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Math-2658 · mathematics · Geometry

Conditional L² Einstein gap for nonnegative curvature

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assume that every connected smooth closed Riemannian four-manifold (N,h) with $Ric_h=3h$ and $sec_h\geq 0$ has universal Riemannian cover isometric to round $S^{4}$ of curvature 1, Fubini–Study $CP^{2}$ with $Ric=3g$, or $S^{2}(1/\surd 3)×S^{2}(1/\surd 3)$. Does there then exist $\varepsilon >0$ such that every simply connected closed smooth Riemannian four-manifold (M,g) with $sec_g\geq 0$ and $\int |Ric_g-(Scal_g/4)g|^{2}dvol_g<\varepsilon $ is diffeomorphic to $S^{4}$, $CP^{2}$ or $S^{2}×S^{2}$, without a prescribed orientation?

Short English statement

Assume that every connected smooth closed Riemannian four-manifold (N,h) with $Ric_h=3h$ and $sec_h\geq 0$ has universal Riemannian cover isometric to round $S^{4}$ of curvature 1, Fubini–Study $CP^{2}$ with $Ric=3g$, or $S^{2}(1/\surd 3)×S^{2}(1/\surd 3)$. Does there then exist $\varepsilon >0$ such that every simply connected closed smooth Riemannian four-manifold (M,g) with $sec_g\geq 0$ and $\int |Ric_g-(Scal_g/4)g|^{2}dvol_g<\varepsilon $ is diffeomorphic to $S^{4}$, $CP^{2}$ or $S^{2}×S^{2}$, without a prescribed orientation?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2658-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2658

A topological Einstein-energy gap: Under exactly ass:classification (all closed Ric=3g nonnegative-sectional-curvature universal covers isometrically standard), the stated result gives the universal scale-invariant L${}^{2}$ threshold and unoriented diffeomorphism conclusion. The original weaker premise was not established.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Conditional L${}^{2}$ Einstein gap for nonnegative curvature

OpenAI's claim

A topological Einstein-energy gap: Assuming the manuscript's normalized classification of nonnegatively curved Einstein four-manifolds, a sufficiently small scale-invariant $L^2$ trace-free Ricci energy is claimed to force a simply connected closed nonnegatively curved four-manifold to be diffeomorphic to $S^4$, $\mathbb{CP}^2$ or $S^2\times S^2$. No additional geometric bounds are imposed.

Mathematical statement of the claimed result

Mathematical claim 1.
Assume the companion classification: every closed connected smooth four-manifold with $\operatorname{Ric}=3g$ and sectional curvature $\ge0$ has universal cover isometric to round $S^4$, Fubini--Study $\mathbb{CP}^2$ normalized by $\operatorname{Ric}=3g$, or $S^2(1/\sqrt3)\times S^2(1/\sqrt3)$. Under that explicit premise, the claim gives an absolute $\epsilon_0>0$ such that any closed connected simply connected smooth four-manifold with $\sec_g\ge0$ and \[\int_M\left|\operatorname{Ric}_g-\tfrac14\operatorname{Scal}_g g\right|^2\,dV_g<\epsilon_0\] is diffeomorphic, without specified orientation, to $S^4$, $\mathbb{CP}^2$ or $S^2\times S^2$. The conclusion is topological rather than isometric, and the theorem is expressly conditional on the classification premise.

Outline of the argument

A topological Einstein-energy gap: A hypothetical countersequence is normalized and shown to satisfy initial geometry and Sobolev bounds. A uniform smoothing interval controls its high-curvature region and propagates the Einstein defect and negative-sectional-curvature error. The limit is a compact nonnegatively curved positive-Einstein space, whose companion classification identifies the topology and contradicts the sequence.

Formal verification

Family 348 describes Lean coverage for the strict-positive-curvature Einstein classification; zero-plane rigidity and the small-energy gap are separate.

A topological Einstein-energy gap: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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