{
  "abstract": "We study the Friedrichs Laplacian associated with the pullback of the round metric on \\(\\mb P^1\\) by a nonconstant meromorphic function \\(φ:X\\to\\mb P^1\\) on a compact Riemann surface. For arbitrary ramification profiles, including several ramification points over the same branch value, we prove the local formula \\[ \\operatorname{Det}_ζ(Δ_{[φ],\\mc F}) =C\\,\\det\\operatorname{Im}B\\,|τ_B|^2 \\prod_{k=1}^Nρ(z_k,\\overline{z_k})^{c_k}. \\] The zero eigenvalue is omitted. Here \\(B\\) is the period matrix, \\(τ_B\\) is the local Bergman tau-function, \\(z_k\\) are the branch-value coordinates, \\(ρ(z,\\overline z)=4(1+|z|^2)^{-2}\\), and \\(c_k=\\frac1{12}\\sum_j(n_{kj}-n_{kj}^{-1})\\), where \\(n_{kj}\\) are the ramification indices over the \\(k\\)-th branch value. The constant \\(C>0\\) is independent of the Hurwitz coordinates, and \\(\\det\\operatorname{Im}B\\) is taken to be \\(1\\) in genus zero. The proof combines smooth trivializations and trace-norm variation of resolvent powers with matrix comparison to spherical conic models. The Davies--Gaffney estimate provides the required uniform high-energy control, while the zero-energy terms are identified through the Schiffer bidifferential and Rauch variational formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19604",
  "author_total": 2,
  "authors": [
    {
      "name": "Jia-Ming"
    },
    {
      "name": "Liou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/avd978e6ebepr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19604",
  "fulltext_url": null,
  "id": "avd978e6ebepr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19604",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7213584728",
  "pdf_url": "https://arxiv.org/pdf/2609.19604",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/avd978e6ebepr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19604",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19604"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification",
  "title_text": "Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/avd978e6ebepr",
  "venue": "arXiv",
  "versions": []
}