首页 | 本学科首页   官方微博 | 高级检索  
     


Planar Harmonic Maps with Inner and Blaschke Dilatations
Authors:Laugesen   Richard Snyder
Affiliation:Department of Mathematics, Johns Hopkins University Baltimore, Maryland 21218-2689, USA. E-mail: laugesen{at}math.jhu.edu
Abstract:A univalent harmonic map of the unit disk {Delta}:={zisinC:|z|<1} isa complex-valued function f(z) on {Delta} that satisfies Laplace'sequation Formula and is injective. The Jacobian Formula of a univalent harmonic map can never vanish [18], and so we might as wellassume that J>0 throughout {Delta}. Then |fz|>0 and a short computationverifies that the analytic dilatation Formula is indeed an analytic function, with |{omega}|<1 sinceJ>0. Clearly {omega}{equiv}0 when f is a conformal map, and in generalthe 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 coordinatemap for a minimal surface, and in that case i/{surd}{omega} equals the stereographicprojection of the Gauss map of the surface.
Keywords:
本文献已被 Oxford 等数据库收录!
设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号