Non-negatively curved Kähler manifolds with average quadratic curvature decay

Abstract

Let (M, g) be a complete noncompact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in [A. Chau and L.-F. Tam. On the complex structure of Kähler manifolds with non-negative curvature, J. Differs. Geom. 73 (2006), 491-530.], we prove that the universal cover M̃ of M is biholomorphic to ℂn provided either that (M, g) has average quadratic curvature decay, or M supports an eternal solution to the Kähler-Ricci flow with non-negative and uniformly bounded holomorphic bisectional curvature. We also classify certain local limits arising from the Kähler-Ricci flow in the absence of uniform estimates on the injectivity radius.