Abstract boundaries and continuous extension of biholomorphisms

TL;DR

This paper explores various geometric constructions of abstract boundaries for certain complex domains and applies these to extend biholomorphic maps beyond their original domains.

Contribution

It introduces new geometric boundary constructions for hyperbolic domains and demonstrates their use in extending biholomorphisms.

Findings

Different constructions of abstract boundaries for hyperbolic domains.

Extension results for biholomorphic maps using these boundaries.

Applications to geometric theory of metric spaces.

Abstract

We present different constructions of abstract boundaries for bounded complete (Kobayashi) hyperbolic domains in ${\mathbb C}^d$, $d \geq 1$. These constructions essentially come from the geometric theory of metric spaces. We also present, as an application, some extension results concerning biholomorphic maps.