Pinched negative, no bounded domain?

The claim, in plain words

On the unit disc, the Poincaré metric has constant negative curvature, and bounded holomorphic functions are plentiful: the coordinate z itself is one. On the whole complex plane, Liouville’s theorem makes every bounded holomorphic function constant, and the plane admits no metric compatible with its complex structure with curvature at most −1. In higher dimensions the unit ball’s natural metric has sectional curvatures between −2 and −1/2, in one normalization: a quarter-pinched range, which our computer’s samples fill out.

So does negative curvature make a simply connected complete Kähler manifold look like a bounded domain? Hung-Hsi Wu asked a version in 1967; Yau proposed the bounded-domain conjecture, and in 2019 Damin Wu and Yau stated it with curvature pinched between two negative constants. Yau’s problem from 1982 asks something weaker, with curvature only bounded above by a negative constant: whether such a manifold has even one nonconstant bounded holomorphic function. A claimed proof for pinched curvature, in dimension three and up, was posted in 2015 and withdrawn in 2016.

Result 359 is two independent OpenAI papers, dated September 25 and made public October 6, 2026. The first, 36 pages, claims a contractible domain in complex three-space with a complete Kähler metric whose sectional curvature is pinched between two negative constants, yet no bounded holomorphic map from it has a nowhere-vanishing Jacobian, so it is not biholomorphic to any bounded domain. The second, 58 pages, claims a domain in some large dimension, shaped like ordinary space, with a complete Kähler metric of curvature at most −1, whose only bounded holomorphic functions are constants.

What it doesn’t claim

Not claimed: the first example has no bounded coordinates but does have bounded functions, and the second example’s curvature is unbounded below. So neither settles whether pinched curvature gives one nonconstant bounded function, and neither treats complex dimension two. The second paper’s dimension is not given explicitly. We found no competing claim.

Where it stands

Posted as a preprint

OpenAI, 36 + 58 pages, dated September 25, 2026

Peer reviewed

Not yet

Formalized in Lean

The first paper’s main theorem and two key propositions; not the second paper (scope notes)

Last checked October 19, 2026. Both papers are unrefereed and unchanged since release. The first paper’s main theorem and two key propositions are formalized in Lean, about 19,000 lines, not run by us. The second paper has no Lean for its theorem. An AI audit of the second paper found no gap but only skimmed two sections. We found no specialist comment and no reported error. OSVerify has not reviewed it.

How the proof goes

1. Discs of carefully varying size. The first construction puts a disc over every point of the two-ball, with radius e^(−φ/2). When φ is the ball’s own potential, the result is exactly the three-ball, which our computer confirms. The paper adds peak terms to φ so that the disc sizes vary in a pattern no holomorphic function can follow. Cauchy estimates turn a bounded map with a nowhere-zero Jacobian into a holomorphic function whose real part must stay under a bound built from φ, and the peaks make that impossible, using positive densities that preserve holomorphic averages yet give a logarithm a negative average. Our one-peak example, with parameters we chose, gives a model log-integral of −0.0076.
2. Curvature bookkeeping, and a tube. Keeping the curvature pinched takes a computation over every real plane. On some planes the main term vanishes exactly, like 4 + 4 + 4 − 12 = 0, which our computer reproduces, and the paper shows the largest error vanishes there too. The second paper builds a tube over a nonpositively curved base, which gets curvature at most −1 for free, then uses huge holomorphic discs placed at grids of polynomial zeros to show every bounded function does not depend on the tube’s vertical variable. It then comes from a bounded entire function on the base, which Liouville makes constant.

The numbers in the video

• Ours: the unit ball in ℂ² has sectional curvatures filling [−2, −1/2] in the first paper's normalization — quarter-pinched.
• Ours: with φ equal to the ball's own potential, the first paper's construction gives exactly the unit 3-ball (checked at 1,762 exact points).
• Ours: a one-peak model log-integral, with parameters we chose, comes out at −0.0076, the sign the first paper needs.
• Lean: 119 files and 18,688 lines in the import closure of the first paper's comparator (counted by us; not run by us).

Read the source

• A negatively pinched Kähler threefold without bounded holomorphic coordinates (PDF), 36 pages.
• One-sided negative sectional curvature and the holomorphic Liouville property (PDF), 58 pages.
• The Lean formalization notes.
• The collection’s catalog, where this is result 359.

The 2 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

• A negatively pinched Kähler threefold without bounded holomorphic coordinates main paper
• One-sided negative sectional curvature and the holomorphic Liouville property companion

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.