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

Abstract

This paper constructs a class of complete K\"{a}hler metrics of positive holomorphic sectional curvature on ${\bf C}^n$ 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 $\beta\geq 0.$