A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities

Abstract

Let $\sigma(t)=\operatorname{sl}(\varpi t)$ be the lemniscatic sine, normalized to have period $2$. We study the zero set $\mathscr{M}$ of $S(x)=\sigma(x_1)\sigma(x_3)-\sigma(x_2)\sigma(x_4)$ 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 $\pi/2$ 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.