{
  "abstract": "Let $σ(t)=\\operatorname{sl}(\\varpi t)$ be the lemniscatic sine, normalized to have period $2$. We study the zero set $\\mathscr{M}$ of $S(x)=σ(x_1)σ(x_3)-σ(x_2)σ(x_4)$ in $\\mathbb{R}^{4}$. We show that $\\mathscr{M}$ is a complete embedded minimal hypersurface with infinitely many isolated singularities, which are exactly the vertices of the regular honeycomb of $\\mathbb{R}^{4}$ by $24$-cells. Near every singular point $\\mathscr{M}$ is an analytic image of the Clifford cone, and $\\mathscr{M}$ is a stationary integral varifold with density $π/2$ at each singularity. The isometry group of $\\mathscr{M}$ is generated by the translations of the root lattice $D_4$ and a hyperoctahedral group of order $384$ fixing the centre of a $24$-cell; it has index three in the symmetry group of the honeycomb, acts transitively on the singular points, and contains a symmetry which is not visible from the defining equation. Seen from the centre of a $24$-cell, $\\mathscr{M}$ is the zero set of a theta series of a translate of the Hurwitz quaternions, which makes the whole symmetry group manifest; the sections of $\\mathscr{M}$ by a family of parallel hyperplanes are triply periodic surfaces with cubic symmetry which change topology through conical points. Finally, $\\mathscr{M}$ is obtained from a union of $2$-planes by attaching cones over $2$-tori, and its quotient by the lattice of periods is a compact minimal hypersurface in a flat $4$-torus with exactly three singular points and infinite Morse index; we show that its regular part is one-sided and compute its Euler characteristic and integral homology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37453",
  "author_total": 1,
  "authors": [
    {
      "name": "Vladimir G. Tkachev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fbka2ht1f2tx2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37453",
  "fulltext_url": null,
  "id": "fbka2ht1f2tx2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37453",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7224109803",
  "pdf_url": "https://arxiv.org/pdf/2610.37453",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fbka2ht1f2tx2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37453",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37453"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities",
  "title_text": "A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/fbka2ht1f2tx2",
  "venue": "arXiv",
  "versions": []
}