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

Authors: Joaquín Pérez and Martin Traizet
Journal: Trans. Amer. Math. Soc. 359 (2007), 965-990
MSC (2000): Primary 53A10; Secondary 49Q05, 53C42
Published electronically: October 16, 2006
MathSciNet review: 2262839
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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.

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Additional Information

Joaquín Pérez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

Martin Traizet
Affiliation: Faculte des Sciences et Techniques, Universite François Rabelais, Parc de Grandmont, 37200, Tours, France

Keywords: Singly periodic minimal surface, Scherk-type end, moduli space
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.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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