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The Solomon–Yau least-volume theorem
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Abstract
We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m-dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.
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Follow-up works
- Jingwen Chen, Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators (arXiv:2610.20453, 2026-10-14). Uses "OpenAI's resolution of the Solomon–Yau conjecture" (cited as [23], "for n≥3") to extend constructions of eternal Allen–Cahn and integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres to all dimensions n ≥ 2; it does not discuss verification of the proof.
Related prior literature mentioned by others
- Jingwen Chen and Pedro Gaspar, Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (arXiv:2107.09140, 2021-07-19). Chen's October 2026 follow-up identifies this construction of eternal Brakke flows from Clifford tori to equatorial spheres in S³ as the earlier work it extends using the Solomon–Yau theorem of this paper.
- Jingwen Chen and Pedro Gaspar, Morse theory for the Allen-Cahn functional (arXiv:2310.17096, 2023-10-26). Chen's October 2026 follow-up identifies this Morse-theoretic construction of connecting Allen–Cahn flows on S³ as another input to its extension.
Possibly relevant literature (found by the agent, not checked)
- Tom Ilmanen and Brian White, Sharp lower bounds on density of area-minimizing cones (arXiv:1010.5068, Camb. J. Math. 3 (2015)). Proves sharp density lower bounds for topologically nontrivial area-minimizing hypercones, with Simons cones as extremal case; possibly relevant to the paper's cone density |Σ|/|S^m| discussion and cone-induction section.
- Hang Chen and Peng Wang, On the index of minimal hypersurfaces in S^{n+1} with λ1<n (arXiv:2405.10843, May 2024). Proves Morse index at least n+4 when λ1 < n, partially answering a conjecture of Perdomo; possibly relevant to the paper's index-rigidity step ("index m+3 forces a Clifford product"), which builds on Perdomo 2004.
Digests and expositions
- Latent Seminar, The Solomon–Yau least-volume problem: Clifford products and min–max (2026-10-07). Explains the manuscript's immersion reduction, curvature and index arguments, Lorentz-distance family, and cone induction, and describes itself as an exposition rather than an independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 349
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Solomon-Yau-least-volume-theorem-September-23-2026,
author = {{OpenAI}},
title = {{The Solomon--Yau least-volume theorem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf}{OAI:The-Solomon-Yau-least-volume-theorem-September-23-2026}},
year = {2026}
}Family 349: The Solomon–Yau least-volume conjecture
Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere $S^{m+1}$ with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.
This is the only paper in this family.