Non-negatively curved Kahler manifolds with average quadratic curvature decay

Let (M, g) be a complete noncompact Kahler 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 Kahler manifolds with non-negative curvature, J. Differs. Geom. 73 (2006), 491-530.], we prove that the universal cover (M) over tilde of M is biholomorphic to C-n provided either that (M, g) has average quadratic curvature decay, or M supports an eternal solution to the Kahler-Ricci flow with non-negative and uniformly bounded holomorphic bisectional curvature. We also classify certain local limits arising from the Kahler-Ricci flow in the absence of uniform estimates on the injectivity radius.