On the generalization of the Riemann mapping theorem in the theory of several complex variables.

Abstract

One of the most powerful results in complex analysis is The Riemann Mapping Theorem which states that every non-empty simply connected domain in the complex plane which is not the entire ₵ is biholomorphically equivalent to the open unit disc. However, this theorem does not hold in higher dimensions. For instance, the open unit ball and the open polydisc are not biholomorphic in ₵n for n 1. Generalizations of the Riemann Mapping Theorem in the theory of several complex variables rely on additional characterizations of the complex structure of the domain. For instance, Stanton built his generalization on specific conditions on the Kobayashi and the Carathéodory metrics defined on a given complex manifold. Whereas Wong-Rosay theorem mainly relies on the group of automorphisms of a given domain. In this work, our basic aim is to study Stanton and Wong-Rosay theorems and their proofs. We will also approach the proof of Wong-Rosay theorem using the scaling method of Pinchuk.