## The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

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- by Joaquín Pérez and Martin Traizet PDF
- Trans. Amer. Math. Soc.
**359**(2007), 965-990 Request permission

## Abstract:

Given an integer $k\geq 2$, let ${\mathcal S}(k)$ be the space of complete embedded singly periodic minimal surfaces in $\mathbb {R}^3$, which in the quotient have genus zero and $2k$ Scherk-type ends. Surfaces in ${\mathcal S}(k)$ can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that ${\mathcal S}(2)$ consists of the $1$-parameter family of singly periodic Scherk minimal surfaces. We prove that for each $k\geq 3$, there exists a natural one-to-one correspondence between ${\mathcal S}(k)$ and the space of convex unitary nonspecial polygons through the map which assigns to each $M\in {\mathcal S}(k)$ the polygon whose edges are the flux vectors at the ends of $M$ (a special polygon is a parallelogram with two sides of length $1$ and two sides of length $k-1$). As consequence, ${\mathcal S}(k)$ reduces to the saddle towers constructed by Karcher.## References

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## Additional Information

**Joaquín Pérez**- Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain
- Email: jperez@ugr.es
**Martin Traizet**- Affiliation: Faculte des Sciences et Techniques, Universite François Rabelais, Parc de Grandmont, 37200, Tours, France
- Email: martin.traizet@lmpt.univ-tours.fr
- Received by editor(s): September 29, 2004
- Published electronically: October 16, 2006
- Additional Notes: The research of the first author was partially supported by a MEC/FEDER grant no. MTM2004-02746.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 965-990 - MSC (2000): Primary 53A10; Secondary 49Q05, 53C42
- DOI: https://doi.org/10.1090/S0002-9947-06-04094-3
- MathSciNet review: 2262839