Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry

The singly periodic Scherk surfaces with higher dihedral symmetry have $2 n$ -ends that come together based upon the value of $φ$ . These surfaces are embedded provided that $\frac{π}{2} - \frac{π}{n} < \frac{n - 1}{n} φ < \frac{π}{2}$ . 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.

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

Vol. 6, No. 4, 2013

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

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors