Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups (Q1849648)

Let \(G\) be a Carnot (i.e. stratified nilpotent Lie) group of homogeneous dimension \(Q\). In [J. Geom. Anal. 7, No. 1, 109--148 (1997; Zbl 0905.30018)] \textit{J. Heinonen} and \textit{I. Holopainen} proved the existence of Green's functions for the \(Q\)-Laplacian with pole at the neutral element of \(G\). The paper under review proves, by means of nonlinear potential theory, that such a Green's function \(u\) is unique (up to additive constants). As an important consequence, it is shown that there exists a positive constant \(\gamma=\gamma(G)\) so that \(N=e^{-\gamma u}\) is a homogeneous norm on \(G\). This allows the authors to establish sharp estimates for the extremal length of annular ring domains measured with respect to the Carnot-Carathéodory metric. These estimates are applied to the study of quasiconformal and bi-Lipschitz mappings on Carnot groups. More precisely, the value of the Hölder exponent of a \(K\)-quasiconformal map between open subsets of \(G\) is determined explicitly, and a geometric characterization of the bi-Lipschitz mappings within the quasiconformal mappings is established.