The two cubic separable minimal surfaces: 𝒟₁𝒟₂𝒟₃=1 and 𝒞₁+𝒞₂+𝒞₃=3 𝒞₁𝒞₂𝒞₃

@misc{finch2026cubic,
  title = {{The two cubic separable minimal surfaces: $\mathcal D_1\mathcal D_2\mathcal D_3=1$ and $\mathcal C_1+\mathcal C_2+\mathcal C_3=3\,\mathcal C_1\mathcal C_2\mathcal C_3$}},
  author = {Steven Finch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14206},
  eprint = {2609.14206},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14206},
}

TY  - UNPB
TI  - The two cubic separable minimal surfaces: 𝒟_1𝒟_2𝒟_3=1 and 𝒞_1+𝒞_2+𝒞_3=3 𝒞_1𝒞_2𝒞_3
AU  - Steven Finch
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14206
UR  - https://arxiv.org/abs/2609.14206
AB  - A minimal surface in ℝ^3 is separable if it is the zero set of f(x)+g(y)+h(z), and 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 D and Schwarz's primitive surface P. In lattice-normalized coordinates their implicit equations are D: 𝒟(x)𝒟(y)𝒟(z)=1, P: 𝒞(x)+𝒞(y)+𝒞(z) =3 𝒞(x)𝒞(y)𝒞(z), where 𝒟(t)=sndn/cn(K[1/4]t,1/4) and 𝒞(t)=cn(2K[3/4]t,3/4). The first equation is classical -- it is due to Schwarz, later Cayley, and appears in Nitsche's Lectures -- we identify Nitsche's transcendental function exactly as sndn/cn at parameter 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 P. Both surfaces are governed by one differential equation with two signs, φ'^ 2=1+2cosh 2φ (D), φ'^ 2=1+2cos 2φ (P), the two signs of the separation constant of the classical reduction; and in that common normalization the two lattice periods are 2K[1/4] and 2K[3/4], whose ratio is Schwarz's 1866 constant K'[1/4]/K[1/4]=1.2792615..., the necessary scale ratio for a conjugate pair. Proofs will appear in [D, P].
ER  -