A Note on the Total Curvature of a Kähler Manifold

Given a complete manifold with non-negative Ricci curvature, it is a very interesting geometric problem of how curvature decays at infinity. While it is not true that the curvature decays in a strong sense, it is possible that the average of the scalar curvature decays at least linearly. Such a statement is certainly consistent with the Cohn-Vossen inequality which holds for surfaces. The significance of such an inequality is also clear because of its relevance with the work of the first author [1] on the attempt to prove the conjecture of the second author that a complete noncompact Kähler manifold with positive bisectional curvature is biholomorphic to the complex euclidean space. The purpose of this note is to prove a weaker version of the conjecture for Kähler manifolds.