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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lawson correspondence and Ribaucour transformations


Authors: M. Lemes, P. Roitman, K. Tenenblat and R. Tribuzy
Journal: Trans. Amer. Math. Soc. 364 (2012), 6229-6258
MSC (2010): Primary 53C20
Published electronically: July 11, 2012
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Abstract: We prove that Darboux transformations commute with the Lawson correspondence and we show that the property of completeness is preserved by this commutativity. We provide examples of these results. Two applications provide families of explicitly parametrized complete surfaces of constant mean curvature 1 and $ -\sqrt {5}/2$ in $ \mathbb{H} ^3$, depending on 2 parameters and 1 parameter respectively. For special choices of the parameters, we get surfaces that are periodic in one variable and in particular complete cmc surfaces or cmc1 surfaces in $ \mathbb{H}^3$, with any finite or infinite number of bubbles, ``segments'' or embedded ends of horosphere type. Moreover, we consider Ribaucour transformations for associated linear Weingarten surfaces in space forms. We show that such a transformation is a Darboux transformation (i.e., it is conformal) if and only if the surfaces have the same constant mean curvature. We prove that Ribaucour transformations for surfaces with constant mean curvature 1 (cmc1) immersed in the hyperbolic space $ \mathbb{H}^3$ produce embedded ends of horosphere type.


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Additional Information

M. Lemes
Affiliation: Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil
Email: max@mat.ufg.br

P. Roitman
Affiliation: Departmento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil
Email: roitman@mat.unb.br

K. Tenenblat
Affiliation: Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil
Email: K.Tenenblat@mat.unb.br

R. Tribuzy
Affiliation: Departamento de Matemática, Universidade Federal do Amazonas, Manaus, AM, Brazil
Email: tribuzy@pq.cnpq.br

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05440-7
PII: S 0002-9947(2012)05440-7
Keywords: Ribaucour transformations, constant mean curvature surfaces, Lawson correspondence, Darboux transformations, horosphere type end.
Received by editor(s): March 12, 2010
Received by editor(s) in revised form: July 30, 2010
Published electronically: July 11, 2012
Additional Notes: This work was partially supported by CNPq, CAPES/PROCAD
Article copyright: © Copyright 2012 American Mathematical Society