Invariant solutions for the asymptotic Plateau problem in $\mathbb{H}^3$

Matheus Pimentel Gomes; Patrícia Kruse Klaser; Jaime Bruck Ripoll; Leonardo Prange Bonorino

Invariant solutions for the asymptotic Plateau problem in $\mathbb{H}^3$

Differential Geometry 2026-07-08 v1

Authors: Matheus Pimentel Gomes , Patrícia Kruse Klaser , Jaime Bruck Ripoll , Leonardo Prange Bonorino

Abstract

In this paper, we present solutions to the asymptotic Plateau problem in the hyperbolic space $\mathbb{H}^3$ . In this context, we exhibit solutions for curves that are invariant under the action of a one-parameter subgroup of isometries of $\mathbb{H}^3$ . To achieve this, we prove the existence of foliations of $\mathbb{H}^3$ by minimal surfaces that are properly embedded, complete, and invariant under these subgroups, which are then used to solve the problem.

Keywords

asymptotic Plateau problem hyperbolic space minimal surfaces invariant solutions foliations

Cite

@article{arxiv.2607.07986,
  title  = {Invariant solutions for the asymptotic Plateau problem in $\mathbb{H}^3$},
  author = {Matheus Pimentel Gomes and Patrícia Kruse Klaser and Jaime Bruck Ripoll and Leonardo Prange Bonorino},
  journal= {arXiv preprint arXiv:2607.07986},
  year   = {2026}
}