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Math-2668 · mathematics · Geometry

Bounded coordinates for negatively pinched Kähler manifolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must a contractible complex manifold with a complete smooth Kähler metric satisfying $-B\leq sec\leq -A<0$ for finite positive $A\leq B$ admit a bounded holomorphic map into $C^{n}$ with everywhere nonzero complex Jacobian, where n is its complex dimension?

Short English statement

Must a contractible complex manifold with a complete smooth Kähler metric satisfying $-B\leq sec\leq -A<0$ for finite positive $A\leq B$ admit a bounded holomorphic map into $C^{n}$ with everywhere nonzero complex Jacobian, where n is its complex dimension?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2668-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2668

Pinched Kähler uniformization obstruction: Two-sided negative sectional pinching and completeness are retained in the threefold example. Its failure of bounded coordinates directly prevents a biholomorphism to a bounded domain, without needing the stronger assertion that every bounded holomorphic function is constant.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Bounded coordinates for negatively pinched Kähler manifolds

OpenAI's claim

Pinched Kähler uniformization obstruction: A contractible complex threefold is claimed to have a complete Kähler metric with sectional curvature between two negative constants but no bounded holomorphic coordinate system. In particular, it is not biholomorphic to a bounded domain.

Mathematical statement of the claimed result

Mathematical claim 1.
The claimed example is a contractible domain $M\subset\mathbb C^3$ with a smooth complete Kähler metric and finite $0<A\le B$ satisfying \[-B\le K_g(\sigma)\le-A<0\] for all real tangent two-planes. Every bounded holomorphic $F:M\to\mathbb C^3$ has a Jacobian determinant vanishing somewhere. In particular $M$ cannot be biholomorphic to any bounded domain in $\mathbb C^3$. Both negative sectional-curvature pinching bounds are present.

Outline of the argument

Pinched Kähler uniformization obstruction: Disk fibers over the complex two-ball have radii prescribed by a smooth base function. The Hessian of a logarithmic fiber potential gives the Kähler metric, with explicit control of completeness and two-sided real sectional curvature. The chosen radius behavior imposes a holomorphic obstruction preventing a bounded map with nonvanishing Jacobian.

Formal verification

Family 359 describes Lean coverage for the two-sided negatively pinched threefold obstruction; the one-sided Liouville construction is separate.

Pinched Kähler uniformization obstruction: The manuscript links PinchedKahler.lean (declaration OAI.PinchedHartogs.main_theorem).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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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