Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type in C3

Abstract

Let $\mathrm{\Omega }$ be a smoothly bounded pseudoconvex domain in ${\mathbb{C}}^3$ and assume that $z_0\in b\mathrm{\Omega }$ is a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curve $\mathrm{\Gamma }\subset \mathrm{\Omega }\cup \{z_0\}$, connecting points $q_0\in \mathrm{\Omega }$ and $z_0\in b\mathrm{\Omega }$, and a quantity $M(z,X)$, along $z\in \mathrm{\Gamma }$, which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.