Browse › Differential geometry › 348 Nonnegative-curvature Einstein classification and an L2 topological gap
Zero-Plane Rigidity for Einstein Four-Manifolds
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Abstract
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.
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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)
- Z. Zhang, Four-dimensional Einstein manifolds with sectional curvature bounded from above (arXiv:1606.01157, 2016). Works under nonnegative sectional curvature with Ric normalized and proves rigidity (S⁴, RP⁴, CP²) under an extra pinching condition; this is a rigidity result in the nonnegative setting of Corollary 1.2 that is not cited.
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:Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026,
author = {{OpenAI}},
title = {{Zero-Plane Rigidity for Einstein Four-Manifolds}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf}{OAI:Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
The exact arithmetic checker requires Python 3.10 or later and only the standard library. From this paper directory, run:
python3 -B verification/verify.py --check
Success means exit status zero and all_passed: true, with 53 identities,
21 univariate certificate rows (15 reported and 6 supplementary), 12 triangle
certificate rows, and 24 exact-value assertions. The checker compares its
results with verification/results.json; its scope is documented in
verification/COVERAGE.md. It checks finite arithmetic, not the geometric
and analytic arguments.
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:
- An L² Einstein Gap for Nonnegatively Curved Four-Manifolds · October 5, 2026
- Positively curved Einstein four-manifolds · September 23, 2026