Harmonic Shears of Slit and Polygonal Mappings

Details Harmonic Shears of Slit and Polygonal Mappings Harmonic Shears of Slit and Polygonal Mappings

2012-01-10

arxiv.org/abs/1201.2015v1

Mathematics

Complex Variables

17 pages, 6 figures, 2 tables

Scientific paper

In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations \omega. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica.

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