Planar Harmonic Maps with Inner and Blaschke Dilatations

A univalent harmonic map of the unit disk :={zC:|z|<1} isa complex-valued function f(z) on that satisfies Laplace'sequation and is injective. The Jacobian of a univalent harmonic map can never vanish [18], and so we might as wellassume that J>0 throughout . Then |f_z|>0 and a short computationverifies that the analytic dilatation is indeed an analytic function, with ||<1 sinceJ>0. Clearly 0 when f is a conformal map, and in generalthe dilatation measures how far f is from being conformal.Also, if happens to be the square of an analytic function,then f ‘lifts’ to give an isothermal coordinatemap for a minimal surface, and in that case i/ equals the stereographicprojection of the Gauss map of the surface.