{
  "abstract": "We define a Bergman barycentric extension of boundary homeomorphisms of the complex unit ball by \\[ E_B(f)(z)=\\barB(f_*σ_z), \\] where \\(\\barB\\) is the Busemann barycenter for the Bergman metric and \\(σ_z\\) is the visual, equivalently Poisson--Szegő, measure based at \\(z\\). We prove well-definedness, prescribed boundary values, full \\(\\Aut(\\B^n)\\)-naturality, and interior real-analyticity. For every CR-quasisymmetric boundary homeomorphism, the Bergman barycentric extension is a quasi-isometry of complex hyperbolic space; moreover \\(E_B(f^{-1})\\) is a coarse inverse of \\(E_B(f)\\). Under sufficiently small positive CR cross-ratio distortion we obtain a sharper Tukia-type theorem: for every \\(M>1\\), the extension is a real-analytic diffeomorphism satisfying \\[ M^{-1}d_B(x,y)\\le d_B(E_B(f)(x),E_B(f)(y))\\le M d_B(x,y). \\] In contrast, for \\(n\\ge2\\) there are smooth CR-orientation-preserving CR-quasisymmetric boundary diffeomorphisms whose barycentric extensions are non-injective and, after normalization, have singular differential. Thus large-scale quasi-isometric control persists on the full CR-quasisymmetric class, whereas local non-degeneracy requires stronger boundary control.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28014",
  "author_total": 3,
  "authors": [
    {
      "name": "David Kalaj"
    },
    {
      "name": "Anton Gjokaj"
    },
    {
      "name": "Vladimir Jacimovic"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9xsh2hx9a81x0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28014",
  "fulltext_url": null,
  "id": "9xsh2hx9a81x0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28014",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7214195955",
  "pdf_url": "https://arxiv.org/pdf/2609.28014",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9xsh2hx9a81x0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28014",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28014"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Bergman barycentric extensions of boundary homeomorphisms of the complex ball",
  "title_text": "Bergman barycentric extensions of boundary homeomorphisms of the complex ball",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/9xsh2hx9a81x0",
  "venue": "arXiv",
  "versions": []
}