Result 359, Differential geometry

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can strong negative curvature guarantee that a complex space can be described using bounded complex-analytic coordinates? These manuscripts report counterexamples, showing that tightly controlled curvature need not provide this kind of analytic description.

What changes?

The first manuscript reports a contractible domain in complex dimension three, a region shrinkable to a point. Its complete Kähler metric is compatible with its complex structure and has no finite-distance edge. Real sectional curvatures lie between two finite negative constants. Nevertheless, no bounded holomorphic, or complex-analytic, map to complex three-dimensional space has an everywhere nonsingular derivative. Thus the domain cannot be identified with a bounded domain by a reversible complex-analytic change of coordinates.

What does that help mathematicians do?

A second manuscript strengthens the function-theoretic obstruction, but under different curvature assumptions. In some sufficiently large fixed finite complex dimension m, it reports a domain diffeomorphic to real 2m-dimensional space with a complete Kähler metric and real sectional curvature at most -1, unbounded below. Every bounded holomorphic function is constant. Thus an upper negative curvature bound alone need not provide even one nonconstant bounded holomorphic function; this stronger obstruction is not asserted under two-sided pinching.

Are there practical applications?

The immediate value is foundational: these examples identify limits on recovering complex-analytic descriptions from curvature. In complex dimension three, a general theorem guaranteeing bounded holomorphic coordinates would need assumptions beyond completeness, contractibility and two-sided negative curvature bounds. The results also distinguish two targets: finding bounded functions and finding enough such functions to supply coordinates.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A negatively pinched Kähler threefold without bounded holomorphic coordinates

September 25, 2026 36 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 25, 2026 58 pages

We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm diffeomorphic to R2m\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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/359.md.

Negative Kähler curvature without bounded holomorphic coordinates

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

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

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.