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Harmonic shears of slit and polygonal mappings

Saminathan Ponnusamy
Organization: Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security)
Tri Quach
Organization: Aalto University
Department: Department of Mathematics and Systems Analysis
Antti Rasila
Organization: Department of Mathematics and Systems Analysis, Aalto University
Journal / Anthology

Applied Mathematics and Computation
Year: 2014
Volume: 233
Page range: 588598

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 x. 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.

*Applied Mathematics