Homeomorphic extension of quasi-isometries for convex domains in $\mathbb C^d$ and iteration theory

TL;DR

This paper investigates the extension of biholomorphisms and quasi-isometries between convex domains in complex space, using coarse geometry and Gromov hyperbolic space techniques, without boundary regularity or boundedness constraints.

Contribution

It introduces a novel approach leveraging coarse geometry to extend quasi-isometries and biholomorphisms between convex domains in complex spaces.

Findings

Established homeomorphic extensions for biholomorphisms.

Extended results to quasi-isometries with Kobayashi distances.

Connected Gromov boundary with domain topological boundary.

Abstract

We study the homeomorphic extension of biholomorphisms between convex domains in $\mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.