A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities
Abstract
Let σ(t)=sl(ϖ t) be the lemniscatic sine, normalized to have period $2$. We study the zero set ℳ of S(x)=σ(x1)σ(x3)−σ(x2)σ(x4) in ℝ4. We show that ℳ is a complete embedded minimal hypersurface with infinitely many isolated singularities, which are exactly the vertices of the regular honeycomb of ℝ4 by $24$-cells. Near every singular point ℳ is an analytic image of the Clifford cone, and ℳ is a stationary integral varifold with density π/2 at each singularity. The isometry group of ℳ is generated by the translations of the root lattice D4 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, 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 ℳ by a family of parallel hyperplanes are triply periodic surfaces with cubic symmetry which change topology through conical points. Finally, 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.
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BibTeX
@misc{tkachev2026periodic,
title = {{A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities}},
author = {Vladimir G. Tkachev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37453},
eprint = {2610.37453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities
AU - Vladimir G. Tkachev
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37453
UR - https://arxiv.org/abs/2610.37453
AB - Let σ(t)=sl(ϖ t) be the lemniscatic sine, normalized to have period $2$. We study the zero set ℳ of S(x)=σ(x_1)σ(x_3)−σ(x_2)σ(x_4) in ℝ^4. We show that ℳ is a complete embedded minimal hypersurface with infinitely many isolated singularities, which are exactly the vertices of the regular honeycomb of ℝ^4 by $24$-cells. Near every singular point ℳ is an analytic image of the Clifford cone, and ℳ is a stationary integral varifold with density π/2 at each singularity. The isometry group of 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, 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 ℳ by a family of parallel hyperplanes are triply periodic surfaces with cubic symmetry which change topology through conical points. Finally, 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.
ER -