On the asymptotic Plateau problem in hyperbolic space

Abstract

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant σ n − 1 \sigma _{n-1} curvature, i.e. the existence of a complete hypersurface in H n + 1 \mathbb {H}^{n+1} satisfying σ n − 1 ( κ ) = σ ∈ ( 0 , n ) \sigma _{n-1}(\kappa )=\sigma \in (0,n) with a prescribed asymptotic boundary Γ \Gamma . The key ingredient is the curvature estimates. Previously, this was only known for σ 0 > σ > n \sigma _0>\sigma >n , where σ 0 \sigma _0 is a positive constant.