Browse › Differential geometry › 359 Negative Kähler curvature without bounded holomorphic coordinates
One-sided negative sectional curvature and the holomorphic Liouville property
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Abstract
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm 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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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.
Possibly relevant literature (found by the agent, not checked)
- P. Klembeck, Kähler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets (Indiana Univ. Math. J. 27 (1978), no. 2, 275–282). The paper's introduction mentions "Klembeck's radial curvature calculations", which Seshadri adapted for a complete negatively curved Kähler metric on C^n; Klembeck's paper itself is not in the bibliography.
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: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}
}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:
- A negatively pinched Kähler threefold without bounded holomorphic coordinates · September 25, 2026