Yau's uniformization conjecture (negative-curvature side) predicts that every complete simply connected Kähler manifold of complex dimension n≥2n\ge 2 whose Riemannian sectional curvature is pinched between two negative constants is biholomorphic to a bounded domain in Cn\mathbb{C}^n - the higher-dimensional analogue of the fact that a simply connected negatively curved Riemann surface is the unit disk. Not even the function-theoretic core is settled: it is not known how to prove the existence of a single nonconstant bounded holomorphic function on any such manifold beyond the classical one-dimensional case, and a 2015 claimed proof of this step (Cao–Shaw) was withdrawn. The primary formulations are those of Siu–Yau (1977), Yau's 1982 problem list (Problem 38), and Wu (1983); the main partial results - Greene–Wu's Bergman-kernel bounds, Wu–Yau's 2020 uniform equivalence of invariant metrics, Hernández–Yau–Zheng rigidity for negatively 1/4-pinched compact quotients - leave the problem open in all dimensions n≥2n\ge 2 as of August 2026.

Background

The classical uniformization theorem for Riemann surfaces states that every simply connected one-dimensional complex manifold is biholomorphic to exactly one of the Riemann sphere, the complex plane C\mathbb{C}, or the unit disk; for a surface carrying a complete Riemannian metric the sign of the Gaussian (sectional) curvature selects the model - positive curvature forces the sphere, curvature bounded above by a negative constant forces the disk. The disk is a bounded domain (an open subset of C\mathbb{C} contained in some ball), and boundedness of the domain is exactly what makes nonconstant bounded holomorphic functions possible: by Liouville's theorem every bounded entire function on C\mathbb{C} is constant, while the disk carries plenty. Whether an analogous curvature-to-complex-structure dictionary holds in higher dimensions is a central theme of complex differential geometry.

A Kähler manifold is a complex manifold with a Hermitian metric gg whose associated (1,1)(1,1)-form ω\omega is closed; its Riemannian sectional curvature sec⁡\sec is the Gaussian curvature of real two-planes, while the holomorphic bisectional curvature is the curvature term computed on pairs of complex tangent lines, the expression naturally adapted to the complex structure. On the positive-curvature side the dictionary is settled: the Frankel conjecture - a compact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to complex projective space - was proved by Mori and by Siu–Yau, and Mok extended it to a full structure theorem for compact Kähler manifolds with nonnegative holomorphic bisectional curvature. On the negative side almost nothing is settled. A complete simply connected Kähler manifold with sectional curvature pinched between two negative constants is expected to be 'hyperbolic' in the complex sense, and Yau conjectured - in the 1977 paper of Siu–Yau and in Wu's 1983 DMV lectures - that such a manifold is biholomorphic to a bounded domain in Cn\mathbb{C}^n; Yau's 1982 problem list (Problem 38) predicts more basically that it should admit many nonconstant bounded holomorphic functions. The name requires care: in much of the Kähler–Ricci-flow literature 'Yau's uniformization conjecture' instead denotes the positive-side conjecture that a complete noncompact Kähler manifold with positive bisectional curvature is Cn\mathbb{C}^n (see the survey of Chau–Tam); the present task concerns the negative-side conjecture.

The class of manifolds in question is genuinely special: P. Yang proved that the polydisk Dd\mathbb{D}^d (the product of dd copies of the unit disk D\mathbb{D}), d≥2d\ge 2, admits no complete Kähler metric with pinched negative bisectional curvature, so negatively pinched Kähler metrics are rigid objects that cannot simply be manufactured on arbitrary complex manifolds. What is known about the conjecture goes through L2L^2 theory and invariant metrics rather than bounded functions. Greene–Wu proved that a simply connected complete Kähler manifold with −B≤sec⁡≤−A<0-B\le\sec\le -A<0 carries a complete Bergman metric - the metric built from square-integrable holomorphic functions through the Bergman kernel, the projection kernel onto the L2L^2 holomorphic functions - whose kernel is comparable to the volume form; Wu–Yau (2020) proved that such a manifold admits a unique complete Kähler–Einstein metric (Ricci curvature equal to minus the metric) and that the Kähler–Einstein, Kobayashi–Royden (an invariant metric on any complex manifold measuring infinitesimal holomorphic disks), and Bergman metrics are all uniformly equivalent to the background metric. None of this produces a single bounded holomorphic function: L2L^2-integrability does not imply boundedness, and no exhaustion, normal-families, or scaling mechanism is known that converts curvature pinching into bounded function theory on a manifold not known to be Stein (a Stein manifold being one that admits enough holomorphic functions to embed as a closed complex submanifold of some CN\mathbb{C}^N). In the compact quotient case with strong pinching there is rigidity: a compact Kähler manifold carrying a negatively 1/4-pinched metric (sectional curvature in [−1,−1/4][-1,-1/4] after rescaling) is a quotient of complex hyperbolic space CHn\mathbb{C}H^n - the unit ball with its symmetric complete Kähler metric of constant negative holomorphic sectional curvature (Hernández; Yau–Zheng) - whose universal cover is a ball; but for general pinching constants even the compact case is open. A related result, Gang Liu's 'Compact Kähler manifolds with nonpositive bisectional curvature' (GAFA 2014), settles the compact nonpositive-bisectional analogue of the uniformization program - a finite cover is a holomorphic and metric fiber bundle over a compact Kähler manifold with negative first Chern class, with flat complex torus fibers, confirming a conjecture of Yau - and recalls the negative-pinched conjecture in its introduction as the contrasting open problem, noting that not even a nontrivial bounded holomorphic function on such manifolds is known.

Problem Statement

Let (Mn,g)(M^n,g) be a complete simply connected Kähler manifold of complex dimension n≥2n\ge 2 whose Riemannian sectional curvature is pinched between two negative constants, i.e. there exist constants 0<A≤B0<A\le B with

−B  ≤  sec⁡g  ≤  −A  <  0-B\;\le\;\sec_g\;\le\;-A\;<\;0

at every point of MM (the pinching constants are arbitrary and form no part of the data). Must MM be biholomorphic to a bounded domain in Cn\mathbb{C}^n (an open subset of Cn\mathbb{C}^n contained in some ball)? In particular - the function-theoretic core, closely related to Problem 38 in Yau's 1982 problem list - must MM carry at least one nonconstant bounded holomorphic function? The bounded-domain question is the target: establishing the biholomorphism, or exhibiting a counterexample, settles the task. The bounded-holomorphic-function question is the strictly weaker, implied sub-question - the bounded-domain statement implies it, since the coordinate functions of a bounded domain pull back to nonconstant bounded holomorphic functions - so proving it alone counts as partial progress, whereas refuting it settles the task: a manifold that admits no nonconstant bounded holomorphic function cannot be biholomorphic to a bounded domain, since a bounded domain's coordinate functions are nonconstant and bounded and pull back along a biholomorphism. Both questions are open for every n≥2n\ge 2; for n=1n=1 the statement is the classical uniformization theorem, the unit disk being a bounded domain.

References

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