{
  "id": "2103.07383",
  "version": "v1",
  "published": "2021-03-12T16:24:42.000Z",
  "updated": "2021-03-12T16:24:42.000Z",
  "title": "On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case",
  "authors": [
    "Daniel Steenebrügge",
    "Nicole Vorderobermeier"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the analyticity of smooth critical points for generalized integral Menger curvature energies $\\mathrm{intM}^{(p,2)}$, with $p \\in (\\tfrac 73, \\tfrac 83)$, subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical $C^1$-curves $\\gamma: \\mathbb{R}/\\mathbb{Z} \\to \\mathbb{R}^n$ of generalized integral Menger curvature $\\mathrm{intM}^{(p,2)}$ subject to a fixed length constraint are not only $C^\\infty$ but also analytic. Our approach is inspired by analyticity results on critical points for O'Hara's knot energies based on Cauchy's method of majorants and a decomposition of the first variation. The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-12T16:24:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A20",
      "35A10",
      "35B65",
      "57K10"
    ],
    "keywords": [
      "critical points",
      "hilbert case",
      "analyticity",
      "fixed length constraint",
      "generalized integral menger curvature energies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}