{
  "id": "2102.08714",
  "version": "v1",
  "published": "2021-02-17T11:56:57.000Z",
  "updated": "2021-02-17T11:56:57.000Z",
  "title": "Some geometric properties of nonparametric $μ$-surfaces in $\\mathbb{R}^3$",
  "authors": [
    "Michael Bildhauer",
    "Matin Fuchs"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Smooth solutions of the equation \\[ \\rm{div}\\, \\Bigg\\{ \\frac{g'\\big(|\\nabla u|\\big)}{|\\nabla u|} \\nabla u \\Bigg\\} = 0 \\] are considered generating nonparametric $\\mu$-surfaces in $\\mathbb{R}^3$, whenever $g$ is a function of linear growth satisfying in addition \\[ \\int_0^\\infty s g''(s) d s < \\infty \\, . \\] Particular examples are $\\mu$-elliptic energy densities $g$ with exponent $\\mu > 2$ (see [1]) and the minimal surfaces belong to the class of $3$-surfaces. Generalizing the minimal surface case we prove the closedness of a suitable differential form $\\hat{N} \\wedge d X$. As a corollary we find an asymptotic conformal parametrization generated by this differential form.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-17T11:56:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q05",
      "53A10",
      "53C42",
      "58E12"
    ],
    "keywords": [
      "geometric properties",
      "nonparametric",
      "asymptotic conformal parametrization",
      "minimal surface case",
      "elliptic energy densities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}