Abstract
We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.
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Kalaj, D., Mateljevic, M. Inner estimate and quasiconformal harmonic maps between smooth domains. J. Anal. Math. 100, 117–132 (2006). https://doi.org/10.1007/BF02916757
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DOI: https://doi.org/10.1007/BF02916757