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 C^n 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 R^n (Theorem 1.3). The conjecture is cited to situate their results within the broader uniformization program.