PLANAR HARMONIC MAPS WITH INNER AND BLASCHKE DILATATIONS

Abstract

A univalent harmonic map of the unit disk Δ[ratio ]={z∈[Copf ][ratio ][mid ]z[mid ]<1} is a complex-valued function f(z) on Δ that satisfies Laplace's equation f_zz[bar]=0 and is injective. The Jacobian J[ratio ]=[mid ]f_z[mid ]² −[mid ]f_z[bar][mid ]² of a univalent harmonic map can never vanish [18], and so we might as well assume that J>0 throughout Δ. Then [mid ]f_z[mid ]>0 and a short computation verifies that the analytic dilatation ω[ratio ] =f[bar]_z[bar]/f_z is indeed an analytic function, with [mid ]ω[mid ]<1 since J>0. Clearly ω≡0 when f is a conformal map, and in general the dilatation ω measures how far f is from being conformal. Also, if ω happens to be the square of an analytic function, then f ‘lifts’ to give an isothermal coordinate map for a minimal surface, and in that case i/√ω equals the stereographic projection of the Gauss map of the surface.