{
  "abstract": "We study the hyperbolically weighted distortion energies \\[ \\mathcal{I}_p(f)=\\int_\\mathbb{D} \\frac{|μ_f|^p}{(1-|μ_f|^2)^{\\max\\{2,p\\}/2}}\\,dA_η\\] for finite-distortion disk homeomorphisms with prescribed boundary values, together with the associated inverse problem in the monotone Sobolev class. For $1<p\\leq2$, the inverse density is polyconvex precisely when $p\\geq p_{\\mathrm c}=1+1/\\sqrt2$. In this range, quasisymmetric data with finite-energy monotone barriers yield minimisers with holomorphic Ahlfors--Hopf differentials and strong local Sobolev convergence of minimising subsequences. For $p_{\\mathrm c}<p<2$, a finite-energy finite-distortion barrier yields a unique absolute monotone minimiser, which is a quasiconformal diffeomorphism and is asymptotically conformal. Together with the boundary-class theorem of Part~I, this characterises the admissible boundary data as $T^p$. The proof combines a selected barrier approximation, uniform scalar gradient estimates and a weighted Reich--Strebel comparison. For $1<p<p_{\\mathrm c}$, finite rank-one comparisons bound distortion on the positive-Jacobian set by a cutoff strictly below the ellipticity cutoff. An attained monotone minimum is either a quasiconformal diffeomorphism or has nonzero rank-one derivatives on a set of positive area. For $0<p\\leq1$, minimisers with finite distortion, or homeomorphic minimisers with a locally Sobolev inverse, are Möbius maps. We also obtain energy-dependent quasiconformal bounds for regular minimisers when $p>2$. \\",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34007",
  "author_total": 2,
  "authors": [
    {
      "name": "Gaven Martin"
    },
    {
      "name": "Cong Yao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4j8kpr7ddejkf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34007",
  "fulltext_url": null,
  "id": "4j8kpr7ddejkf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34007",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7223849083",
  "pdf_url": "https://arxiv.org/pdf/2610.34007",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4j8kpr7ddejkf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34007",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34007"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Integrable distortion II: the p-Weil-Petersson variational problem",
  "title_text": "Integrable distortion II: the p-Weil-Petersson variational problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/4j8kpr7ddejkf",
  "venue": "arXiv",
  "versions": []
}