{
  "id": "2404.02215",
  "version": "v1",
  "published": "2024-04-02T18:07:52.000Z",
  "updated": "2024-04-02T18:07:52.000Z",
  "title": "An $\\infty$-Laplacian for differential forms, and calibrated laminations",
  "authors": [
    "Aidan Backus"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichm\\\"uller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \\to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is already known to give rise to laminations by minimal hypersurfaces; we show that the limiting $p$-harmonic conjugates converge to calibrations $F$ of the laminations. Moreover, we show that the laminations which are calibrated by $F$ are exactly those which arise from the $1$-Laplacian. We also explore the limiting dual problem as a model problem for the optimal Lipschitz extension problem, which exhibits behavior rather unlike the scalar $\\infty$-Laplacian. In a companion work, we will apply the main result of this paper to associate to each class in $H^{d - 1}$ a lamination in a canonical way, and study the duality of the stable norm on $H_{d - 1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-02T18:07:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J94",
      "35J92",
      "53C38",
      "49N15",
      "49Q20"
    ],
    "keywords": [
      "differential forms",
      "calibrated laminations",
      "optimal lipschitz extension problem",
      "harmonic conjugates converge",
      "limiting dual problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}