Pairs of domains with quasiconformality coefficient unity (Q1335898)

Consider the family \(\mathfrak F\) of domains in \(\mathbb{R}^ n\), \((n\geq 3)\) quasiconformally equivalent to a ball. For two such domains \(D\) and \(D^*\) in \(\mathfrak F\), we define the coefficient of quasiconformality of \(D\) with respect to \(D^*\) as the number \(K(D,D^*)= \inf K(f)\), where the infimum is calculated over all quasiconformal maps \(f: D\to D^*\). See [(*) \textit{J. Väisälä}, ``Lectures on \(n\)-dimensional quasiconformal mappings'', (1971; Zbl 0221.30031)]. The quantity \(d(D,D^*)= \log(k(D,D^*))\) provides a pseudometric for \(\mathfrak F\). \textit{F. W. Gehring} and \textit{J. Väisälä},``The coefficients of quasiconformality of domains in space'', Acta Math. 114, 1-70 (1965; Zbl 0134.297)] proved that the pseudometric space \(\mathfrak F\) is complete and non-separable. Furthermore, factorization by the equivalence relation \((D\sim D^*)\leftrightarrow (d(D,D^*)= 0)\) renders \(\mathfrak F\) a metric space. One then asks about the relation of domains in an equivalence class: Does the constraint \(d(D,D^*)= 0\) imply that \(D^*\) is a Möbius image of \(D\)? The question reduces to that of the existence of an extremal quasiconformal mapping \(\widetilde f: D\to D^*\) with \(K(\widetilde f)= K(D,D^*)\). \textit{J. Väislälä} [(*)] established established the result that an extremal mapping \(\widetilde f\) exists in the two cases (i) \(D\) or \(D^*\) is a ball or (ii) \(\partial D\) has exactly \(k\) components, where \(2\leq k<\infty\). The author [``On inner quasiconformality coefficient for a pair of dihedral wedges'', Sib. Math. J. 29, No. 6, 884-887 (1988); translation from Sib. Mat. Zh. 29, No. 6(172), 12-16 (1988; Zbl 0695.30016), and ``On extremal mappings of dihedral wedges'', Sov., Math., Dokl. 43, No. 1, 162-165 (1991); translation from Dokl. Akad. Nauk SSSR 316, No. 4, 788-791 (1991; Zbl 0758.30017)] later proved for certain pairs of dihedral wedge domains with \(K(D,D^*)> 1\) that no extremal mapping exists. In the paper currently under review, the author settles the remaining case for extremal maps. He constructs a pair of domains \(D\) and \(D^*\) with \(K(D,D^*)= 1\) for which there is no Möbius map \(\mu\) satisfying \(\mu(D)= D^*\). In fact, the constructed domains satisfy \(L(D,D^*)= 1\), so an analogous bi-Lipschitz theorem can be stated as well. (Here, \(L(D,D^*)= \inf L(f)\), where the infimum is taken over all bi-Lipschitz homeomorphisms \(f: D\to D^*\) and \(L(f)\) is the bi- Lipschitz coefficient of \(f\)).