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

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

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Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Solomon-Yau least-volume conjecture for minimal hypersurfaces of round spheres · Solomon-Yau least-volume conjecture

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

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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 $m\ge2$ 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.

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

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Ge and Li, A lower bound for L2 length of second fundamental form on minimal hypersurfaces (Conjecture 1.12)

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Ge and Li, A lower bound for L2 length of second fundamental form on minimal hypersurfaces (Conjecture 1.12) evidence for this event

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