Koebe and Caratheódory type boundary behavior results for harmonic mappings

Prime ends in metric spaces and boundary extensions of mappings

By using the inner diameter distance condition we define and investigate new, in such a generality, class $\mathcal{F}$ of homeomorphisms between domains in metric spaces and show that, under additional assumptions on domains, $\mathcal{F}$ contains (quasi)conformal, bi-Lipschitz and quasisymmetric mappings as illustrated by examples. Moreover, we employ a prime ends theory in metric spaces and provide conditions allowing continuous and homeomorphic extensions of mappings in $\mathcal{F}$ to topological closures of domains, as well as homeomorphic extensions to the prime end boundary. Domains …