Nonnegative-curvature Einstein classification and an L² topological gap

The explanation

Start with two round spheres of radius 1/√3, about 0.577. Each sphere bends by 1/r², which is 3. Now take their product, the four-dimensional space of pairs of points, one on each sphere. A flat piece of a plane inside it can lie along one sphere, or take one direction from each. Along one sphere the bending, called sectional curvature, is 3. Across the two spheres it is 0. Ricci curvature in a direction adds the sectional curvatures of the three planes through it. For a direction along the first sphere this gives 3 + 0 + 0 = 3. Every direction gives 3. A space where Ricci curvature is the same constant in every direction is called Einstein. The round four-sphere of radius 1 is also Einstein: each plane bends by 1, and 1 + 1 + 1 = 3.

Which closed four-dimensional Einstein spaces have no negative bending at all? Closed means finite in size with no edge. In 2000 Yang listed three models for strictly positive bending as a conjectureSource 1. They are the round sphere, the round real projective space, and the complex projective plane with its Fubini–Study metric. After scaling to Ricci curvature 3, Fubini–Study bends between 0.5 and 2. Earlier proofs needed extra conditions. Gursky and LeBrun needed a condition on the topologySource 2. Cheng needed an Euler characteristic of at most 3Source 3. The third paper proves Yang's classification with no such conditionSource 4. It needs no orientability, so mirror-reversing spaces are includedSource 5.

The main obstacle is in four dimensions only. There the curvature splits into two independent pieces called Weyl blocks. The Einstein equation does not force either block to vanishSource 6. The paper shows that one block must vanish, and classical results then name the space. The first paper studies the boundary case, where some plane has zero bending, as in the two-sphere example. An estimate proved for strictly positive bending does not settle that case by itselfSource 7. It claims that one zero-bending plane forces the universal cover to be the product of two spheres of radius 1/√3Source 8. The universal cover is the space unwrapped so that no loop is stuck. Together this gives three models up to covering: the sphere, the complex projective plane, and the product of two spheresSource 9.

The second paper asks about spaces that are nearly Einstein. It measures the trace-free Ricci energy, the squared gap between Ricci curvature and its average, summed over the space. In four dimensions this number does not change when the space is scaled. For spheres of radii 1 and 2, the product has energy 16π²(2 − 1/2)², about 355.3. Radii 10 and 20 give the same 355.3. The paper claims a universal number ε₀ above zero. A simply connected space with no negative bending and energy below ε₀ has the smooth shape of one of the three modelsSource 10. It needs no bound on volume, diameter or curvature sizeSource 11. The conclusion is about the smooth shape, not the metricSource 12. The proof smooths the metric with Ricci flow, an equation that evens out curvature over timeSource 13.

How sure is this? The Lean scope note says the formalization classifies closed Einstein four-manifolds with strictly positive sectional curvature. That covers the third paper's theorem. The zero-plane theorem and the energy gap are not in that note. No specialist review has been published.

What it doesn't show. This does not give a value for the gap constant ε₀, and the gap result says nothing about the metric itself, only the smooth shape. Only the strictly positive classification has a Lean check.