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Richard Snyder Laugesen, Planar Harmonic Maps with Inner and Blaschke Dilatations, Journal of the London Mathematical Society, Volume 56, Issue 1, August 1997, Pages 37–48, https://doi.org/10.1112/S0024610797005346
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Abstract
A univalent harmonic map of the unit disk Δ:={z∈C:∣z∣<1} is a complex-valued function f(z) on Δ that satisfies Laplace's equation and is injective. The Jacobian of a univalent harmonic map can never vanish [18], and so we might as well assume that J>0 throughout Δ. Then ∣fz∣>0 and a short computation verifies that the analytic dilatation is indeed an analytic function, with ∣ω∣<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.
