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Yau's uniformization conjecture

Claims to prove Yau's uniformization conjecture: positive bisectional curvature makes a complete Kähler manifold ℂⁿ.

1 paper September 23, 2026 Explained by AI

A Kähler manifold is a curved space that looks like complex n-space near each point. In 1982 Yau asked if such a space, complete, without edge and positively curved in a complex sense, must be ℂⁿ itself. Earlier proofs needed extra conditions, such as bounded curvature or fast volume growth. This paper claims to remove all of them, in every dimension. It identifies the complex structure with ℂⁿ, not the metric.

First step
1982
Shing-Tung Yau, in his problem survey
Significance
Landmark
It claims a well-known conjecture of Yau from 1982 that many earlier results proved only under extra conditions.
Trust
Claimed, not checked
Field
Differential geometry
1 paper

The explanation

Start with a bowl. Take the surface z = x² + y² in ordinary space. It goes on forever, so it is noncompact. A path along it never falls off an edge, so it is complete. Its Gaussian curvature, the amount it bends like a sphere, is 4/(1 + 4r²)² at distance r from the axis, measured flat. At r = 0 this is 4. At r = 1 it is 0.16. At r = 10 it is about 0.000025. The curvature stays positive but tends to zero. Even so, a classical theorem says this bowl has the same complex shape as the flat plane. Here complex shape means the angles between curves, which a map can keep while it stretches lengths.

Yau asked whether the same holds in every dimension. A complex manifold of dimension n looks like ℂⁿ, the space of n complex numbers, near each point. A Kähler metric is a way to measure lengths on it that fits the complex structure. Holomorphic bisectional curvature is a complex version of the bowl's curvature. Two such spaces are biholomorphic if a map between them keeps the complex structure both ways. Yau asked this in his 1982 problem survey. The paper claims a yes: every such complete noncompact space with strictly positive bisectional curvature is biholomorphic to ℂⁿ.

Before this, each proof added a condition. Chau and Tam needed bounded curvature and maximal volume growth. Maximal volume growth means large balls hold about as much volume as flat balls. Liu removed the curvature bound but kept that growth condition. For surfaces, Datar, Pingali and Seshadri needed the stronger positive sectional curvature. The paper adds no volume, noncollapsing, pinching or topological condition. It needs no uniform lower bound and no upper bound on curvature.

There were two obstacles. The curvature can be unbounded, and some regions can be very thin. So standard flow theorems cannot be used directly. Also, shrinking coordinate charts may only show the space is a proper part of ℂⁿ. The paper runs the Kähler–Ricci flow, an equation that smooths the metric over time. It proves Harnack inequalities, which bound how fast curvature can change. These give charts that shrink in every direction. Polynomial maps then correct the charts so that their limit is a map from the space onto all of ℂⁿ. The conclusion is about the complex structure. It does not say the metric is flat.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof is long and relies on delicate estimates for the flow. No specialist review of the paper has been published.

What it doesn't show. This does not say the given metric is flat or Euclidean; it only identifies the complex manifold with ℂⁿ. It has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark-level claim if the proof holds: it would settle a 1982 conjecture of Yau. The proof is long, there is no Lean check, and no specialist review is published.

Words used

Complex manifold
A space that looks like ℂⁿ, the space of n complex numbers, near each point.
Kähler metric
A way to measure lengths on a complex manifold that fits its complex structure.
Complete
Every path of finite length that keeps going stays inside the space; there is no missing edge.
Noncompact
Not closed up and bounded; the space goes on forever.
Holomorphic bisectional curvature
A complex version of surface curvature, measured on pairs of complex directions.
Biholomorphic
Matched by a map that keeps the complex structure in both directions.
Kähler–Ricci flow
An equation that changes a Kähler metric over time, smoothing out its curvature.
Harnack inequality
A bound on how fast a positive quantity, here curvature, can change across space and time.

The original paper

All quoted lines (11)
  1. A complete noncompact Riemann surface of positive Gaussian curvature is conformally the plane, by Cohn-Vossen's topology theorem and the parabolicity theorem of Blanc–Fiala–Huber.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  2. In his 1982 problem survey, Yau asked whether a complete noncompact K\"ahler manifold with positive holomorphic bisectional curvature must be biholomorphic to complex Euclidean space.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  3. If MM admits a smooth complete K\"ahler metric with strictly positive holomorphic bisectional curvature, then MM is biholomorphic to Cn\C^n.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  4. Chau–Tam established uniformization under bounded nonnegative bisectional curvature and maximal volume growth.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  5. Liu proved uniformization under nonnegative bisectional curvature and maximal volume growth, without an upper curvature bound, using a Gromov–Hausdorff approach and retracting holomorphic vector fields.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  6. Datar, Pingali, and Seshadri prove that every complete noncompact K\"ahler surface with positive real sectional curvature is biholomorphic to C2\mathbb C^2.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  7. No volume-growth, noncollapsing, Ricci-pinching, or topological hypothesis is added.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  8. In particular, the theorem assumes neither a uniform positive lower bound nor a finite upper bound for the curvature.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  9. Two issues must be addressed in the unrestricted problem.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  10. This proves that the limiting coordinate map is onto all of Cn\C^n.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text
  11. Its conclusion identifies the entire complex manifold with Cn\C^n; it does not assert that the given metric is Euclidean.
    Uniformization of complete Kähler manifolds with positive bisectional curvature Back to text

Other sources

  1. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

Browse Differential geometry Named conjecturesCurvature and geometric flowsOpen since the 1980sClaimed, not yet checkedLandmark results