Result 349, Differential geometry

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 Sm+1S^{m+1} with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.

Proof

The bigger picture

Why it matters

How small can a closed surface be if its curvature balances everywhere inside a round sphere? The manuscript reports a sharp answer in every dimension at least two, once equatorial spheres are excluded.

What changes?

For every dimension m at least two, the manuscript claims that a closed, connected minimal immersion into the unit round sphere of dimension m+1, whose image is not totally geodesic, has volume at least the smallest m-dimensional minimal Clifford product. Here closed means compact without boundary; minimal means zero mean curvature, not necessarily least volume. Totally geodesic images are equatorial spheres. Clifford products combine two round spheres. Immersions allow overlaps, and volume counts covering multiplicity rather than just the image.

What does that help mathematicians do?

The claimed bound rules out any non-equatorial minimal hypersurface below a specific geometric benchmark, including immersed examples rather than only embedded ones. For an immersion satisfying the stated assumptions, an independent volume estimate below that benchmark would force its image to be totally geodesic. The smallest Clifford product attains the benchmark, but the supplied statement does not classify all equality cases.

Are there practical applications?

The immediate value is foundational in differential geometry: the result would give a universal constraint on the volume of closed minimal hypersurfaces in round spheres. It separates the local condition of balanced curvature from a global restriction on size. The supplied material describes no practical deployment or computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Solomon–Yau least-volume theorem

September 23, 2026 32 pages

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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Cite (BibTeX) 
@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}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.