Convexity and linear distortion : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics, Institute of Natural and Mathematical Science, Massey University of Albany, New Zealand

Abstract

This thesis is primarily concerned with the convexity properties of distortion functionals (particularly the linear distortion) defined on quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal applica-tion is in identifying the lower semi-continuity of distortion on uniformly convergent limits of sequences of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality - discussed in this thesis - it is known that if {fn}n=1 is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function f, then the limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasi-conformality (via the linear distortion H(f) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this thesis is to show that this failure of lower semicontinuity is actually far more common, perhaps even generic in the sense that it might be true that under mild restrictions on a quasiconformal f, there may be a sequence {fn}n=1 with fn → f uniformly and with lim supn→∞ H(fn) < H(f). The main result of this thesis is to show this is true for a wide class of linear mappings.