White's cone over the Möbius band is area-minimising

Abstract

We prove that the cone $C(\Sigma )$ in ${\mathbb{R}}^4$ over a suitable minimal Möbius band $\Sigma \subset {\mathbb{S}}^3$ is locally mass-minimising with coefficients in ${\mathbb{Z}}_2$. This proves a conjecture by Brian White in the context of models of boundary singularities of Plateau's problem in ambient dimension $4$ or higher. The key insight is that the cylindrical slices $C(\Sigma )\cap \{(z,w)\in {\mathbb{R}}^2\times {\mathbb{R}}^2:|w|=r\}$ are minimal surfaces in a flat rank-one translation quotient of ${\mathbb{R}}^3$. In fact, they correspond to non-orientable minimising helicoids previously studied by Antonio Ros. We also show that, up to rigid motions, the minimising cones $C(\Sigma )\times {\mathbb{R}}^{n-4}\subset {\mathbb{R}}^n$ are the only non-planar minimal hypercones with boundary ${\mathbb{R}}^{n-2}$ which are smooth away from the boundary that satisfy the minimal cylindrical slice property.

arXiv comment14 pages, 1 figure