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

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 $\mathrm{\Lambda }$ in the asymptotic boundary of ${\mathbb{H}}^3$, we show that uniqueness of the minimal surfaces with asymptotic boundary $\mathrm{\Lambda }$ 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) $\mathrm{\Lambda }$ is the asymptotic boundary of a minimal surface $\mathrm{\Sigma }$ 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.