## Involve: A Journal of Mathematics

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

### Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry

#### 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 $π2−πn. 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

#### 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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