Let Sigma⊂ S^3 be the minimal Möbius band specified in White's conjecture, and let C(Sigma)=tx:t≥ 0, x∈Sigma⊂ R^4. The conjecture asserts that C(Sigma) is locally mass-minimizing with coefficients in Z_2: for every relatively compact open set U⊂ R^4 and every 2-dimensional Z_2-current T agreeing with C(Sigma) outside U, one has mathbf M(T llcorner U)≥mathbf M(C(Sigma)llcorner U) whenever the comparison is defined.

White’s Möbius-band cone conjecture

Let $\Sigma\subset\mathbb S^3$ be the minimal Möbius band specified in White's conjecture, and let $C(\Sigma)=\{tx:t\geq 0,\ x\in\Sigma\}\subset\mathbb R^4$ . The conjecture asserts that $C(\Sigma)$ is locally mass-minimizing with coefficients in $\mathbb Z_2$ : for every relatively compact open set $U\subset\mathbb R^4$ and every $2$ -dimensional $\mathbb Z_2$ -current $T$ agreeing with $C(\Sigma)$ outside $U$ , one has $\mathbf M(T\llcorner U)\geq\mathbf M(C(\Sigma)\llcorner U)$ whenever the comparison is defined.

References

Primary source

arXiv

Additional references

White's cone over the Möbius band is area-minimising — arXiv — Marco A. M. Guaraco, Davide Parise

geometric-measure-theory minimal-surfaces

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture by proving the proposed cone is minimizing and classifying its higher-dimensional extensions, but this has not been independently verified.

White’s conjecture proposes a singular cone over the Möbius band as a model for Plateau-type boundary problems in dimensions four and above. Guaraco and Parise claim to prove the conjecture and derive the associated higher-dimensional classification.

September 21, 2026 claimed proof

Guaraco and Parise claim that the specified cone is locally mass-minimizing with coefficients in $\mathbb{Z}_2$ . They also claim that $C(\Sigma)\times\mathbb{R}^{n-4}$ is area-minimizing for $n\geq 4$ , and that the stated cylindrical-slice hypotheses force either a half-hyperplane or this product cone. No independent verification, proof-gap report, withdrawal, or counterexample was found in the retrieved sources.

Current status (as of September 2026): A preprint claims the conjecture is solved modulo $2$ , including the higher-dimensional classification, but the result remains unverified.

Sources

No solutions have been posted yet.