{
  "abstract": "We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and $\\infty$, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors--Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to $\\mathbb{C}$ have the form $ν(z)=a(\\bar z/z)^{n-1}+ν_0(z)$, where $a\\in(-1,1)$, $n\\ge1$ is an integer, and $ν_0\\in L^r(\\mathbb{C})$ for $0<r<2$. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12021",
  "author_total": 2,
  "authors": [
    {
      "name": "Yong Han"
    },
    {
      "name": "Renxin Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fe5pn3akvzh0t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12021",
  "fulltext_url": null,
  "id": "fe5pn3akvzh0t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12021",
  "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.CV"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7221082858",
  "pdf_url": "https://arxiv.org/pdf/2610.12021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fe5pn3akvzh0t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Quasiconformal deformations of chordal Loewner chains",
  "title_text": "Quasiconformal deformations of chordal Loewner chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/fe5pn3akvzh0t",
  "venue": "arXiv",
  "versions": []
}