A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvature

Chen, Bing-Long; Tang, Siu-Hung; Zhu, Xi-Ping

A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvature

Authors

Chen, Bing-Long

Tang, Siu-Hung

Zhu, Xi-Ping

Description

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then $M$ is biholomorphic to $\C^2$. During the proof, we also discover an interesting gap phenomenon which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.
57 pages