Uniform domains and quasiconformal mappings on the Heisenberg group (Q1895707)

Let \(H\) be the Heisenberg group and \(f: H\to H\) a quasiconformal mapping [\textit{A. Korányi}, \textit{H. M. Reimann}, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Advances in Math. (to appear)]. It is shown that the image of a uniform on an NTA (non- tangentially accessible) domain is again a uniform or an NTA domain. For a study of this phenomenon in \(\mathbb{R}^ n\) see [\textit{O. Martio} and \textit{J. Sarvas}, Ann. Acad. Sci. Fenn., Ser. A I 4, 383-401 (1979; Zbl 0427.30021)]. The proof makes use of the Carnot-Carathéodory distance. Several examples of quasiconformal mappings in \(H\) are constructed with a special attention to volume preserving mappings.