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The Solomon-Yau least-volume conjecture for minimal hypersurfaces of round spheres

The equator is the least-volume closed minimal hypersurface of the unit sphere Sm+1S^{m+1}. The minimal Clifford products Sk(k/m)×Sm−k((m−k)/m)S^k(\sqrt{k/m})\times S^{m-k}(\sqrt{(m-k)/m}), 1≤k<m1\le k<m, are the natural next examples; let ama_m be the least of their volumes. Cheng, Li and Yau (1984) proved a dimension-dependent volume gap above the equator, and for m=2m=2 the sharp bound 2π22\pi^2 follows from the Marques-Neves proof of the Willmore conjecture. Yau's problem list (Problem 31) asks for the next volume level, and Ge and Li (2021, Conjecture 1.12(ii)) state it as the Solomon-Yau conjecture. For every m≥2m\ge2, does every closed minimal hypersurface of Sm+1S^{m+1} that is not totally geodesic have volume at least ama_m?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Minimal hypersurfaces in spheres; volume gaps
Posed by
S.-T. Yau, Open problems in geometry (1994), Problem 31; formulated as the Solomon-Yau conjecture by J. Ge and F. Li (2021, Conjecture 1.12(ii))
Year posed
1994
Years open
32y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for m≥2m\ge2, a closed connected smooth minimal immersion F:Mm→Sm+1F:M^m\to S^{m+1} with non-totally-geodesic image has Vol(M,F∗g)≥am\mathrm{Vol}(M,F^*g)\ge a_m, attained by a minimizing Clifford product. Intermediate results: ∫Σ∣A∣2≥m∣Σ∣\int_\Sigma|A|^2\ge m|\Sigma| for closed embedded non-equatorial minimal hypersurfaces (the lower-bound part of Perdomo's conjecture, also in Nifa 2026) and Clifford rigidity at index m+3m+3. It does not classify hypersurfaces at volume ama_m beyond attainment, and says nothing about higher codimension or the next volume levels.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Yau's Problem 31 and Ge-Li Conjecture 1.12(ii). It covers every m≥2m\ge2 and smooth closed connected minimal immersions, with volume counted on the domain; the reduction from immersed to embedded uses known multiplicity bounds (Nguyen 2023, Ge-Li 2022). The release has no Lean formalization for this family (lean/docs/349.md does not exist at the pinned commit). The proof is a min-max argument with a dimension induction on cone regularity; it was not refereed here. The same paper proves the lower bound in Perdomo's average-curvature conjecture as a step, which Nifa (2026 preprint) also proves.

Sources

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