Hypersurfaces of constant curvature in Hyperbolic space

Abstract

We show that for a very general and natural class of curvature functions, the problem of finding a complete strictly convex hypersurface satisfying f({\kappa}) = {\sigma} over (0,1) with a prescribed asymptotic boundary {\Gamma} at infinity has at least one solution which is a "vertical graph" over the interior (or the exterior) of {\Gamma}. There is uniqueness for a certain subclass of these curvature functions and as {\sigma} varies between 0 and 1, these hypersurfaces foliate the two components of the complement of the hyperbolic convex hull of {\Gamma}.