On the asymptotic Plateau problem in hyperbolic space

TL;DR

This paper solves the asymptotic Plateau problem in hyperbolic space for hypersurfaces with constant _{n-1} curvature, establishing existence results with prescribed asymptotic boundary conditions.

Contribution

It extends the existence theory for the asymptotic Plateau problem to the case of constant _{n-1} curvature, previously known only for a restricted range.

Findings

Established curvature estimates for the problem.

Proved existence of hypersurfaces with prescribed asymptotic boundary.

Extended known results to a broader curvature range.

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.