Yau’s uniformization conjecture for complete noncompact Kähler manifolds

Establish that every complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space C^n.

Background

The paper studies a real analogue of Kähler and Chern–Ricci flows on noncompact affine/Hessian manifolds and draws parallels with uniformization results in complex geometry. In this context, the authors recall Yau’s longstanding uniformization conjecture for complete noncompact Kähler manifolds with positive holomorphic bisectional curvature, which motivated subsequent work on flow-based approaches to uniformization.

They note related progress: Liu proved biholomorphism to Cn under nonnegative bisectional curvature with maximal volume growth, and Lee–Tam obtained similar conclusions via the Chern–Ricci flow. The present paper proves a corresponding real (Hessian) version: under nonnegative Hessian bisectional curvature and maximal volume growth of the tangent bundle, the manifold is diffeomorphic to Rn (Theorem 1.3). The conjecture is cited to situate their results within the broader uniformization program.

References

A longstanding conjecture of Yau [35] states that a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to Cn.

— A geometric flow on noncompact affine Riemannian manifolds  (2401.13261 - Jiao et al., 2024) in Section 1, Introduction (preceding Theorem 1.3)

Yau's uniformization conjecture asks whether a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space.

— Conditional Uniformization of Kähler Surfaces  (2609.18506 - Wu, 16 Sep 2026) in Section 1, Introduction

Obtaining strong Steinness from $BK>0$ and SCI alone remains unresolved.

— Conditional Uniformization of Kähler Surfaces  (2609.18506 - Wu, 16 Sep 2026) in Section 5, Consequences for uniformization

In the K\"ahler setting, Yau's uniformization conjecture asserts that a complete noncompact K\"ahler manifold with positive holomorphic bisectional curvature should be biholomorphic to complex Euclidean space .

— An $m$-Hessian approach to Yau uniformization conjecture  (2609.18171 - Dat, 16 Sep 2026) in Section 1, Introduction