Beltrami equation for the harmonic diffeomorphisms between surfaces

Abstract

In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefore is a harmonic function. The real part of the logarithm of the Beltrami function satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is an elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unified way. The harmonic maps are therefore classified by the classification of the solutions of the sinh-Gordon equation.