Negative Kähler curvature without bounded holomorphic coordinates

2 papersLean

Constructs a contractible domain in ℂ³ with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

A negatively pinched Kähler threefold without bounded holomorphic coordinates

OpenAI · September 25, 2026

We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ³ with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.

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Cite

@misc{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026,
  author = {{OpenAI}},
  title = {{A negatively pinched K{\"a}hler threefold without bounded holomorphic coordinates}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf}{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026}},
  year = {2026}
}

One-sided negative sectional curvature and the holomorphic Liouville property

OpenAI · September 25, 2026

We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂ^m diffeomorphic to \(\mathbb R^{2m}\) that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.

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Cite

@misc{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026,
  author = {{OpenAI}},
  title = {{One-sided negative sectional curvature and the holomorphic Liouville property}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf}{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

A negatively pinched Kähler threefold without bounded holomorphic coordinates

Scope

The formalized result constructs a nonempty contractible complex threefold with a complete Kähler metric whose real sectional curvatures lie between two fixed negative constants. Nevertheless, it has no system of bounded holomorphic coordinates and is not biholomorphic to a bounded domain. The metric is obtained from an explicitly controlled potential. This is the negatively pinched construction; the different companion with only one-sided curvature control is separate.

Comparator links

Result	Comparator statement
Negatively pinched Kähler threefold without bounded coordinates	PinchedKahler.lean

Scope note on GitHub →