Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space

Abstract

We prove several results on the number of solutions to the asymptotic Plateau problem in ${\mathbb{H}}^3$. Firstly we discuss criteria that ensure uniqueness. Given a Jordan curve $\Lambda$ in the asymptotic boundary of ${\mathbb{H}}^3$, we show that uniqueness of the minimal surfaces with asymptotic boundary $\Lambda$ is equivalent to uniqueness in the smaller class of stable minimal disks. Then we show that if a quasicircle (or more generally, a Jordan curve of finite width) $\Lambda$ is the asymptotic boundary of a minimal surface $\Sigma$ with principal curvatures less than or equal to 1 in absolute value, then uniqueness holds. In the direction of non-uniqueness, we construct an example of a quasicircle that is the asymptotic boundary of uncountably many pairwise distinct stable minimal disks.

Revised version, large parts of the paper have been rewritten, arguments simplified and improved, details added, main results unchanged