Nonnegative-curvature Einstein classification and an L^{2} topological gap

Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S⁴, Fubini–Study \mathbb{CP}^2, or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L² norm of its trace-free Ricci curvature lies below a universal positive constant. (Lean)

Manuscripts (3)

Zero-Plane Rigidity for Einstein Four-Manifolds

October 4, 2026 · source · cite as \cite{OAI:Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026}

We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization \mathop{\mathrm{Ric}}\nolimits =3g, both spheres have radius 1/\sqrt3. The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.

An L² Einstein Gap for Nonnegatively Curved Four-Manifolds

October 5, 2026 · source · cite as \cite{OAI:An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026}

We prove a universal, scale-invariant L² gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S⁴, \mathbb{CP}^2, or S^2\times S^2. The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result. No auxiliary curvature, volume, diameter, injectivity-radius, or Sobolev bound is required. The conclusion concerns the smooth manifold, not an isometry of the original metric.

Positively curved Einstein four-manifolds

September 23, 2026 · source · cite as \cite{OAI:Positively-curved-Einstein-four-manifolds-September-23-2026}

We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.

Result 348 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.