A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

We present a two-parameter family of minimal surfaces constructed by lifting a family of planar harmonic mappings. In the process, we use the Clunie and Sheil-Small shear construction for planar harmonic mappings convex in one direction. This family of minimal surfaces, through a continuous transfor...

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Main Authors: Michael Dorff, Stacey Muir
Format: Article
Language:English
Published: Hindawi Limited 2014-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2014/476061
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spelling doaj-5bd1a84eba764199a52c6180c654aab52020-11-24T22:02:07ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/476061476061A Family of Minimal Surfaces and Univalent Planar Harmonic MappingsMichael Dorff0Stacey Muir1Department of Mathematics, Brigham Young University, Provo, UT 84602, USADepartment of Mathematics, University of Scranton, Scranton, PA 18510, USAWe present a two-parameter family of minimal surfaces constructed by lifting a family of planar harmonic mappings. In the process, we use the Clunie and Sheil-Small shear construction for planar harmonic mappings convex in one direction. This family of minimal surfaces, through a continuous transformation, has connections with three well-known surfaces: Enneper’s surface, the wavy plane, and the helicoid. Moreover, the identification process used to recognize the surfaces provides a connection to surfaces that give tight bounds on curvature estimates first studied in a 1988 work by Hengartner and Schober.http://dx.doi.org/10.1155/2014/476061
collection DOAJ
language English
format Article
sources DOAJ
author Michael Dorff
Stacey Muir
spellingShingle Michael Dorff
Stacey Muir
A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
Abstract and Applied Analysis
author_facet Michael Dorff
Stacey Muir
author_sort Michael Dorff
title A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
title_short A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
title_full A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
title_fullStr A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
title_full_unstemmed A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
title_sort family of minimal surfaces and univalent planar harmonic mappings
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2014-01-01
description We present a two-parameter family of minimal surfaces constructed by lifting a family of planar harmonic mappings. In the process, we use the Clunie and Sheil-Small shear construction for planar harmonic mappings convex in one direction. This family of minimal surfaces, through a continuous transformation, has connections with three well-known surfaces: Enneper’s surface, the wavy plane, and the helicoid. Moreover, the identification process used to recognize the surfaces provides a connection to surfaces that give tight bounds on curvature estimates first studied in a 1988 work by Hengartner and Schober.
url http://dx.doi.org/10.1155/2014/476061
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AT staceymuir afamilyofminimalsurfacesandunivalentplanarharmonicmappings
AT michaeldorff familyofminimalsurfacesandunivalentplanarharmonicmappings
AT staceymuir familyofminimalsurfacesandunivalentplanarharmonicmappings
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