Browse › Differential geometry › 359 Negative Kähler curvature without bounded holomorphic coordinates
A negatively pinched Kähler threefold without bounded holomorphic coordinates
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Abstract
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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Notes
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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.
Formalization
- OpenAI math release, Lean scope note 359: Negative Kähler curvature without bounded holomorphic coordinates (release Lean documentation, 2026-10-06). The release's Lean scope note covers the main construction (a contractible threefold with a complete Kähler metric pinched between two negative constants and no bounded holomorphic coordinates), with the comparator statement in
lean/ComparatorChallenges/PinchedKahler.lean; it explicitly separates the one-sided companion paper from this scope.
Possibly relevant literature (found by the agent, not checked)
- F. Bracci, H. Gaussier, A. Zimmer, The geometry of domains with negatively pinched Kähler metrics (arXiv:1810.11389, 2018, revised 2022). Studies how a complete Kähler metric with pinched negative holomorphic bisectional curvature on a domain in C^d constrains its boundary, relevant to the paper's construction of a domain M ⊂ C^3 carrying such a metric.
- J. Cao, M.-C. Shaw, Bounded holomorphic functions on negatively curved Kähler manifolds of dimension ≥ 3 (arXiv:1512.00368, 2015, withdrawn 2016). This withdrawn preprint announced nonconstant bounded holomorphic functions under two-sided negative pinching in complex dimension at least 3, the setting in which the paper notes its example has two nonconstant bounded base-coordinate functions; it is cited in the companion paper but not in this paper's bibliography.
- Nikolay Shcherbina and Liyou Zhang, A Kobayashi and Bergman complete domain without bounded representations (arXiv:2003.06728, 2020-03-15). Their strictly pseudoconvex domain in complex dimension two has complete Kobayashi and Bergman metrics and no nonconstant bounded holomorphic functions, a comparison point for the paper's distinction between intrinsic-metric completeness and bounded holomorphic coordinates.
Digests and expositions
- Latent Seminar, Negative Kähler curvature: bounded coordinates and the holomorphic Liouville property (2026-10-07). This exposition compares the two constructions and their curvature hypotheses, outlines the Jacobian obstruction and tube disk-testing argument, and states that it does not independently verify the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 359
Independent sites; their content is not reviewed here.
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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/359.md. The Lean statements have not been checked by people against the paper's claims.
Family 359: Negative Kähler curvature without bounded holomorphic coordinates
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.
Other papers in this family:
- One-sided negative sectional curvature and the holomorphic Liouville property · September 25, 2026