Non-negatively curved Kähler manifolds with average quadratic curvature decay

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Let (M, g) be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in CT3, we prove that the universal cover M of M is biholomorphic to ⁿ 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.