The Kobayashi metric, extremal discs, and biholomorphic mappings

TL;DR

This paper investigates extremal discs related to the Kobayashi metric on strongly pseudoconvex domains, introduces a biholomorphic invariant, and presents new results on domain equivalence and automorphism groups.

Contribution

It extends Lempert's work to strongly pseudoconvex domains, introduces a new biholomorphic invariant, and provides novel insights into domain equivalence and automorphism groups.

Findings

Results on extremal discs for strongly pseudoconvex domains

Introduction of a biholomorphic invariant inspired by Kobayashi and Carathéodory metrics

New theorems on automorphism groups of complex domains

Abstract

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 Carath\'{e}odory) metric, and prove several new results about biholomorphic equivalence of domains. Some useful results about automorphism groups of complex domains are also established.