Harmonic shears of slit and polygonal mappings

Saminathan Ponnusamy; Tri Quach; Antti Rasila

doi:10.1016/j.amc.2014.01.076

Harmonic shears of slit and polygonal mappings

Saminathan Ponnusamy, Tri Quach^*, Antti Rasila

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

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 ω. 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.

Original language	English
Pages (from-to)	588-598
Number of pages	11
Journal	Applied Mathematics and Computation
Volume	233
DOIs	https://doi.org/10.1016/j.amc.2014.01.076
State	Published - 1 Mar 2014
Externally published	Yes

Keywords

Convex along real directions
Convex functions
Harmonic shear
Harmonic univalent mappings
Minimal surfaces
Polygonal mappings
Slit mappings

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title = "Harmonic shears of slit and polygonal mappings",

abstract = "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 ω. 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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AU - Quach, Tri

AU - Rasila, Antti

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N2 - 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 ω. 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.

AB - 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 ω. 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.

KW - Convex along real directions

KW - Convex functions

KW - Harmonic shear

KW - Harmonic univalent mappings

KW - Minimal surfaces

KW - Polygonal mappings

KW - Slit mappings

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