{
  "id": "1709.02178",
  "version": "v1",
  "published": "2017-09-07T10:53:00.000Z",
  "updated": "2017-09-07T10:53:00.000Z",
  "title": "Complete flat fronts as hypersurfaces in Euclidean space",
  "authors": [
    "Atsufumi Honda"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean $3$-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\\geq 3)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-07T10:53:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "57R45"
    ],
    "keywords": [
      "complete flat fronts",
      "euclidean space",
      "non-empty singular set",
      "vertex type theorem",
      "complete flat hypersurface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}