Abstract boundaries and continuous extension of biholomorphisms

Abstract

We present different constructions of abstract boundaries for bounded complete (Kobayashi) hyperbolic domains in C d , d ≥ 1. These constructions essentially come from the geometric theory of metric spaces. We also present, as an application, some extension results concerning biholomorphic maps. CONTENTS 1. Introduction 1 2. Abstract boundaries and extension 2 3. The Carathéodory boundary 3 4. The Gromov boundary 4 5. The Horosphere boundary 9 6. The Busemann (or horofunctions) boundary 9 Appendix A. D'Addezio's Lemmas 11 References 12