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

TL;DR

This paper investigates conditions for the uniqueness and non-uniqueness of minimal surfaces in hyperbolic space with given asymptotic boundaries, providing criteria for when solutions are unique or multiple.

Contribution

It establishes new criteria linking boundary curve properties to the uniqueness of minimal surfaces and constructs examples demonstrating non-uniqueness in hyperbolic space.

Findings

Uniqueness is equivalent to uniqueness among stable minimal disks.

Curves with principal curvatures ≤ 1 ensure uniqueness.

Existence of quasicircles with uncountably many minimal disks.

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.