{
  "id": "2302.01320",
  "version": "v1",
  "published": "2023-02-02T18:47:52.000Z",
  "updated": "2023-02-02T18:47:52.000Z",
  "title": "Generic uniqueness for the Plateau problem",
  "authors": [
    "Gianmarco Caldini",
    "Andrea Marchese",
    "Andrea Merlo",
    "Simone Steinbrüchel"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Given a complete Riemannian manifold $\\mathcal{M}\\subset\\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{3,\\beta}$, without boundary and an oriented, closed submanifold $\\Gamma \\subset \\mathcal M$ of dimension $m-1$, of class $C^{3,\\alpha}$ with $\\alpha<\\beta$, which is a boundary in integral homology, we construct a complete metric space $\\mathcal{B}$ of $C^{3,\\alpha}$-perturbations of $\\Gamma$ inside $\\mathcal{M}$ with the following property. For the typical element $b\\in\\mathcal B$, in the sense of Baire categories, every $m$-dimensional integral current in $\\mathcal{M}$ which solves the corresponding Plateau problem has an open dense set of boundary points with density $1/2$. We deduce that the typical element $b\\in\\mathcal{B}$ admits a unique solution to the Plateau problem. Moreover we prove that, in a complete metric space of integral currents without boundary in $\\mathbb{R}^{m+n}$, metrized by the flat norm, the typical boundary admits a unique solution to the Plateau problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-02T18:47:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q05",
      "49Q15"
    ],
    "keywords": [
      "plateau problem",
      "generic uniqueness",
      "complete metric space",
      "unique solution",
      "open dense set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}