On the asymptotic Plateau problem in hyperbolic space

Abstract

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