The Solomon–Yau least-volume conjecture

The explanation

Start with the ordinary round sphere of radius 1 in three-dimensional space. Its equator is a great circle. Now go up one dimension. Inside the round three-sphere, which sits in four-dimensional space, the equator is a round two-sphere of area 4π, about 12.57. Another surface there is the Clifford torus. It is the set of points whose first two coordinates and last two coordinates both have length 1/√2. It is a product of two circles of radius 1/√2, so its area is (2π/√2)², which is 2π², about 19.74.

Both surfaces are minimal. A minimal hypersurface is balanced like a soap film: a small push does not shrink it to first order. A hypersurface has one dimension less than the sphere around it. The equator is the smallest closed minimal hypersurface. The question is which one comes next. The natural guesses are Clifford products, which multiply two round spheres of well-chosen radii. Two dimensions higher, the smallest product is two round two-spheres of radius 1/√2. Its volume is 4π², about 39.48, against 8π²/3, about 26.32, for the equator.

The claim that the smallest Clifford product is the next level is Yau's Problem 31Source 1. Ge and Li call it the Solomon–Yau conjectureSource 2. The paper claims to prove it in every dimension m of at least 2Source 3. Volume is counted on the original surface, so a surface that wraps twice counts twiceSource 4. The smallest Clifford product reaches the bound exactlySource 5.

What was known before? Cheng, Li and Yau found some gap above the equator, but not the sharp oneSource 6. In dimension two the sharp bound of 2π² follows from Marques and Neves' proof of the Willmore conjectureSource 7. Surfaces that cross themselves are easy. A point covered q times forces volume at least q times the equatorSource 8. Since the Clifford bound is below twice the equator, a counterexample would have no self-crossings.

The main difficulty is to take a smallest counterexample and then find an even smaller oneSource 9. Limits of surfaces can develop singular points, and a family of surfaces can collapse to an equatorSource 10. The paper first proves an average curvature bound that Perdomo had conjecturedSource 11. From it, a counterexample must have an extra direction in which its area can drop. It then builds a large family of surfaces using Lorentz boosts, the changes of frame from relativitySource 12. A topological test stops this family from collapsing. Induction on the dimension rules out singular points below the Clifford levelSource 13. Nifa's recent preprint also contains the average curvature boundSource 14.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof combines curvature estimates, a large topological family and an induction on dimension. No specialist review of the paper has been published.

What it doesn't show. This does not classify all minimal hypersurfaces, or say which ones attain the bound apart from a Clifford product. It has no Lean check.