View source for The Solomon–Yau least-volume conjecture

The '''Solomon–Yau least-volume conjecture''' asserts that, after the totally geodesic equator, the smallest volume of a closed minimal hypersurface of the unit round sphere ''S''<sup>''m''+1</sup> is attained by a minimal Clifford product ''S''<sup>''k''</sup>(√(''k''/''m'')) × ''S''<sup>''m''−''k''</sup>(√((''m''−''k'')/''m'')). A manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims a proof in every dimension. This page summarizes the published claim and lists publicly available follow-up work.

== Background ==
* According to the manuscript's introduction, the question is Problem 31 in Yau's 1994 problem collection, and Ge and Li's Conjecture 1.12(ii) calls it the Solomon–Yau conjecture. Cheng, Li and Yau obtained a dimension-dependent volume gap above the equator. In dimension two the sharp bound, the area of the Clifford torus, together with its rigidity, follows from Marques and Neves's proof of the Willmore conjecture.
* The same introduction lists earlier higher-dimensional progress: rotational results of Cheng, Wei and Zeng, Viana's bound when both complementary regions are antipodally invariant, and related spectral and density results. It also recalls, from work of Nguyen and of Ge and Li, that a non-embedded counterexample is impossible, so the question reduces to embedded hypersurfaces.

== The original result ==
* '''Claim.''' For every ''m'' ≥ 2, a smooth minimal immersion of a closed connected ''m''-manifold into the unit sphere ''S''<sup>''m''+1</sup> whose image is not totally geodesic has volume at least ''a''<sub>''m''</sub>, the least volume of the minimal Clifford products in that dimension (Theorem 1.1). Volume is measured on the domain, so covering multiplicities are included, and the bound is attained.
* '''Source.''' [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026 ''The Solomon–Yau least-volume theorem''] in the openai/math repository (commit <code>1581d4c</code>).
* '''Formalization.''' The repository's contents list gives no Lean scope note for this family, and no Comparator statement is listed for it.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 16 October 2026)
|-
| 2026-10-15 (listed) || J. Chen, ''Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators'' ([https://arxiv.org/abs/2610.20453 arXiv:2610.20453]) || Use || "Using OpenAI's resolution of the Solomon–Yau conjecture", extends earlier joint work with P. Gaspar to all dimensions ''n'' ≥ 2: eternal Allen–Cahn flows on ''S''<sup>''n''+1</sup> connecting approximations of any prescribed least-area Clifford hypersurface and an equatorial sphere, and infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres. || arXiv preprint.
|}

== Related work ==
* A. Nifa, ''Sharp Schrödinger inequalities for embedded hypersurfaces'' (Zenodo, [https://doi.org/10.5281/zenodo.22999024 doi:10.5281/zenodo.22999024], version 1). The OpenAI manuscript cites it: it states that Nifa obtains a stronger quadratic-form inequality and derives both the average-curvature bound and the embedded least-index classification, and that its own diagonal and distinct-point contact calculations are closely related to Nifa's.

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]