Beltrami equation for the harmonic diffeomorphisms between surfaces

Abstract

We study harmonic maps between surfaces, that are solutions to a nonlinear elliptic PDE. In Refs. Minsky (1992); Wolf (1989) it was proved that harmonic diffeomorphisms, with nonvanishing Hopf differential, satisfy a Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami coefficient coincides with the imaginary part of the logarithm of the Hopf differential. Therefore, it is a harmonic function. The real part of the logarithm of the Beltrami coefficient satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is the elliptic sinh-Gordon equation.In this paper we also prove the converse: if the imaginary part of the logarithm of the Beltrami coefficient is a harmonic function, then the target surface can be equipped with a metric, conformal to the original one, and the solution of the Beltrami equation is a harmonic map. Therefore, solving a certain Beltrami equation is equivalent to solving the harmonic map problem. Harmonic maps to a constant curvature surface are therefore classified by the classification of the solutions of the elliptic sinh-Gordon equation.The general problem of solving the sinh-Gordon equation is a nonlinear problem and is still open. Different well-known harmonic maps to the hyperbolic plane are proved to be related to the one-soliton solutions of the elliptic sinh-Gordon equation. Moreover, an example is proposed which does not belong to the one-soliton solution of the elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unified way, for positive, negative and zero curvature of the target surface.