Quasiconformal mappings on certain classes of domains in metric spaces

Abstract

First, we give some results about certain properties of domains and balls in a geodesic metric space. We shall prove that a domain in a geodesic metric space is arc connected and any ball in a geodesic metric space is arc connected. Moreover, if the metric space is GC-geodesic then we shall prove that any ball in a geodesic metric space is locally arc connected on the boundary. Next, our framework is the Ahlfors Q-regular metric measure spaces, with 1 ≤ Q < ∞. The Ahlfors Q-regular spaces are a natural setting for the theory of quasiconformal mappings since in these spaces the three definitions of quasiconformal in Euclidean spaces of dimension at least two, can be formulated, but they are not equivalent. We consider in this note the notion of geometric quasiconformal. In the Euclidean space, Gehring [3] introduced the definition of a global quasiconformal collared domain on the boundary. We give an analogous definition for this domain in a Q-regular metric measure space. We shall prove that if a domain D in a Q-regular GC-geodesic metric space is globally quasiconformally on the boundary, then D is locally arc connected on the boundary. We shall prove that any bounded Q-Loewner domain in a Q-regular geodesic metric space is arc strictly quasiconformally accessible (Definition 10[1]) on the boundary. At the end of this note, we give several boundary extension theorems for quasiconformal mappings.