. C V ] 5 S ep 2 00 9 The Kobayashi Metric , Extremal Discs , and Biholomorphic Mappings 12 Steven

We study extremal discs for the Kobayashi metric. Inspired by work of Lempert on strongly convex domains, we present results on strongly pseudoconvex domains. We also consider a useful biholomorphic invariant, inspired by the Kobayashi (and Carathéodory) metric, and prove several new results about biholomorphic equivalence of domains. Some useful results about automorphism groups of complex domains are also established. 0 Introduction Throughout this paper, a domain in C is a connected, open set. Usually our domains will be bounded. It is frequently convenient to think of a domain Ω (with smooth boundary) as given by Ω = {z ∈ Ω : ρ(z) < 0} , where ρ is a C function and ∇ρ 6= 0 on ∂Ω. We say in this circumstance that ρ is a C defining function for Ω. It follows from the implicit function theorem that ∂Ω is a C manifold in a natural sense. See [KRA1] for more on these matters. Throughout the paper D denotes the unit disc in the complex plane C and B denotes the unit ball in complex space C. If Ω1,Ω2 are domains in complex space then we let Ω1(Ω2) denote the holomorphic mappings from Ω2 to Ω1. In