An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron
Abstract
Smyth (1984) proved that every tetrahedron T⊂ℝ3 contains three embedded minimal disks meeting ∂ T orthogonally, obtained as conjugate surfaces of Plateau solutions in a skew quadrilateral Q whose edges are ciνi, where φ1,φ2,φ3,φ4 are faces of T, ci=Area(φi) and νi are outward normal vectors. For the 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 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 cannot be algebraic. Numerically the representation extends to 28 digits -- Area/ℓ2=0.2172341554075574483760009126… -- confirming a 2013 Surface Evolver computation of Ken Brakke.
Related
- Canonical differential equations for Feynman integrals from 𝒜-hypergeometric systems in the Schwinger representation
- Gaussian Boundary Inference in a Hypergeometric Heavy-Tailed Family
- The Casimir on the K-fixed tensor square of the minimal representation of G2(2)
- Free boundary minimal surfaces with connected boundary and arbitrary genus
- Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
BibTeX
@misc{finch2026exact,
title = {{An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron}},
author = {Steven Finch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00434},
eprint = {2610.00434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00434},
}
RIS
TY - UNPB
TI - An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron
AU - Steven Finch
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00434
UR - https://arxiv.org/abs/2610.00434
AB - Smyth (1984) proved that every tetrahedron T⊂ℝ^3 contains three embedded minimal disks meeting ∂ 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=Area(φ_i) and ν_i are outward normal vectors. For the 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 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 cannot be algebraic. Numerically the representation extends to 28 digits -- Area/ℓ^2=0.2172341554075574483760009126… -- confirming a 2013 Surface Evolver computation of Ken Brakke.
ER -