On the complex structure of Kähler manifolds with nonnegative curvature

TL;DR

This work proves that complete noncompact Kähler manifolds with nonnegative and bounded holomorphic bisectional curvature and maximal volume growth are biholomorphic to $\mathbb{C}^n$, addressing Yau's uniformization conjecture in this setting. The authors analyze the Kähler-Ricci flow to obtain long-time existence and detailed asymptotics, showing the Ricci eigenvalues converge and the metric approaches an expanding gradient Kähler-Ricci soliton; they then construct a global biholomorphism by patching local charts via a Rosay–Rudin/Jonsson–Varolin-type scheme. Key steps include a tangent-space decomposition into invariant subspaces, Lyapunov-regular behavior near infinity, and an iterative holomorphic-gluing procedure that yields a global embedding of M into $\mathbb{C}^n$. The results further indicate the volume-growth condition can be dropped under average quadratic scalar curvature decay with a positive curvature operator, and extend to universal-cover scenarios under curvature assumptions.

Abstract

We study the asymptotic behavior of the K\"ahler-Ricci flow on K\"ahler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact K\"ahler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclidean space $\C^n$. We also show that the volume growth condition can be removed if we assume $(M, g)$ has average quadratic scalar curvature decay (see Theorem 2.1) and positive curvature operator.

Theorems & Definitions (43)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Corollary 2.1
• Proposition 2.1
• proof
• Corollary 2.2
• proof