Smyth在正四面体中的自由边界极小圆盘的精确超几何Weierstrass表示

中文摘要 AI 辅助

Smyth（1984）证明了每个四面体$T\subset\mathbb{R}^3$包含三个嵌入的极小圆盘，它们与$\partial T$正交，这些圆盘是通过斜四边形$Q$中Plateau解的共轭曲面获得的，其中$Q$的边为$c_i\nu_i$，这里$\varphi_1,\varphi_2,\varphi_3,\varphi_4$是$T$的面，$c_i=\operatorname{Area}(\varphi_i)$，$\nu_i$是外法向量。对于正四面体，我们明确给出Weierstrass数据。Gauss映射是角度为$(\pi/4,\\ \pi/2,\\ \pi\beta/2)$的圆弧三角形的Schwarz三角形函数，其中$\beta=\arccos(-1/3)/\pi$；超几何参数是$\beta$的线性函数，且没有辅助参数和自由参数。我们记录了$\kappa$的$\Gamma$值闭式、$g'$的Wronskian闭式、精确总曲率$2\pi(2\beta-1)$、数据的对称平方结构，以及一个刚性陈述：由于$\beta/2$是无理数，投影单值群是稠密的，数据不可能是代数的。数值上，该表示精确到28位数字——$\operatorname{Area}/\ell^{2}=0.2172341554075574483760009126\ldots$——证实了Ken Brakke在2013年用Surface Evolver进行的计算。

英文摘要

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.

发表机构

• MIT Sloan School of Management（麻省理工学院斯隆管理学院）

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23 pages