On the uniform convergence of quasiconformal mappings

Abstract

Let D be a domain in extended Euclidean n-space with “smooth” boundary and let { f j } \{ {f_j}\} be a sequence of K-quasiconformal mappings of D into R n {R^n} which converges uniformly on compact sets in D to a quasiconformal mapping. This paper considers the question: When does the sequence { f j } \{ {f_j}\} converge uniformly on all of D? Geometric conditions on the domains f j ( D ) {f_j}(D) are given which are sufficient and, in many cases, necessary for uniform convergence. The particular case where D is the unit ball in R n {R^n} is examined to obtain analogues to classical convergence theorems for conformal mappings in the plane.