{
  "abstract": "We study a Schwarz Lemma on the matrix ball \\[ \\mathbb D^+=\\{Z\\in M_2(\\mathbb C);\\, \\|Z\\|<1\\} \\simeq U(2,2)/(U(2)\\times U(2)), \\] which arises naturally from the complexification of the quaternions. We prove Schwarz and Schwarz--Pick estimates on $\\mathbb D^+$, construct its $U(2,2)$-invariant Riemannian Poincaré metric, and describe a Cayley transform to a tube-domain realization. The intrinsic spectral metric is the Kobayashi metric and is contracted by holomorphic maps. Comparing it with the Poincaré metric gives the sharp estimate $d(F(P),F(Q))\\le \\sqrt2\\,d(P,Q)$ for every holomorphic self-map $F$ of $\\mathbb D^+$; in general the constant $1$ is false. We also prove an infinitesimal rigidity statement and record a Burns--Krantz boundary rigidity theorem at the smooth part of $\\partial\\mathbb D^+$. Finally, we develop a conformally natural barycenter construction on the Shilov boundary $U(2)$. Every admissible probability measure has a unique barycenter in $\\mathbb D^+$, the construction is $U(2,2)$-natural, and it gives a barycentric extension of suitable boundary maps. Under the natural admissibility hypothesis, a holomorphic self-map extending to $U(2)$ is the barycentric extension of its boundary values. Many aspects of this paper are motivated by quaternionic analysis, however, no prior knowledge of quaternionic analysis is required from the reader.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02161",
  "author_total": 2,
  "authors": [
    {
      "name": "Chris Connell"
    },
    {
      "name": "Matvei Libine"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1qsa566v8b5gq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02161",
  "fulltext_url": null,
  "id": "1qsa566v8b5gq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02161",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1qsa566v8b5gq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Schwarz Lemma in Quaternionic Setting and $\\mathbb C^4$",
  "title_text": "Schwarz Lemma in Quaternionic Setting and ℂ^4",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/1qsa566v8b5gq",
  "venue": "arXiv",
  "versions": []
}