The Mathematics of F.W. Gehring

Abstract

By widespread consensus, the modern theory of quasiconformal mappings had its birth in a 1954 Journal D’Analyse article of Lars Ahlfors that presented the first systematic treatment of various definitions for quasiconformal mappings in the correct generality; i.e., with no a priori smoothness imposed on the mappings Before the Ahlfors paper, authors had largely elected to side-step the smoothness issue by considering quasiconformality only within the category of diffeomorphism. From the seeds planted by Ahlfors in that paper have developed three main ways of introducing quasiconformal mappings in the setting of Euclidean n-space, giving rise to the so-called metric, analytic and geometric definitions. (We refer the reader to Vaisala ‘s article in this volume for further details on this subject.) The metric definition, which indeed makes sense for homeomorphisms between arbitrary metric spaces, stipulates uniform control over the infinitesimal distortion of spheres. It is arguably the most natural of the three definitions — and usually the easiest to verify in practice — but would appear, on the surface, to be less restrictive than either the analytic or the geometric definition. It was an open problem throughout most of the 1950s whether a homeomorphism between plane domains that satisfied the conditions of the metric definition would meet the requirements of the other two, seemingly stronger, definitions. In his first paper dealing with quasiconformal mappings [13], Gehring settled this question in the affirmative. The ingenious methods he pioneered in that paper remained for many years the only route to deriving regularity properties of quasiconformal mappings starting from minimal smoothness assumptions. They played a crucial part, for example, in Mostow’s work on the rigidity of hyperbolic space forms. More importantly as far as the present narrative is concerned, [13] was the public announcement of Fred’s baptism into the quasiconformal persuasion.