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On the Poisson Integral of Step Functions and Minimal Surfaces

Published:2002-03-01
Printed: Mar 2002
• Allen Weitsman
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Abstract

Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.
 Keywords: harmonic mappings, dilatation, minimal surfaces
 MSC Classifications: 30C62 - Quasiconformal mappings in the plane 31A05 - Harmonic, subharmonic, superharmonic functions 31A20 - Boundary behavior (theorems of Fatou type, etc.) 49Q05 - Minimal surfaces [See also 53A10, 58E12]