Quasiconformal concordance
Two maps \(f_ 0,f_ 1: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) are called quasiconformally concordant if there is a quasiconformal homeomorphism F: \({\mathbb{R}}^ n\times I\to {\mathbb{R}}^ n\times I\) such that \(F(x,0)=(f_ 0(x),0)\) and \(F(x,1)=(f_ 1(x),1)\). The main result of the paper gives the following necessary and sufficient condition for the quasiconformal concordance of \(f_ 0\) and \(f_ 1:\) (1) \(f_ 0\) and \(f_ 1\) are quasisymmetric, (2) \(f_ 0\), \(f_ 0^{-1}\), \(f_ 1\), \(f_ 1^{-1}\) are uniformly continuous, (3) \(f_ 0-f_ 1\) is bounded. If \(n\geq 2\), (1) means that \(f_ 0\) and \(f_ 1\) are quasiconformal. A bilipschitz version is also given.
scientific article; zbMATH DE number 4122282