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Math-2666 · mathematics · Geometry

Uniformization with positive bisectional curvature

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is every connected noncompact complex n-manifold, $n\geq 1$, carrying a complete smooth Kähler metric with strictly positive holomorphic bisectional curvature biholomorphic to $C^{n}$?

Short English statement

Is every connected noncompact complex n-manifold, $n\geq 1$, carrying a complete smooth Kähler metric with strictly positive holomorphic bisectional curvature biholomorphic to $C^{n}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2666-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2666

The uniformization theorem applies to every connected noncompact complex $n$-manifold, $n\ge1$, with a smooth complete Kähler metric of strictly positive holomorphic bisectional curvature. It claims biholomorphism with $\mathbb C^n$ without extra curvature bounds or volume-growth conditions.

Uniformization with positive bisectional curvature

OpenAI's claim

Kähler uniformization: For a complete noncompact Kähler metric whose bisectional curvature is strictly positive, the claimed conclusion identifies the complex manifold with Euclidean complex space by a biholomorphism. No complex-dimension bound is imposed.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a complex dimension $n\ge1$ and a complex manifold $M$ which is connected but not compact. If it has a smooth complete Kähler metric whose holomorphic bisectional curvature is strictly positive, then the claimed uniformization is \[M\simeq_{\mathrm{bihol}}\mathbb C^n.\] Positivity is pointwise; no bounded-curvature or maximal-volume-growth assumption appears in the theorem.

Outline of the argument

Kähler uniformization: Complete Hermitian approximations on an exhaustion produce a global Kähler--Ricci flow without an initial global curvature bound. A weighted scalar comparison is proved before the completeness needed later is available. Holomorphic-disc minimization on the cotangent bundle then gives a viscosity inequality and a Harnack mechanism, supplying the flow geometry used to construct the global holomorphic uniformization.

Formal verification

Kähler uniformization: The pinned catalogue lists no Lean formalization for this result.

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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