Carath\'eodory-type extension theorem with respect to prime end boundaries

TL;DR

This paper extends classical boundary extension theorems to prime end boundaries in metric spaces, providing new insights into homeomorphisms and quasiconformal mappings under specific geometric conditions.

Contribution

It establishes a Carathéodory-type extension theorem for BQS homeomorphisms and quasiconformal maps in metric spaces with prime end boundaries, under Ahlfors regularity and Poincaré inequality assumptions.

Findings

Extension of BQS homeomorphisms to prime end closures

Extension of quasiconformal mappings under geometric conditions

Examples illustrating prime end boundary properties

Abstract

We prove a Carath\'eodory-type extension of BQS homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carath\'eodory-type extension of geometric quasiconformal mappings between two such domains provided the two domains are both Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality when equipped with their respective Mazurkiewicz metrics. We also provide examples to demonstrate the strengths and weaknesses of prime end closures in this context.