Interior curvature estimates and the asymptotic plateau problem in hyperbolic space

Abstract

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic boundary $\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if $\Gamma$ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if $\Gamma$ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a "global interior curvature estimate".