On the subinvariance of uniform domains in Banach spaces
Abstract
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D'$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Väisälä in the affirmative.
 Publication:

arXiv eprints
 Pub Date:
 September 2012
 arXiv:
 arXiv:1209.4541
 Bibcode:
 2012arXiv1209.4541H
 Keywords:

 Mathematics  Metric Geometry;
 Primary: 30C65;
 30F45;
 Secondary: 30C20
 EPrint:
 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1105.4684