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

TL;DR

This paper proves that quasiconformal harmonic mappings between certain smooth surfaces are Lipschitz continuous and provides explicit Lipschitz constants for minimal surfaces, extending known results from plane domains to surfaces.

Contribution

It establishes Lipschitz continuity for quasiconformal harmonic mappings between $\mathscr C^{1,\mu}$ surfaces and derives explicit Lipschitz constants for minimal surface parametrizations.

Findings

Quasiconformal harmonic mappings are Lipschitz continuous between $\mathscr C^{1,\mu}$ surfaces.

Explicit Lipschitz constants are provided for minimal surface coordinates.

Results extend known plane domain theorems to surface mappings.

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.