{
  "id": "1707.09176",
  "version": "v1",
  "published": "2017-07-28T10:23:14.000Z",
  "updated": "2017-07-28T10:23:14.000Z",
  "title": "Construction of embedded periodic surfaces in $\\mathbb{R}^n$",
  "authors": [
    "Karsten Grosse-Brauckmann",
    "Susanne Kürsten"
  ],
  "comment": "27 pages, 5 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "We construct embedded minimal surfaces which are $n$-periodic in $\\mathbb{R}^n$. They are new for codimension $n-2\\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz reflections, we can characterize those Jordan curves for which the complete surface is embedded. For example, for $n=4$ exactly five such Jordan curves generate embedded surfaces. Our results apply to surface classes other than minimal as well, for instance polygonal surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-28T10:23:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53A07",
      "49Q05"
    ],
    "keywords": [
      "embedded periodic surfaces",
      "construction",
      "jordan curves generate embedded surfaces",
      "instance polygonal surfaces",
      "plateau minimal disk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}