An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron

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 $c_i{\nu }_i$, where ${\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4$ are faces of $T$, $c_i=\mathrm{Area}({\phi }_i)$ and ${\nu }_i$ are outward normal vectors. For the $\text{regular}$ tetrahedron we give the Weierstrass data explicitly. The Gauss map is a Schwarz triangle function for a circular-arc triangle with angles $(\pi /4,\pi /2,\pi \beta /2)$, where $\beta =\arccos (-1/3)/\pi$; the hypergeometric parameters are linear in $\beta$ and there is $\text{no accessory parameter and no free parameter}$. We record a closed form for $\kappa$ in $\Gamma$-values, a Wronskian closed form for $g^{\prime }$, the exact total curvature $2\pi (2\beta -1)$, the symmetric-square structure of the data, and a rigidity statement: since $\beta /2$ is irrational, the projective monodromy is dense and the data $\text{cannot}$ be algebraic. Numerically the representation extends to 28 digits -- $\mathrm{Area}/{\ell }^2=0.2172341554075574483760009126\dots$ -- confirming a 2013 Surface Evolver computation of Ken Brakke.

arXiv comment23 pages