{ "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" } } }