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

We prove several results on the number of solutions to the asymptotic problem in ℍ 3. Firstly, we discuss crite- ria that ensure uniqueness. Given a Jordan curve Λ in the asymptotic boundary of ℍ 3, we show that unique- ness of the minimal surfaces with asymptotic boundary Λ is equivalent to uniqueness in the smaller class of sta- ble minimal disks. Then we show that if a quasicircle (or more generally, a Jordan curve of finite width) Λ is the asymptotic boundary of a minimal surface Σ 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