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 problem in . Firstly, we discuss criteria that ensure uniqueness. Given a Jordan curve in the asymptotic boundary of , we show that uniqueness of the minimal surfaces with asymptotic boundary 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) is the asymptotic boundary of a minimal surface 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.