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.

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.