Quasidisks, John Disks And Apollonian Metric

Quasiconformal mappings are generalizations of conformal mappings. Becauseof the close relationships with Kleinian groups, dynamics of complex analytic func-tions and Riemann surfaces etc, quasiconformal mappings become one of the mainresearch topics in complex analysis.The main aim of this dissertation is to discuss some problems of quasiconformalmappings with respect to quasidisks, John disks and the Apollonian metric.We use the internal distanceλ_D and the reflection RD to obtain several neces-sary and sufficient conditions for John disks and quasidisks. In fact, we prove thatD (?) (?)² is a John disk if and only if D has the decomposable property, and thatD (?) (?)² is a quasidisk if and only if for any z₁, z₂∈(?)D, there exists c≥1, suchthat 1／cλ_D(z₁,z₂)≤λ_D^*(z₁,z₂)≤cλ_D(z_l,z₂).We obtain that the Apollonian metricα_D of a proper subdomain D (?) (?)ⁿ isa quasi-invariant metric under quasiconformal mappings, that the strictly uniformdomains of (?)ⁿ are invariant under quasiconformal mappings, and that a Jordandomain D (?)² is a quasidisk if and only if D is a strictly uniform domain. As ap-plications of our obtained results, we get some relations among Apollonian boundarycondition, quasiconformal mappings and locally Lipschitz mappings.Two generalized hyperbolic metrics j_G^P andδ_G^P defined on a domain G (?) Rⁿare investigated. We prove the Gromov hyperbolicity of the metrics j_G^P andδ_G^P. ForG (?) Rⁿ(n≥3), we prove that the isometries of both metrics j_G^P andδ_G^P are MSbiustransformations. For the latter, G should satisfy that Card (?)G≥2.