Yau's uniformization conjecture: is every complete noncompact Kähler manifold with positive bisectional curvature biholomorphic to C^n?

Yau's uniformization conjecture, posed in the 1970s, predicts that every complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space $\mathbb{C}^n$. Under the additional hypothesis of maximal volume growth (Euclidean-order volume growth, $\mathrm{Vol}(B(o,r))\geq\kappa r^{2n}$) the conjecture is now settled in all dimensions (Gang Liu, 2016/2019, with an independent Chern–Ricci flow proof by Lee–Tam, 2020). However, the conjecture in its original generality — with no assumption whatsoever on the geometry at infinity — remains open in every complex dimension $n\geq 2$, including dimension two. Recent work shows the classical embedding strategy provably yields nothing once volume growth is not maximal, and the strongest recent advance resolves only the analogous statement under the stronger hypothesis of positive sectional curvature for Kähler surfaces. The problem therefore remains open, with a precise and nonempty open core.

Background

A Kähler manifold $(M^n,\omega)$ is a complex $n$-manifold carrying a Hermitian metric whose associated two-form $\omega$ is closed. Its holomorphic bisectional curvature is $BK(X,Y)=R(X,JX,Y,JY)/(\lvert X\rvert^2\lvert Y\rvert^2)$, where $J$ is the complex structure and $R$ the Riemann curvature tensor; positivity of $BK$ implies positive Ricci curvature and positive holomorphic sectional curvature, but is weaker than positive sectional curvature of the underlying Riemannian metric, since it constrains only the curvature of holomorphic planes. In the compact world this positivity rigidly determines the complex structure: a compact Kähler manifold with positive bisectional curvature is biholomorphic to complex projective space (the Frankel conjecture, proved in the 1980s). Yau conjectured in the 1970s (recorded in print in his 1991 review) the noncompact analogue: a complete noncompact Kähler manifold with positive bisectional curvature is biholomorphic to $\mathbb{C}^n$. For $n=1$ this follows from classical uniformization together with the parabolicity results of Cohn-Vossen and Huber (a Riemann surface is parabolic when it carries no nonconstant negative subharmonic function). The statement is not vacuous: $\mathbb{C}^n$ itself carries many complete Kähler metrics with positive bisectional, even sectional, curvature (rotationally symmetric examples, and gradient-soliton examples — metrics that evolve only by rescaling under the natural curvature flow), so the conjecture concerns rigidity of the complex structure, not of the metric.

Most progress has required hypotheses on the geometry at infinity. A manifold has maximal volume growth if $\mathrm{Vol}(B(o,r))\geq\kappa r^{2n}$ for some $\kappa>0$, all $r>0$. Mok (1984) embedded such manifolds onto affine algebraic varieties using holomorphic functions of polynomial growth (holomorphic functions whose modulus is bounded by a fixed power of the distance from a base point), under bounded curvature and decay assumptions; Chen–Tang–Zhu (2004) handled complex dimension two under bounded positive bisectional curvature with maximal volume growth (discovering an automatic quadratic curvature-decay gap); Chau–Tam (2006) proved all dimensions under bounded nonnegative bisectional curvature with maximal volume growth via the Kähler–Ricci flow (the parabolic evolution $\partial_t\omega(t)=-\mathrm{Ric}\,\omega(t)$, which deforms the Kähler metric along its Ricci curvature). Gang Liu's 'On Yau's uniformization conjecture' (arXiv:1606.08958; Cambridge J. Math. 2019), removed boundedness: with nonnegative bisectional curvature and maximal volume growth, $M$ is biholomorphic to $\mathbb{C}^n$ in every dimension, constructed from $n$ polynomial-growth holomorphic functions. Its method studies pointed Gromov–Hausdorff limits (limits of pointed metric spaces with respect to the Gromov–Hausdorff distance) of Kähler manifolds with bisectional curvature lower bound: such limits are homeomorphic to normal complex analytic spaces whose singular set has complex codimension at least four, and the treatment of isolated singularities passes through McLean's contact-topological criterion (a normal threefold isolated singularity whose link is contactomorphic to the standard contact sphere is smooth; the link is the intersection of the singular variety with a small sphere around the singular point, and the standard contact sphere is that sphere equipped with its canonical maximally non-integrable tangent hyperplane field) and is related to Shokurov's conjecture on log-terminal singularities in algebraic geometry (log-terminal singularities being a mild, discrepancy-measured class of singularities of algebraic varieties), currently known only up to dimension three. Lee–Tam (2020) later gave an alternative proof of the same maximal-volume-growth theorem using the Chern–Ricci flow (a variant of the Kähler–Ricci flow defined for arbitrary Hermitian metrics via the curvature of the Chern connection).

Removing all conditions at infinity is the hard core. A theorem synthesized from the works of Ni (2004) and Ni–Tam (2003), among others (stated with references in Y. Shi's 2025/2026 paper), asserts that for a complete Kähler manifold with nonnegative bisectional curvature whose universal cover does not split as a product, maximal volume growth is equivalent to the existence of a single nonconstant holomorphic function of polynomial growth. Hence, once volume growth is not maximal, the entire embedding strategy has no functions to work with; volume growth can be as slow as order $r^n$, and this bound is sharp (attained by known soliton-type examples on $\mathbb{C}^n$). Chen–Zhu (2018, surveyed in 2021) obtained partial results in this regime: finiteness of $\int_M\mathrm{Ric}^n$ forces $M$ to be quasi-projective (biregularly isomorphic to an open subset of a projective algebraic variety), and under bounded curvature and finite Chern number conditions one gets partial uniformization statements. Most recently, Datar–Pingali–Seshadri (2025, 2026) resolved, for complete Kähler surfaces, the analogous statement under the stronger hypothesis of positive sectional curvature, with no assumptions at infinity — introducing new weighted $L^p$ and Monge–Ampère techniques — and proved new obstructions to the existence of complete Kähler metrics with nonnegative bisectional curvature; their introduction emphasizes that under mere positive bisectional curvature without global hypotheses essentially nothing is known, even whether $\mathbb{C}\times\mathbb{C}^*$ admits such a metric.

Problem Statement

Yau's uniformization conjecture: let $(M^n,\omega)$ be a complete noncompact Kähler manifold whose holomorphic bisectional curvature is positive everywhere. Prove that $M$ is biholomorphic to $\mathbb{C}^n$, or refute the conjecture by exhibiting a counterexample. Since the case of maximal volume growth is settled (Liu 2019; Lee–Tam 2020), the open core is the complete absence of hypotheses on the geometry at infinity: no volume-growth lower bound, no curvature-decay rate, no curvature upper bound, and no integral condition such as $\int_M\mathrm{Ric}^n<\infty$ may be assumed, and every complex dimension $n\geq 2$ — including $n=2$ — must be covered. The closely related stronger variant assumes only nonnegative (rather than strictly positive) bisectional curvature, together with the condition that the universal cover does not split as a nontrivial product; strictly positive bisectional curvature implies both of these hypotheses, so a proof of the nonnegative variant would settle the strictly positive conjecture, and a counterexample to the strictly positive conjecture would refute the nonnegative variant as well. The nonnegative variant is stated for context only: a proof or a counterexample for the strictly positive conjecture suffices to resolve this task, and no separate result on the variant is required.

References

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14. V. Datar, V. P. Pingali, H. Seshadri, Uniformisation of complete Kähler surfaces with positive sectional curvature, arXiv:2604.11220 (2026), https://arxiv.org/abs/2604.11220