## On the complex structure of Kähler manifolds with nonnegative curvature.(English)Zbl 1161.53351

Summary: We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclidean space $$\mathbb{C}^n$$. We also show that the volume growth condition can be removed if we assume $$(M, g)$$ has average quadratic scalar curvature decay (see Theorem 2.1) and positive curvature operator.

### MSC:

 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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