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

Zheng Huang; Ben Lowe; Andrea Seppi

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arXiv · 2309.00599

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

Zheng Huang·Ben Lowe·Andrea Seppi

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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.

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BibTeX RIS

Zheng Huang, Ben Lowe, Andrea Seppi. 2023-09-01. Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space. https://arxiv.org/abs/2309.00599

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