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 $\mathbb{H}^{n+1}$ satisfying $f(k) = \sigma \in (0, 1)$ 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) star-shaped, the solution is unique and also (Euclidean) star-shaped, 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.”