Vol. 6, No. 4, 2013

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Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry

Valmir Bucaj, Sarah Cannon, Michael Dorff, Jamal Lawson and Ryan Viertel

Vol. 6 (2013), No. 4, 383–392
Abstract

The singly periodic Scherk surfaces with higher dihedral symmetry have 2n-ends that come together based upon the value of φ. These surfaces are embedded provided that π 2 π n < n1 n φ < π 2 . Previously, this inequality has been proved by turning the problem into a Plateau problem and solving, and by using the Jenkins–Serrin solution and Krust’s theorem. In this paper we provide a proof of the embeddedness of these surfaces by using some results about univalent planar harmonic mappings from geometric function theory. This approach is more direct and explicit, and it may provide an alternate way to prove embeddedness for some complicated minimal surfaces.

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Keywords
minimal surfaces, harmonic mappings, Scherk, univalence
Mathematical Subject Classification 2010
Primary: 30C45, 49Q05, 53A10
Milestones
Received: 23 May 2012
Revised: 24 July 2012
Accepted: 25 July 2012
Published: 8 October 2013

Communicated by Frank Morgan
Authors
Valmir Bucaj
Computer Science, Information Systems and Mathematics
Texas Lutheran University
Seguin, TX 78155
United States
Sarah Cannon
Department of Mathematics
Tufts University
Medford, MA 02155
United States
Michael Dorff
Department of Mathematics
Brigham Young University
Provo, UT 84602
United States
Jamal Lawson
Mathematical Sciences
Loyola University New Orleans
New Orleans, LA 70118
United States
Ryan Viertel
Department of Mathematics
Brigham Young University
Provo, UT 84602
United States