{
  "id": "2309.00599",
  "version": "v1",
  "published": "2023-09-01T17:35:14.000Z",
  "updated": "2023-09-01T17:35:14.000Z",
  "title": "Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space",
  "authors": [
    "Zheng Huang",
    "Ben Lowe",
    "Andrea Seppi"
  ],
  "comment": "38 pages, 6 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove a number of results on the number of solutions to the asymptotic Plateau problem in $\\mathbb H^3$. 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. Moreover, 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, and, when $\\Lambda$ is invariant by a Kleinian group, to uniqueness in the even smaller class of group invariant stable minimal disks. Finally, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-01T17:35:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "57K32"
    ],
    "keywords": [
      "asymptotic plateau problem",
      "hyperbolic space",
      "asymptotic boundary",
      "distinct stable minimal disks",
      "non-uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}