Conformal energy of quasisymmetric and quasimöbius mappings : this dissertation is submitted for the degree of Master of Science - Mathematics, New Zealand Institute for Advanced Study, Massey University

Abstract

Conformal energy was introduced to solve minimisation problems concerning the mean distortion of homeomorphisms of the disk. Interesting new connections were observed between conformal energy and extremal quasiconformal mappings associated with quasisymmetric homeomorphisms of the circle. This thesis outlines the mathematical foundations of quasiconformality and explores its connection with quasisymmetry further, giving explicit formulae for lowest energy quasisymmetric mappings, asymptotics for bilipschitz mappings and other related estimates.