0 Se p 20 12 INTERIOR CURVATURE ESTIMATES AND THE ASYMPTOTIC PLATEAU PROBLEM IN HYPERBOLIC SPACE

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(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if Γ 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”.