{
  "abstract": "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].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14206",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Finch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bbv2aw9nwm65.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14206",
  "fulltext_url": null,
  "id": "0bbv2aw9nwm65",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14206",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213241937",
  "pdf_url": "https://arxiv.org/pdf/2609.14206",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bbv2aw9nwm65.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14206",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14206"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The two cubic separable minimal surfaces: $\\mathcal D_1\\mathcal D_2\\mathcal D_3=1$ and $\\mathcal C_1+\\mathcal C_2+\\mathcal C_3=3\\,\\mathcal C_1\\mathcal C_2\\mathcal C_3$",
  "title_text": "The two cubic separable minimal surfaces: 𝒟_1𝒟_2𝒟_3=1 and 𝒞_1+𝒞_2+𝒞_3=3 𝒞_1𝒞_2𝒞_3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/0bbv2aw9nwm65",
  "venue": "arXiv",
  "versions": []
}