A Quasiconformal Mapping?

Quasiconformalmappings are generalizations of conformalmappings. They can be considered not only on Riemann surfaces, but also on Riemannian manifolds in all dimensions, andevenon arbitrarymetric spaces. Quasiconformal mappings occur naturally in various mathematicalandoftenaprioriunrelatedcontexts. The importance of quasiconformal mappings in complex analysis was realized by Ahlfors and Teichmüller in the 1930s. Ahlfors used quasiconformal mappings in his geometric approach to Nevanlinna’s value distribution theory. He also coined the term “quasiconformal” in his 1935work onÜberlagerungsflächen that earned him one of the first two Fields medals. Teichmüller used quasiconformal mappings to measure a distance between two conformally inequivalent compact Riemann surfaces, starting what isnowcalledTeichmüller theory. There are three main definitions for quasiconformal mappings in Euclidean spaces: metric, geometric, and analytic. We begin with the metric definition, which is the easiest to state andwhichmakes sense in arbitrary metric spaces. It describes the property that “infinitesimal balls are transformed to infinitesimal ellipsoidsofboundedeccentricity”. Let f : X → Y be a homeomorphism between two metric spaces. Forx ∈ X and r > 0 let