{
  "id": "1601.07023",
  "version": "v1",
  "published": "2016-01-26T13:50:29.000Z",
  "updated": "2016-01-26T13:50:29.000Z",
  "title": "The Gradient Flow of the Möbius energy: $\\varepsilon$-regularity and consequences",
  "authors": [
    "Simon Blatt"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article we study the gradient flow of the M\\\"obius energy introduced by O'Hara in 1991. We will show a fundamental $\\varepsilon$-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He. Ruling out blow-ups for planar curves, we will prove that the flow transforms every planar curve into a round circle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T13:50:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44",
      "35S10"
    ],
    "keywords": [
      "gradient flow",
      "möbius energy",
      "consequences",
      "planar curve",
      "infinity norm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160107023B"
    }
  }
}