{
  "id": "2007.11384",
  "version": "v1",
  "published": "2020-07-22T12:50:08.000Z",
  "updated": "2020-07-22T12:50:08.000Z",
  "title": "The isoperimetric problem for regular and crystalline norms in $\\mathbb H^1$",
  "authors": [
    "Valentina Franceschi",
    "Roberto Monti",
    "Alberto Righini",
    "Mario Sigalotti"
  ],
  "categories": [
    "math.DG",
    "math.MG",
    "math.OC"
  ],
  "abstract": "We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $\\phi$ on the horizontal distribution. We first prove a representation formula for the $\\phi$-perimeter of regular sets and, assuming some regularity on $\\phi$ and on its dual norm $\\phi^*$, we deduce a foliation property by sub-Finsler geodesics of $\\mathrm C^2$-smooth surfaces with constant $\\phi$-curvature. We then prove that the characteristic set of $\\mathrm C^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize $\\mathrm C^2$-smooth $\\phi$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $\\phi$, that such sub-Finsler candidate isoperimetric sets are indeed $\\mathrm C^2$-smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where $\\phi$ is crystalline).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-22T12:50:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "52B60",
      "53C17"
    ],
    "keywords": [
      "isoperimetric problem",
      "crystalline norms",
      "sub-finsler candidate isoperimetric sets",
      "smooth surfaces",
      "anisotropic left-invariant perimeter measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}