Non-negatively curved Kähler manifolds with average quadratic curvature decay

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," we prove that the universal cover $\tilde M$ of $M$ is biholomorphic to $\Bbb{C}^2$ 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.

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Published 1 January 2007