Family 348Differential geometryNo LeanNot yet digested
Nonnegative-curvature Einstein classification and an L 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 S, Fubini–Study , 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 L norm of its trace-free Ricci curvature lies below a universal positive constant.
Manuscripts in this family · 3
Zero-Plane Rigidity for Einstein Four-Manifolds
Oct 4, 2026·einstein-boundary.pdf
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 , both spheres have radius . 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
Oct 5, 2026·einstein-gap.pdf
We prove a universal, scale-invariant L 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 S, , or . 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
Sep 23, 2026·paper.pdf
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.