{
  "id": "2208.07317",
  "version": "v1",
  "published": "2022-08-15T16:30:42.000Z",
  "updated": "2022-08-15T16:30:42.000Z",
  "title": "Curl through spin on three-manifold",
  "authors": [
    "S. Montiel"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In the last decades, many mathematicians have studied the {\\em curl operator} on compact (both with or without empty boundary) three-manifolds, mainly the behaviour of its spectrum and some iso\\-pe\\-ri\\-me\\-tric problems associated with it. In this paper, we reveal an (unexpected?) relation between this curl operator and the Dirac operator correspondingto any of the spin$^c$ structures on the manifold. Then, we make the ellipticity of $D$ (curl is not) and the many facts already known about the spectrum of $D$ to recuperate with almost immediate proofs some results above curl and obtain others unknown for me. {\\em For example, we will find that the eigenvalues of curl, removing the point spectrum zero, are always, up to a fixed constant, lower bounded by those of the Dirac and the equality characterize the round three-sphere}. Also, we also show that {\\em there do not exist $L^2$-solutions for the isoperimetric problem associated to curl}, as Cantarella, de Turck, Gluck y Teytel \\cite{CdTGT} had conjectured, while other authors proved properties for these unknown solutions (adding always whether optimal domains exist) perhaps by thinking in the case of the successful maximization of the {\\em helicity.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-15T16:30:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "three-manifold",
      "curl operator",
      "dirac operator correspondingto",
      "point spectrum zero",
      "empty boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}