Volume Growth and Curvature Decay of Complete Positively Curved Kähler Manifolds

Abstract

This paper constructs a class of complete Kähler metrics of positive holomorphic sectional curvature on ${\mathbf{C}}^n$ and finds that the constructed metrics satisfy the following properties: As the geodesic distance $\rho \to \infty ,$ the volume of geodesic balls grows like $O({\rho }^{\frac{2(\beta +1)n}{\beta +2}})$ and the Riemannian scalar curvature decays like $O({\rho }^{-\frac{2(\beta +1)}{\beta +2}}),$ where $\beta \ge 0.$