Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball (Q600646)

Plane harmonic mappings and their connections to quasiconformal mappings have recently been extensively studied, see [\textit{P. Duren}, Harmonic mappings in the plane. Cambridge Tracts in Mathematics 156. Cambridge: Cambridge University Press (2004; Zbl 1055.31001)] and [\textit{D. Kalaj}, Math. Z. 260, No. 2, 237--252 (2008; Zbl 1151.30014)]. Little is known of the situation in higher dimensions. The authors prove a regularity result for small \(K\): If \(f\) is a \(K\)-quasiconformal harmonic mapping of the unit ball \(B^n, \, n >2,\) onto itself, then \(f\) is bi-Lipschitz provided that \(K<2^{n-1}\). There is a similar result of \textit{L.-F. Tam} and \textit{T. Y. H. Wan} [Pac. J. Math. 182, No. 2, 359--383 (1998; Zbl 0892.58017)] where hyperbolic harmonic mappings with respect to the hyperbolic metric were studied. Although Möbius transformations do not provide an invariant class of mappings for harmonic functions in space, the proof makes use of these and the method of an earlier paper of the first author where the plane case was considered.