Quasiconformal harmonic mappings between $\,\mathscr C^{1,μ}$ Euclidean surfaces

Abstract: The conformal deformations are contained in two classes of mappings: quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every $K$ quasiconformal harmonic mapping between $\,\mathscr C^{1,\mu}$ ($0<\alpha\le 1$) surfaces with $\,\mathscr C^{1,\mu}$ boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.