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

TL;DR

This paper constructs a class of complete Kähler metrics with positive holomorphic sectional curvature on complex Euclidean space, analyzing their volume growth and curvature decay as the distance from a point increases.

Contribution

It introduces new complete Kähler metrics with positive curvature and characterizes their volume growth and scalar curvature decay rates.

Findings

Volume of geodesic balls grows polynomially with distance

Scalar curvature decays polynomially with distance

Metrics exhibit controlled curvature decay and volume growth

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 $\beta\geq 0.$