{
"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"
}
}
}