Bruce Palka

F.W. Gehring: A Biographical Sketch

Frederick William Gehring was born in Ann Arbor, Michigan on August 7, 1925. His father Carl Gehring was of German ancestry but the family had been established for many years in Cleveland, Ohio. Carl loved music and was an amateur composer. Although he started out studying engineering, he soon switched to journalism and later worked for the Ann Arbor News as state news editor and music critic. Fred’s mother Hester Reed Gehring was the daughter of a physics professor at the University of Michigan (UM) who later became Dean of the Literary College. She and Carl met as undergraduates at UM. She went on to complete a Ph.D. in German and became a foreign language examiner for the UM Graduate School.

The Mathematics of F.W. Gehring

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.

Cone conditions and quasiconformal mappings

Let f be a quasiconformal mapping of the open unit ball B n = {x ∈ R n : | x | < l× in euclidean n-space R n onto a bounded domain D in that space. For dimension n= 2 the literature of geometric function theory abounds in results that correlate distinctive geometric properties of the domain D with special behavior, be it qualitative or quantitative, on the part of f or its inverse. There is a more modest, albeit growing, body of work that attempts to duplicate in dimensions three and above, where far fewer analytical tools are at a researcher’s disposal, some of the successes achieved in the plane along such lines. In this paper we contribute to that higher dimensional theory some observations relating the behavior of f and f -1 to one of the venerable geometric conditions in analysis, the cone condition prominent in potential theory, geometric measure theory, and elsewhere. We first demonstrate that, when D obeys a specific interior cone condition along its boundary, f must satisfy a uniform Holder condition in B n . With regard to f -1, the dual result one might anticipate — that an exterior cone condition satisfied by D at its boundary would lead to a uniform Holder estimate for f -l in D — is not, in general, true. We show, however, that in the presence of a certain auxiliary condition on D, one which is implied by an exterior cone condition when D is a Jordan domain in the plane, such a cone condition does exert a definite influence on the modulus of continuity of f -1.