# Quasiconformal Mappings Onto John Domains

@article{Heinonen1989QuasiconformalMO, title={Quasiconformal Mappings Onto John Domains}, author={Juha M. Heinonen}, journal={Revista Matematica Iberoamericana}, year={1989}, volume={5}, pages={97-123} }

In this paper we study quasiconformal homeomorphisms of the unit ball B = Bn = {x I Rn: |x| < 1} of Rn onto John domains. We recall that John domains were introduced by F. John in his study of rigidity of local quasi-isometries [J]; the term John domain was coined by O. Martio and J. Sarvas seventeen years later [MS]. From the various equivalent characterizations we shall adapt the following definition based on diameter carrots, cf. [V4], [V5], [NV].

## 35 Citations

The quasiconformal subinvariance property of John domains in $${\mathbb {R}}^n$$Rn and its applications

- Mathematics
- 2015

The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in…

Quasihyperbolic geodesics in John domains in R^n

- Mathematics
- 2011

In this paper, we prove that if $D\subset R^n$ is a John domain which is homeomorphic to a uniform domain via a quasiconformal mapping, then each quasihyperbolic geodesic in $D$ is a cone arc, which…

Gromov hyperbolic John is quasihyperbolic John I

- Mathematics
- 2021

In this paper, we introduce a concept of quasihyperbolic John spaces and provide a necessary and sufficient condition for a space to be quasihyperbolic John. Using this criteria, we exhibit a simple…

Gromov hyperbolic John is quasihyperbolic John II

- Mathematics
- 2021

In this paper, we introduce a concept of quasihyperbolic John spaces (with center) and provide a criteria to determine spaces to be quasihyperbolic John. As an application, we provide a simple proof…

Quasiconformal maps in metric spaces with controlled geometry

- Mathematics
- 1998

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both…

The quasiconformal subinvariance property of John domains in $\protect \IR^n$ and its application

- Mathematics
- 2011

The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in…

ON THE QUASISYMMETRY OF QUASICONFORMAL MAPPINGS AND ITS APPLICATIONS

- Mathematics
- 2012

Suppose that D is a proper domain in R n and that f is a quasicon- formal mapping from D onto a John domain D 0 in R n . First, we show that if D and D 0 are bounded, and D is a broad domain, then…

Applications of the quasihyperbolic metric

- Mathematics
- 2018

A lower bound is given for the αdimensional Hausdorff content of the set of points in the boundary of Ω which can be joined to z by a John curve with a suitable John constant depending only on α, in terms of the distance of z to ∂Ω.

Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality

- Mathematics
- 2017

Using the metric duality theory developed by Vaisala, we characterize generalized John domains in terms of higher dimensional homological bounded turning for its complement under mild assumptions.…