Schwarz 原始曲面椭圆参数化中参数的消去

中文摘要 AI 辅助

设 $\Omega$ 为由四条圆弧 $|\zeta\mp1|=\sqrt{2}$、$|\zeta\mp i|=\sqrt{2}$ 围成的曲线正方形，并设 $W=\sqrt{1+14\zeta^{4}+\zeta^{8}}$。令 \\[ \theta(\zeta)=\arcsin\frac{2(1+i)\zeta} {\sqrt{1+4i\zeta^{2}-\zeta^{4}}}, \\] \\[ f=\tfrac14\bigl(-iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr),\quad \quad g=\tfrac14\bigl(\\ \\ iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr), \\] \\[ \mathfrak h=(2-\sqrt3)\\, F\bigl[\arcsin\bigl(i(2+\sqrt3)\zeta^{2}\bigr),(2-\sqrt3)^{4}\bigr]. \\] 这些正是我们关于金刚石曲面 $\mathrm{D}$ 的姊妹论文 $\mathrm{[D]}$ 中的三个函数，逐字相同：两篇论文从同一点出发，仅在于取哪个实部不同。令 $X=(\kappa\\,\operatorname{Re}f,\\ \kappa\\,\operatorname{Re}g,\\ \tfrac12+\kappa\\,\operatorname{Im}\mathfrak{h})$ 且 $\kappa=3/(2K[1/9])$，则映射 $X$ 参数化了 Schwarz 原始曲面 $\mathrm{P}$（$\mathrm{D}$ 的共轭曲面）的一个基本片。第一部分构建代数工具……归一化常数被精确确定：$\varpi:=\kappa^{-1}=K[-3]=\tfrac12 K[3/4]=\tfrac23 K[1/9]$。第二部分进行消去并完成 \\[ (\star)\qquad \operatorname{sn}\bigl(\varpi(x{+}y),-3\bigr)\\,\operatorname{sn}\bigl(\varpi(x{-}y),-3\bigr) =\tfrac13\operatorname{sn}\bigl(3\varpi(z-\tfrac12),\tfrac19\bigr) \\] 的证明。第三部分表明 $(\star)$ 终究是可加分离的……关于本文和 $\mathrm{[D]}$ 的概述，请参见 arXiv:2609.14206。

英文摘要

Let $Ω$ be the curvilinear square bounded by the four circular arcs $|ζ\mp1|=\sqrt{2}$, $|ζ\mp i|=\sqrt{2}$, and let $W=\sqrt{1+14ζ^{4}+ζ^{8}}$. Put \[ θ(ζ)=\arcsin\frac{2(1+i)ζ} {\sqrt{1+4iζ^{2}-ζ^{4}}}, \] \[ f=\tfrac14\bigl(-iF[θ,\tfrac14]+F[θ,\tfrac34]\bigr),\quad \quad g=\tfrac14\bigl(\ \ iF[θ,\tfrac14]+F[θ,\tfrac34]\bigr), \] \[ \mathfrak h=(2-\sqrt3)\, F\bigl[\arcsin\bigl(i(2+\sqrt3)ζ^{2}\bigr),(2-\sqrt3)^{4}\bigr]. \] These are, verbatim, the three functions of our companion paper $\mathrm{[D]}$ on the diamond surface $\mathrm{D}$: the two papers start from the same point, and differ only in which real parts are taken. With $X=(κ\,\operatorname{Re}f,\ κ\,\operatorname{Re}g,\ \tfrac12+κ\,\operatorname{Im}\mathfrak{h})$ and $κ=3/(2K[1/9])$, the map $X$ parametrizes a fundamental patch of Schwarz's primitive surface $\mathrm{P}$, the conjugate of $\mathrm{D}$. Part I builds the algebraic apparatus $\ldots$ the normalizing constant is identified exactly: $\varpi:=κ^{-1}=K[-3]=\tfrac12 K[3/4]=\tfrac23 K[1/9]$. Part II carries out the elimination and completes the proof of \[ (\star)\qquad \operatorname{sn}\bigl(\varpi(x{+}y),-3\bigr)\,\operatorname{sn}\bigl(\varpi(x{-}y),-3\bigr) =\tfrac13\operatorname{sn}\bigl(3\varpi(z-\tfrac12),\tfrac19\bigr). \] Part III shows that $(\star)$ is, after all, additively separable $\ldots$ for an overview of this paper and $\mathrm{[D]}$, please see arXiv:2609.14206.

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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