Invariant metrics on pseudoconvex domains

Abstract

Let Ω be a bounded domain in ℂn. Each of the metrics of Bergman, Caratheodory, and Kobayashi assigns a positive number to a given non-zero tangent vector X above a point z in Ω. This assignment is invariant in the sense that if f is a biholomorphism of Ω onto another bounded domain Ω′, then the metric applied to X equals the value of the metric on Ω′ applied to the tangent vector df(X) at the point f(z). Although it is very difficult to calculate the precise value of the above metrics in all but a few special cases, it is sometimes possible to compute a formula for the asymptotic behavior of the metric as the point z approaches the boundary of Ω. When Ω is a smoothly bounded strongly pseudoconvex domain, asymptotic formulas for the Bergman metric were obtained by Diederich [3] and later in much more precise form by Fefferman [4]. Formulas for the asymptotic behavior of the Caratheodory and Kobayashi metric on the same domains were obtained by Graham [5]. In this note we shall consider the case of pseudoconvex domains of finite type in ℂ2 Instead of determining an asymptotic formula for the above metrics, we obtain only a formula that expresses the approximate size of the metrics. In a sense which we shall make precise, these metrics are all equivalent for the given class of domains. We also obtain a formula for the approximate size of the Bergman kernel K(z,\( \bar{\text z} \)) of the domain Ω