两个三次可分离极小曲面：$\mathcal D_1\mathcal D_2\mathcal D_3=1$ 与 $\mathcal C_1+\mathcal C_2+\mathcal C_3=3\\,\mathcal C_1\mathcal C_2\mathcal C_3$
The two cubic separable minimal surfaces: $\mathcal D_1\mathcal D_2\mathcal D_3=1$ and $\mathcal C_1+\mathcal C_2+\mathcal C_3=3\,\mathcal C_1\mathcal C_2\mathcal C_3$
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中文总结 AI 辅助
本文证明在相似与刚体运动下恰有两个非平面各向同性可分离极小曲面，即 Schwarz 菱形曲面 D 与原始曲面 P，二者共轭，并给出其隐式方程、统一微分方程及晶格周期比。
中文摘要 AI 辅助
若 $\mathbb{R}^3$ 中的极小曲面是 $f(x)+g(y)+h(z)$ 的零集，则称其为可分离的；若进一步有 $f=g=h$，则称其为各向同性的。我们宣布，在相似变换和刚体运动意义下，恰好存在两个非平面的各向同性可分离极小曲面，它们互为共轭，并且它们就是 Schwarz 的菱形曲面 $\mathrm{D}$ 和 Schwarz 的原始曲面 $\mathrm{P}$。在晶格归一化坐标下，它们的隐式方程为 \\[\mathrm{D}:\\; \mathcal{D}(x)\mathcal{D}(y)\mathcal{D}(z)=1,\\] \\[\mathrm{P}:\\; \mathcal{C}(x)+\mathcal{C}(y)+\mathcal{C}(z) =3\\,\mathcal{C}(x)\mathcal{C}(y)\mathcal{C}(z),\\] 其中 $\mathcal{D}(t)=\frac{\operatorname{sn}\operatorname{dn}}{\operatorname{cn}}(K[\frac{1}{4}]t,\frac{1}{4})$ 且 $\mathcal{C}(t)=\operatorname{cn}(2K[\frac{3}{4}]t,\frac{3}{4})$。第一个方程是经典的——它归功于 Schwarz，后来 Cayley 也研究过，并出现在 Nitsche 的《讲义》中——我们将 Nitsche 的超越函数精确地识别为参数为 $\frac{1}{4}$ 时的 $\operatorname{sn}\operatorname{dn}/\operatorname{cn}$。第二个方程似乎是新的；它是 Kim 和 Ogata (2024) 的两参数族中的对称成员，并且证明了他们关于该族包含 $\mathrm{P}$ 的断言。两个曲面都由一个带两个符号的微分方程控制：\\[\varphi'^{\\,2}=1+2\cosh 2\varphi\\ \\ (\mathrm{D}),\\] \\[\varphi'^{\\,2}=1+2\cos 2\varphi\\ \\ (\mathrm{P}),\\] 这两个符号对应于经典约化中分离常数的两个符号；在该共同归一化下，两个晶格周期为 $2K[\frac{1}{4}]$ 和 $2K[\frac{3}{4}]$，其比值是 Schwarz 1866 年的常数 $K'[\frac{1}{4}]/K[\frac{1}{4}]=1.2792615...$，这是共轭对所必需的尺度比。证明将出现在 [D, P] 中。
英文摘要
A minimal surface in $\mathbb{R}^3$ is $\textit{separable}$ if it is the zero set of $f(x)+g(y)+h(z)$, and $\textit{isotropic}$ if moreover $f=g=h$. We announce that there are exactly two non-planar isotropic separable minimal surfaces up to homothety and rigid motion, that they are conjugate, and that they are Schwarz's diamond surface $\mathrm{D}$ and Schwarz's primitive surface $\mathrm{P}$. In lattice-normalized coordinates their implicit equations are \[\mathrm{D}:\; \mathcal{D}(x)\mathcal{D}(y)\mathcal{D}(z)=1,\] \[\mathrm{P}:\; \mathcal{C}(x)+\mathcal{C}(y)+\mathcal{C}(z) =3\,\mathcal{C}(x)\mathcal{C}(y)\mathcal{C}(z),\] where $\mathcal{D}(t)=\frac{\operatorname{sn}\operatorname{dn}}{\operatorname{cn}}(K[\frac{1}{4}]t,\frac{1}{4})$ and $\mathcal{C}(t)=\operatorname{cn}(2K[\frac{3}{4}]t,\frac{3}{4})$. The first equation is classical -- it is due to Schwarz, later Cayley, and appears in Nitsche's $\textit{Lectures}$ -- we identify Nitsche's transcendental function exactly as $\operatorname{sn}\operatorname{dn}/\operatorname{cn}$ at parameter $\frac{1}{4}$. The second appears to be new; it is the symmetric member of the two-parameter family of Kim and Ogata (2024), and it proves their assertion that the family contains $\mathrm{P}$. Both surfaces are governed by one differential equation with two signs, \[φ'^{\,2}=1+2\cosh 2φ \ (\mathrm{D}),\] \[φ'^{\,2}=1+2\cos 2φ \ (\mathrm{P}),\] the two signs of the separation constant of the classical reduction; and in that common normalization the two lattice periods are $2K[\frac{1}{4}]$ and $2K[\frac{3}{4}]$, whose ratio is Schwarz's 1866 constant $K'[\frac{1}{4}]/K[\frac{1}{4}]=1.2792615...$, the necessary scale ratio for a conjugate pair. Proofs will appear in [D, P].
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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