{
  "id": "2106.06318",
  "version": "v1",
  "published": "2021-06-11T11:35:15.000Z",
  "updated": "2021-06-11T11:35:15.000Z",
  "title": "On the Size of Minimal Surfaces in $\\mathbb{R}^4$",
  "authors": [
    "Ari Aiolfi",
    "Marc Soret",
    "Marina Ville"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "The Gauss map $g$ of a surface $\\Sigma$ in $\\mathbb{R}^4$ takes its values in the Grassmannian of oriented 2-planes of $\\mathbb{R}^4$: $G^+(2,4)$. We give geometric criteria of stability for minimal surfaces in $\\mathbb{R}^4$ in terms of $g$. We show in particular that if the spherical area of the Gauss map $|g(\\Sigma)|$ of a minimal surface is smaller than $2\\pi$ then the surface is stable by deformations which fix the boundary of the surface.This answers a question of Barbosa and Do Carmo in $\\mathbb{R}^4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-11T11:35:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10"
    ],
    "keywords": [
      "minimal surface",
      "gauss map",
      "geometric criteria",
      "spherical area",
      "grassmannian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}