Abstract
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 satisfying the bounded turning condition, locally and finitely connected at the boundary and the structure of prime end boundaries for such domains play a crucial role in our investigations. We apply our results to show the Koebe theorem on arcwise limits for mappings in {mathcal {F}}. Furthermore, relations between the Royden boundary and the prime end boundary are presented. Our work generalizes results due to Carathéodory, Näkki, Väisälä and Zorič.
Highlights
The extension problem for mappings between two open domains has been studied in various settings and for various kinds of extension properties
We illustrate the above definition in Examples 2–5, where we show that, under additional assumptions on domains, conformal, quasiconformal, bi-Lipschitz and quasisymmetric mappings belong to class F
Last section is devoted to studies of two applications of extension results: in Sect. 6.1 we present a variant of the Koebe theorem, see Theorem 6
Summary
The extension problem for mappings between two open domains has been studied in various settings and for various kinds of extension properties. Carathéodory [11] created and studied an abstract type of a boundary, the so-called prime end boundary and proved that for planar -connected domains a homeomorphic extension of a conformal map is possible with respect to the prime end closure of the target domain. Prime ends have been employed to investigate other topics, e.g. the theory of continua, see Carmona–Pommerenke [12], local connectivity of sets, see Rempe [45], the dynamical systems, see Koropecki–Le Calvez–Nassiri [29], the boundary behavior of solutions to elliptic PDEs, see Ancona [4] and the studies of the p-harmonic Dirichlet problem in metric spaces, see Björn [7], Björn–Björn–Shanmugalingam [9], Estep–Shanmugalingam [15] and [2] Another direction of related studies arises from quasiconformal mappings (qcmappings for short). We obtain an upper estimate for a number of components of fibers in the Royden compactification for John domains, see Corollary 6
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