{
  "id": "2407.16333",
  "version": "v1",
  "published": "2024-07-23T09:35:01.000Z",
  "updated": "2024-07-23T09:35:01.000Z",
  "title": "Quadratic differentials and function theory on Riemann surfaces",
  "authors": [
    "Dragomir Saric"
  ],
  "comment": "51 pages, 18 figures",
  "categories": [
    "math.DS",
    "math.CV",
    "math.FA",
    "math.GT"
  ],
  "abstract": "A finite-area holomorphic quadratic differentials on an arbitrary Riemann surface $X=\\mathbb{H}/\\Gamma$ is uniquely determined by its horizontal measured foliation. By extending our prior result for $\\Gamma$ of the first kind to arbitrary Fuchsian group $\\Gamma$, we obtain that a measured foliation $\\mathcal{F}$ is realized by the horizontal foliation of a finite-area holomorphic quadratic differential on $X$ if and only if $\\mathcal{F}$ has finite Dirichlet integral. We determine the image of this correspondence when the infinite Riemann surface has bounded geometry -- an extension of the realization result of Hubbard and Masur for compact surfaces. A corollary is that a planar surface $X$ with bounded pants decomposition and with (at most) countably many ends is parabolic, i.e., does not support Green's function, in notation $X\\in O_G$ where $G$ is Green's function. The class of harmonic functions with finite Dirichlet integral is denoted by $HD$. We give a geometric proof that the class $O_{HD}$ of the Riemann surfaces (that do not support non-constant $HD$-functions) is invariant under quasiconformal maps. Lyons proved that the $O_{HB}$ class (surfaces that do not support non-constant bounded harmonic functions) is not invariant under quasiconformal maps, and it is well-known that the $O_G$ class is invariant. Therefore, the noninvariant class $O_{HB}$ is between two invariant classes: $O_G\\subset O_{HB}\\subset O_{HD}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-23T09:35:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann surface",
      "function theory",
      "finite-area holomorphic quadratic differential",
      "finite dirichlet integral",
      "support non-constant bounded harmonic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}