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

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