Lipschitz conditions in conformally invariant metrics (Q1177190)

Let \(\gamma_ n\) and \(\tau_ n\) denote the capacities of the Grötzsch and Teichmüller rings in \(\mathbb{R}^ n\), respectively, and let \[ \varphi_{k,n}(r)=1/\gamma_ n^{-1}(k\gamma_ n(1/r)),\qquad \theta_{k,n}(r)=1/\tau_ n^{-1}(k\gamma_ n(1/r)), \] for \(r\in(0,1)\), \(k\geq 1\). Using these functions the authors obtain estimates for two conformally invariant metrics \(\mu_ G\), \(\lambda_ G^{-1/n}\), \(G\subset\mathbb{R}^ n\) [cf. \textit{M. Vuorinen}, Conformal geometry and quasi-regular mappings, Lect. Notes Math. 1319 (1988; Zbl 0646.30025)]. It is shown that Lipschitz mappings with respect to these metrics have similar estimates for the modulus of continuity as quasiconformal maps. In particular, these maps satisfy a Schwarz lemma but may have infinite linear dilatation at a point (and thus may fail to be quasiconformal). In some special cases \(\lambda_ G\)- and \(\mu_ G\)-isometries are proved to be conformal but the general case is open.