{
  "id": "1712.05405",
  "version": "v1",
  "published": "2017-12-15T00:11:05.000Z",
  "updated": "2017-12-15T00:11:05.000Z",
  "title": "On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions",
  "authors": [
    "Victor Kalvin"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1612.08660",
  "categories": [
    "math.AP",
    "math.DG",
    "math.SP"
  ],
  "abstract": "Let $\\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\\to {\\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\\mathsf m$ or $\\{\\infty\\}$. The pullback $f^*\\mathsf m$ of $\\mathsf m$ under $f$ has conical singularities of angles $4\\pi$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\\mathsf m$. We study the $\\zeta$-regularized determinant $\\operatorname{Det}' \\Delta_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^*\\mathsf m)$ as a functional on the moduli space of pairs $(X, f)$ and obtain an explicit formula for $\\operatorname{Det}' \\Delta_F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-15T00:11:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact riemann surface",
      "meromorphic functions",
      "conical metrics",
      "determinant",
      "conical singularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}