{
  "abstract": "The Ahlfors--Weill construction explicitly extends a univalent function $f$ with Schwarzian norm less than $1/2$ to a quasiconformal homeomorphism of the Riemann sphere, an extension that can be understood geometrically through Epstein surfaces. For asymptotically conformal curves, the uniformization map can be extended to a larger disk, and Loewner theory can be used to verify that this extension is quasiconformal. In this paper, we construct a local quasireflection with such an extension. Using this local quasireflection, we modify classical estimates of Ahlfors and show that domains bounded by asymptotically conformal curves have asymptotically similar hyperbolic and quasihyperbolic metrics. Furthermore, by combining these estimates with a geometric reinterpretation of the extension through Epstein surfaces, we show that the Epstein-Poincaré surfaces of the complementary domains are arbitrarily close to one another near the curve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02171",
  "author_total": 1,
  "authors": [
    {
      "name": "Ming Hong Tee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ezfje0kc5zh8z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02171",
  "fulltext_url": null,
  "id": "ezfje0kc5zh8z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02171",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7216014357",
  "pdf_url": "https://arxiv.org/pdf/2610.02171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ezfje0kc5zh8z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.02171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02171"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Ahlfors--Weill Extension, Asymptotically Conformal Curves and Epstein--Poincaré Surfaces",
  "title_text": "Ahlfors--Weill Extension, Asymptotically Conformal Curves and Epstein--Poincaré Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/ezfje0kc5zh8z",
  "venue": "arXiv",
  "versions": []
}