{
  "abstract": "We show that Hitchin-Wolf's harmonic map parametrization of Teichmüller space, the Schwarzian holonomy map, and Uhlenbeck's almost-Fuchsian holonomy map all arise as restrictions of a single locally biholomorphic map into the $\\mathrm{PSL}(2,\\mathbb{C})$ character variety. The map admits a right inverse defined on an open subset of quasi-Fuchsian space, encoding harmonic maps together with the Bers embedding. The construction goes through complex harmonic maps to the space of oriented geodesics of $\\mathbb{H}^3$. Along the way, we further develop the theory of complex harmonic maps, with a focus on the natural $\\mathbb{C}^*$-action and the relation to complex projective structures. This leads to an independent application: we obtain a new geometric interpretation of certain real twistor lines in the rank-two Deligne-Hitchin moduli space and we recover a theorem of Gothen-Silva.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08885",
  "author_total": 2,
  "authors": [
    {
      "name": "Christian El Emam"
    },
    {
      "name": "Nathaniel Sagman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cc739a6vc9pa3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08885",
  "fulltext_url": null,
  "id": "cc739a6vc9pa3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08885",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220965161",
  "pdf_url": "https://arxiv.org/pdf/2610.08885",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cc739a6vc9pa3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.08885",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08885"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Quadratic differentials in Teichmüller theory and complex harmonic maps",
  "title_text": "Quadratic differentials in Teichmüller theory and complex harmonic maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/cc739a6vc9pa3",
  "venue": "arXiv",
  "versions": []
}