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In this paper we develop a global correspondence between immersedhorospherically convex hypersurfaces in hyperbolic space and complete conformalmetrics on domains in the sphere. We establish results on when the hyperbolicGauss map is injective and when an immersed horospherically convex hypersurfacecan be unfolded along the normal flow into an embedded one. These results allowus to establish general Alexandrov reflection principles of elliptic problemsof both Weingarten hypersurfaces and complete conformal metrics and relationsbetween them. Consequently, we are able to obtain, for instance, a strongBernstein theorem for a complete, immersed, horospherically convex hypersurfaceof constant mean curvature in hyperbolic space.

Hypersurfaces in hyperbolic space with support function