https://openai-aftermath.mathnt.net/index.php?action=history&feed=atom&title=The_Solomon%E2%80%93Yau_least-volume_conjecture 2026-10-16T09:02:40Z Revision history for this page on the wiki MediaWiki 1.46.2 https://openai-aftermath.mathnt.net/index.php?title=The_Solomon%E2%80%93Yau_least-volume_conjecture&diff=185&oldid=prev 2026-10-16T03:34:56Z

<p>New page: OpenAI&#039;s Solomon-Yau least-volume claim (manuscript; no Lean scope note), J. Chen&#039;s Allen-Cahn/Brakke flow use, Nifa&#039;s cited related preprint</p> <p><b>New page</b></p><div>The &#039;&#039;&#039;Solomon–Yau least-volume conjecture&#039;&#039;&#039; asserts that, after the totally geodesic equator, the smallest volume of a closed minimal hypersurface of the unit round sphere &#039;&#039;S&#039;&#039;&lt;sup&gt;&#039;&#039;m&#039;&#039;+1&lt;/sup&gt; is attained by a minimal Clifford product &#039;&#039;S&#039;&#039;&lt;sup&gt;&#039;&#039;k&#039;&#039;&lt;/sup&gt;(√(&#039;&#039;k&#039;&#039;/&#039;&#039;m&#039;&#039;)) × &#039;&#039;S&#039;&#039;&lt;sup&gt;&#039;&#039;m&#039;&#039;−&#039;&#039;k&#039;&#039;&lt;/sup&gt;(√((&#039;&#039;m&#039;&#039;−&#039;&#039;k&#039;&#039;)/&#039;&#039;m&#039;&#039;)). A manuscript dated 23 September 2026 in OpenAI&#039;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.<br /> <br /> == Background ==<br /> * According to the manuscript&#039;s introduction, the question is Problem 31 in Yau&#039;s 1994 problem collection, and Ge and Li&#039;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&#039;s proof of the Willmore conjecture.<br /> * The same introduction lists earlier higher-dimensional progress: rotational results of Cheng, Wei and Zeng, Viana&#039;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.<br /> <br /> == The original result ==<br /> * &#039;&#039;&#039;Claim.&#039;&#039;&#039; For every &#039;&#039;m&#039;&#039; ≥ 2, a smooth minimal immersion of a closed connected &#039;&#039;m&#039;&#039;-manifold into the unit sphere &#039;&#039;S&#039;&#039;&lt;sup&gt;&#039;&#039;m&#039;&#039;+1&lt;/sup&gt; whose image is not totally geodesic has volume at least &#039;&#039;a&#039;&#039;&lt;sub&gt;&#039;&#039;m&#039;&#039;&lt;/sub&gt;, 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.<br /> * &#039;&#039;&#039;Source.&#039;&#039;&#039; [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026 &#039;&#039;The Solomon–Yau least-volume theorem&#039;&#039;] in the openai/math repository (commit &lt;code&gt;1581d4c&lt;/code&gt;).<br /> * &#039;&#039;&#039;Formalization.&#039;&#039;&#039; The repository&#039;s contents list gives no Lean scope note for this family, and no Comparator statement is listed for it.<br /> <br /> == Public follow-up work ==<br /> {| class=&quot;wikitable&quot;<br /> ! Date !! Work !! Relation !! Public claim !! Public status (checked 16 October 2026)<br /> |-<br /> | 2026-10-15 (listed) || J. Chen, &#039;&#039;Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators&#039;&#039; ([https://arxiv.org/abs/2610.20453 arXiv:2610.20453]) || Use || &quot;Using OpenAI&#039;s resolution of the Solomon–Yau conjecture&quot;, extends earlier joint work with P. Gaspar to all dimensions &#039;&#039;n&#039;&#039; ≥ 2: eternal Allen–Cahn flows on &#039;&#039;S&#039;&#039;&lt;sup&gt;&#039;&#039;n&#039;&#039;+1&lt;/sup&gt; 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.<br /> |}<br /> <br /> == Related work ==<br /> * A. Nifa, &#039;&#039;Sharp Schrödinger inequalities for embedded hypersurfaces&#039;&#039; (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&#039;s.<br /> <br /> == Notes ==<br /> * The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.<br /> <br /> [[Category:OpenAI results]]</div>

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