Volume Growth and Curvature Decay of Positively Curved K\"{a}hler manifolds

Abstract

In this paper we obtain three results concerning the geometry of complete noncompact positively curved K\"{a}hler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact K\"{a}hler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex dimension). The second one states that the curvature of a complete noncompact K\"{a}hler manifold with positive bisectional curvature decays at least linearly in the average sense. The third result is concerned with the relation between the volume growth and the curvature decay. We prove that the curvature decay of a complete noncompact K\"{a}hler manifold with nonnegative curvature operator and with the maximal volume growth is precisely quadratic in certain average sense.