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 fzz[bar]=0 and is injective. The Jacobian
J[ratio ]=[mid ]fz[mid ]2
−[mid ]fz[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 ]fz[mid ]>0 and a short computation
verifies that the analytic dilatation ω[ratio ]
=f[bar]z[bar]/fz
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.