Cluster sets and quasiconformal mappings

Certain classical results on cluster sets and boundary cluster sets of analytic functions, due to Iversen, Lindelof, Noshiro, Tsuji, Ohtsuka, Pommerenke, Carmona, Cufi and others, are extended to n-dimensional quasiconformal mappings. Unlike what is usually the case in the context of analytic functions, our considerations are not restricted to mappings of a disk or ball only. It is shown, for instance, that quasiconformal cluster sets and boundary cluster sets, taken at a non-isolated boundary point of an arbitrary domain, coincide. More refined versions are established in the special case where the domain is the open unit ball. These include cluster set considerations of the induced radial…

Quasiconformal distortion on arcs

Boundary angles, cusps and conformal mappings

Let f be a conformal mapping of a bounded Jordan domain D in the complex plane onto the unit disk . This paper examines the consequences for the local geometry of D near a boundary point z 0 of the mapping f-or, to be more precise, of the homeomorphic extension of this mapping to the closure of D—satisfying a Holder condition at z 0 or, alternatively, of its inverse satisfying a Holder condition at the point f(z 0). In particular, the compatibility of Holder conditions with the presence of cusps in the boundary of D is investigated.

Extremal length and Hölder continuity of conformal mappings

Boundary modulus of continuity and quasiconformal mappings

Let D be a bounded domain in R n , n ‚ 2, and let f be a continuous mapping of D into R n which is quasiconformal in D. Suppose that jf(x) i f(y)j • !(jx i yj) for all x and y in @D, where ! is a non-negative non-decreasing function satisfying !(2t) • 2!(t) for t ‚ 0. We prove, with an additional growth condition on !, that jf(x) i f(y)jC maxf!(jx i yj);jx i yj fi g

Asymptotic values and hölder continuity of quasiconformal mappings

Lipschitz conditions,b-arcwise connectedness and conformal mappings

Boundary Hölder Continuity and Quasiconformal Mappings

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 observati…

Boundary regularity and the uniform convergence of quasiconformal mappings

Quasiconformally Bi-Homogeneous Compacta in the Complex Plane