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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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