On the asymptotic Plateau problem in hyperbolic space

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 $\mathrm{\Gamma }$ . The key ingredient is the curvature estimates. Previously, this was only known for ${\sigma }_0>\sigma >n$ , where ${\sigma }_0$ is a positive constant.