Bounded holomorphic functions under one-sided negative Kähler curvature
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Must a complete simply connected Kähler manifold with real sectional curvature bounded above by a negative constant admit a nonconstant bounded holomorphic function? No lower curvature bound is assumed.
Why it matters and what counts as progress
Why it matters: The question tests whether one-sided negative curvature forces a basic complex-analytic function theory.
What counts as progress: A function-existence theorem or a complete negatively upper-bounded Kähler example with only constant bounded holomorphic functions, retaining any dimension restriction; a bounded-coordinate obstruction is weaker.
How the problem developed
- 1979 · Open question [1]
Shing-Tung Yau Asked whether a simply connected complete Kähler manifold with sectional curvature bounded above by a negative constant must admit a nonconstant bounded holomorphic function.
- 2008 · Partial result [1]
Jianguo Cao and Shu-Cheng Chang Derived a bounded contact structure under the negative-curvature hypothesis; this did not produce the bounded holomorphic function sought in Yau’s question.
- 25 Sept 2026 · Reported theorem [2]
OpenAI The manuscript reports a holomorphic Liouville result under a one-sided negative sectional-curvature hypothesis, with the theorem’s stated completeness and curvature assumptions.
AI contribution
One-sided negative sectional curvature and the holomorphic Liouville property
The OpenAI manuscript claims a domain in some sufficiently large fixed finite complex dimension, diffeomorphic to real Euclidean space, with a complete Kähler metric of real sectional curvature at most −1 and only constant bounded holomorphic functions. The curvature is not bounded below.
Sources
- Cao and Chang, The modified Calabi–Yau problems for CR-manifolds and applications
- OpenAI, one-sided curvature and holomorphic Liouville property
- One-sided negative sectional curvature and the holomorphic Liouville property
- OpenAI curvature manuscript map
- Formalization scope for the separate threefold
- OpenAI mathematics release
Website timeline
Publication dates and editorial changes.
OpenAI reports a one-sided curvature counterexample to Yau’s question