The Characterizations Of Free Quasiconformal Mappings In Metric Spaces

In the theory of Vaisala’s freely quasiconformal mappings, a homeomor-phism between two Banach spaces is said to be freely quasiconformal if this homeomorphism and its inverse are fully semisolid. This shows that the semisol-id or the fully semisolid homeomorphisms play a key role in Vaisala’s free theory. In fact, in a series of his related papers, Vaisala has emphasized very much on the discussions of properties, particularly, the characterizations of semisolid or fully semisolid homeomorphisms between Banach spaces.The purpose of this thesis is to generalize Vaisala’s discussions on the characterizations of fully semisolid homeomorphisms into the setting of metric spaces. First, we get several sufficient conditions for a homeomorphism to be fully semisolid homeomorphisms between two suitable metric spaces. Then, as the main result of this paper, we show that, in suitable Boman metric spaces, all the obtained sufficient conditions for fully semisolid homeomorphisms are also necessary. As a direct consequence, we get six alternative characterizations for free quasiconformality of a homeomorphism and an implication from the free quasiconformality of a homeomorphism to the quasiconformality of this homeomorphism and its inverse. In the end, as an application of our main results, we prove that the composition of two locally weakly quasisymmetric mappings is locally quasisymmetric and also quasiconformal.The thesis consists of six chapters. The arrangement is as follows. In Chapter 1. we mainly state the background of our study problems and give the statement of our main results which include four theorems. They are Theorem 1.2.1, Theorem 1.2.2, Theorem 1.2.3 and 1.2.4. In Chapter 2, we introduce some related concepts, particularly the quasihyperbolic metric together with some basic results. In the next four chapters, we prove our four main theorems, respectively.