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Journal of the London Mathematical Society 1997 56(1):37-48; doi:10.1112/S0024610797005346
© 1997 by London Mathematical Society
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© The London Mathematical Society

Planar Harmonic Maps with Inner and Blaschke Dilatations

Richard Snyder Laugesen

Department of Mathematics, Johns Hopkins University Baltimore, Maryland 21218-2689, USA. E-mail: laugesen{at}math.jhu.edu

Received 23 February 1995. Revision received 30 June 1995.

A univalent harmonic map of the unit disk {Delta}:={zisinC:|z|<1} is a complex-valued function f(z) on {Delta} that satisfies Laplace's equation Formula and is injective. The Jacobian Formula of a univalent harmonic map can never vanish [18], and so we might as well assume that J>0 throughout {Delta}. Then |fz|>0 and a short computation verifies that the analytic dilatation Formula is indeed an analytic function, with |{omega}|<1 since J>0. Clearly {omega}{equiv}0 when f is a conformal map, and in general the dilatation {omega} measures how far f is from being conformal. Also, if {omega} 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/{surd}{omega} equals the stereographic projection of the Gauss map of the surface.

Department of Mathematics, University of Illinois, Urbana Illinois 61801, USA


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