On Pavlovic's theorem in space (Q905619)

In his well-known paper [Ann. Acad. Sci. Fenn., Math. 27, No. 2, 365--372 (2002; Zbl 1017.30014)], \textit{M. Pavlović} proved that harmonic quasiconformal (qc) maps of the unit disk onto itself are bi-Lipschitz. The authors study counterparts of this theorem in higher dimensions and prove the following result. Theorem. Suppose that \(f : B^3 \to B^3\) is a harmonic qc map and that \(f= \nabla u\) for some harmonic function \(u\) of the unit ball \(B^3\). Then \(f\) is a bi-Lipschitz map.