Browse › Differential geometry › 348 Nonnegative-curvature Einstein classification and an L2 topological gap
An L² Einstein Gap for Nonnegatively Curved Four-Manifolds
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Abstract
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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- S.-Y. A. Chang, M. J. Gursky, P. Yang, A conformally invariant sphere theorem in four dimensions (arXiv:math/0309287; Publ. Math. IHÉS 98 (2003)). Gives a diffeomorphism characterization of S⁴ from an L² Weyl-tensor inequality and positive Yamabe invariant; this is an earlier integral-smallness-implies-diffeomorphism-type result comparable to Theorem 1.1, and it is not cited.
- S.-Y. A. Chang, M. Gursky, S. Zhang, A conformally invariant gap theorem characterizing CP² via the Ricci flow (arXiv:1809.05918, 2018). Proves an integral gap theorem concluding diffeomorphism to CP², using Ricci flow to pass from integral bounds to pointwise curvature control, which parallels the Ricci-flow smoothing step here.
- J. Streets, A concentration-collapse decomposition for L² flow singularities (arXiv:1311.0961, 2013). Includes compactness and diffeomorphism-finiteness theorems for four-manifolds under L² curvature pinching and volume noncollapsing, related to the paper's passage from small critical-curvature energy to a diffeomorphism conclusion.
Digests and expositions
- Latent Seminar, Einstein four-manifolds: curvature rigidity and an L² topological gap (7 October 2026). Explains the three manuscripts’ dependencies, coupled Weyl estimates and Ricci-flow gap argument, and distinguishes its reported arithmetic checks from verification of the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 348
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026,
author = {{OpenAI}},
title = {{An $L^2$ Einstein Gap for Nonnegatively Curved Four-Manifolds}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf}{OAI:An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/348.md. The Lean statements have not been checked by people against the paper's claims.
Family 348: Nonnegative-curvature Einstein classification and an L2 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.
Other papers in this family:
- Zero-Plane Rigidity for Einstein Four-Manifolds · October 4, 2026
- Positively curved Einstein four-manifolds · September 23, 2026