On complete Kähler manifolds with fast curvature decay
The author gives characterization of \({\mathbb{C}}^ n\). Let (M,o) be an n- dimensional complete Kähler manifold with a pole o. Suppose that there exist \(C^{\infty}\)-functions k,K: [0,\(\infty)\to [0,\infty)\) satisfying the following conditions: -K(r(x)) \(\leq\) all the radial curvatures at \(x\leq k(r(x))\), \(\int^{\infty}_{0}sK(s)ds<\infty\), \(\int^{\infty}_{0}s k(s) ds\leq 1\). Then there exists a positive constant \(\gamma_ o\) depending only on K(s) such that moreover if \(\int^{\infty}_{0}s k(s) ds<\gamma_ 0\), then M is biholomorphic to \({\mathbb{C}}^ n\).
scientific article; zbMATH DE number 3925758