{
  "abstract": "Smyth (1984) proved that every tetrahedron $T\\subset\\mathbb{R}^3$ contains three embedded minimal disks meeting $\\partial T$ orthogonally, obtained as conjugate surfaces of Plateau solutions in a skew quadrilateral $Q$ whose edges are $c_iν_i$, where $φ_1,φ_2,φ_3,φ_4$ are faces of $T$, $c_i=\\operatorname{Area}(φ_i)$ and $ν_i$ are outward normal vectors. For the $\\textit{regular}$ tetrahedron we give the Weierstrass data explicitly. The Gauss map is a Schwarz triangle function for a circular-arc triangle with angles $(π/4,\\ π/2,\\ πβ/2)$, where $β=\\arccos(-1/3)/π$; the hypergeometric parameters are linear in $β$ and there is $\\textit{no accessory parameter and no free parameter}$. We record a closed form for $κ$ in $Γ$-values, a Wronskian closed form for $g'$, the exact total curvature $2π(2β-1)$, the symmetric-square structure of the data, and a rigidity statement: since $β/2$ is irrational, the projective monodromy is dense and the data $\\textit{cannot}$ be algebraic. Numerically the representation extends to 28 digits -- $\\operatorname{Area}/\\ell^{2}=0.2172341554075574483760009126\\ldots$ -- confirming a 2013 Surface Evolver computation of Ken Brakke.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00434",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Finch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cqm8g07zpav1c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00434",
  "fulltext_url": null,
  "id": "cqm8g07zpav1c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00434",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7215726260",
  "pdf_url": "https://arxiv.org/pdf/2610.00434",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cqm8g07zpav1c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00434",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00434"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron",
  "title_text": "An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/cqm8g07zpav1c",
  "venue": "arXiv",
  "versions": []
}