Geometric Properties of Quasiconformal Maps and Special Functions

Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors. In particular, we adopt results from the book authored by Anderson-Vamanamurthy-Vuorinen [AVV6]. Part 1. Quasiconformal maps and spheres Part 2. Conformal invariants and special functions Part 3. Recent results on special functions 1991 Mathematics Subject Classification: 30C62, 30C65 1 Quasiconformal maps and spheres Some current trends in multi-dimensional quasiconformal analysis are reviewed in [G6], [G8], [I2], [V6], [V7], [Vu5]. 1.1. Categories of homeomorphisms. Below we shall discuss homeomorphisms of a domain of Rn onto another domain in Rn, n ≥ 2. Conformal maps provide a well-known subclass of general homeomorphisms. By Riemann’s mapping theorem this class is very flexible and rich for n = 2 whereas Liouville’s theorem shows that, for n ≥ 3, conformal maps are the same as Möbius transformations, i.e., their class is very narrow. Thus the unit ball Bn = {x ∈ Rn : |x| < 1} can be mapped conformally only onto a halfspace or a ball if the dimension is n ≥ 3. Quasiconformal maps constitute a