu (M)$ is contained in the closure of a domain enclosed by a totally umbilical spacelike hypersurface of $\mathbb {S}_1^{n+1}$ determined by some vector $a$ of the Minkowski space $\mathbb {L}^{n+2}$, with the tangential component of $a$ with respect to $M^n$ having Lebesgue integrable norm.