Involve: A Journal of Mathematics

  • Involve
  • Volume 6, Number 4 (2013), 383-392.

Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry

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

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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 π2πn<n1nφ<π2. 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.

Article information

Source
Involve, Volume 6, Number 4 (2013), 383-392.

Dates
Received: 23 May 2012
Revised: 24 July 2012
Accepted: 25 July 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733606

Digital Object Identifier
doi:10.2140/involve.2013.6.383

Mathematical Reviews number (MathSciNet)
MR3115973

Zentralblatt MATH identifier
1356.31001

Subjects
Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 49Q05: Minimal surfaces [See also 53A10, 58E12] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Keywords
minimal surfaces harmonic mappings Scherk univalence

Citation

Bucaj, Valmir; Cannon, Sarah; Dorff, Michael; Lawson, Jamal; Viertel, Ryan. Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry. Involve 6 (2013), no. 4, 383--392. doi:10.2140/involve.2013.6.383. https://projecteuclid.org/euclid.involve/1513733606


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