White's cone over the Möbius band is area-minimising
Abstract
We prove that the cone C(Σ) in ℝ4 over a suitable minimal Möbius band Σ⊂ 𝕊3 is locally mass-minimising with coefficients in ℤ2. 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(Σ)∩ {(z,w)∈ ℝ2 × ℝ2 : |w|=r} are minimal surfaces in a flat rank-one translation quotient of ℝ3. 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(Σ)× ℝn−4⊂ ℝn are the only non-planar minimal hypercones with boundary ℝn−2 which are smooth away from the boundary that satisfy the minimal cylindrical slice property.
Related
- Pin Equivariance and Local Dirac Boundary Problems on the Flat Mobius Band
- Primes and the Möbius function over quadratic polynomials
- Chowla Conjecture for the truncated Möbius function over 𝔽q[t]
- Short sums of the Möbius function over function fields
- A Formal Proof of Riemann Hypothesis from Existence Alone, the Sign Excepted by Theorem: The Constructed Witness, the Identity Cone over the Fixed Locus, the Grounded Counter-Model, and the Cure Theorem Proved in Lean 4 and Re-executed in Fortran
BibTeX
@misc{guaraco2026whites,
title = {{White's cone over the Möbius band is area-minimising}},
author = {Marco A. M. Guaraco and Davide Parise},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24909},
eprint = {2609.24909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24909},
}
RIS
TY - UNPB
TI - White's cone over the Möbius band is area-minimising
AU - Marco A. M. Guaraco
AU - Davide Parise
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24909
UR - https://arxiv.org/abs/2609.24909
AB - We prove that the cone C(Σ) in ℝ^4 over a suitable minimal Möbius band Σ⊂ 𝕊^3 is locally mass-minimising with coefficients in 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(Σ)∩ {(z,w)∈ ℝ^2 × ℝ^2 : |w|=r} are minimal surfaces in a flat rank-one translation quotient of 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(Σ)× ℝ^n−4⊂ ℝ^n are the only non-planar minimal hypercones with boundary ℝ^n−2 which are smooth away from the boundary that satisfy the minimal cylindrical slice property.
ER -