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Transactions of the American Mathematical Society

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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
MathSciNet review: 2958934
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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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  • [Bi] L. Bianchi: Le transformazioni di Ribaucour dei sistemi $ \textup {n}^{pli}$ ortogonali e il teorema generale di permutabilitá, Annali di Matematica (3), 27 (1918), 183-253 e (3), 28 (1919), 187-233.
  • [Bi1] L. Bianchi: Complementi alle ricerche sulle superficie isoterme. Ann. Mat. Pura Appl. 12 (1905), 19-54.
  • [Bi2] L. Bianchi: Lezioni de Geometria Differenziale Vol II, Bologna Nicola Zanichelli Editore 1927.
  • [Bu] F. Burstall: Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems. AMS/IP Studies in Advanced Mathematics, Volume 36, 2006. MR 2222512 (2008b:53006)
  • [CFT1] A. V. Corro, W. Ferreira and K. Tenenblat: Minimal surfaces obtained by Ribaucour transformations. Geom. Dedicata 96 (2003), 117-150. MR 1956836 (2004d:53009)
  • [CFT2] A. V. Corro, W. Ferreira and K. Tenenblat: Ribaucour transformations for constant mean curvature and linear Weingarten surfaces, Pacific J. Math. 212 (2003), 265-296. MR 2038049 (2005c:53007)
  • [DPW] J. Dorfmeister, F. Pedit and H. Wu: Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633-668. MR 1664887 (2000d:53099)
  • [ET] R. S. Earp and E. Toubiana: On the geometry of constant mean curvature one surfaces in hyperbolic space. Illinois J. Math. 45 (2001), 371-401. MR 1878610 (2002m:53098)
  • [HJ] U. Hetrich-Jeromin: Introduction to Möbius Differential Geometry. London Mathematical Society, Lecture Note Series 300. Cambridge Univ. Press, 2003. MR 2004958 (2004g:53001)
  • [HJP] U. Hetrich-Jeromin, F. Pedit: Remarks on the Darboux transformation of isothermic surfaces, Doc. Math 2 (1997), 313-333. MR 1487467 (99k:53006)
  • [Ko] S. Kobayashi: Asymptotics of ends of constant mean curvature surfaces with bubbletons. Proc. Amer. Math. Soc. 136 (2008), 1433-1443. MR 2367117 (2009b:53009)
  • [La] B. Lawson, Complete minimal surfaces in $ S^3$. Ann. of Math. (2) 92 (1970), 335-374. MR 0270280 (42:5170)
  • [LT] M. V. Lemes and K. Tenenblat: On Ribaucour transformations and minimal surfaces. Mat. Contemp. 29 (2005), 13-40. MR 2196777 (2006j:53007)
  • [LR] L. L. de Lima and W. Rossman: On the index of constant mean curvature $ 1$ surfaces in hyperbolic space. Indiana Univ. Math. 47 (1998), 685-723. MR 1647877 (2000a:53010)
  • [MRR] W. H. Meeks III, A. Ros, and H. Rosenberg: The Global Theory of Minimal Surfaces in Flat Spaces, Lecture Notes in Mathematics #1775, Springer-Verlag, 2002. MR 1901611 (2003i:53012)
  • [RUY] W. Rossman, M. Umehara and K. Yamada: Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus. Tohoku Math. J. 49 (1997), 449-484. MR 1478909 (99a:53025)
  • [Sc] N. Schmitt: CMCLab software,
  • [Si] H. Sievert: Über die Zentraflächen der Enneperschen Flächen konstanten krümmungsmasses, Diss. Tübingen, 1886.
  • [SW] I. Sterling and H. C. Wente: Existence and classification of cmc multibubbleton of finite and infinite type, Indiana Univ. Math. J. 42 (1993), 1239-1266. MR 1266092 (95a:53015)
  • [TW1] K. Tenenblat and Q. Wang: Ribaucour Transformations for Hypersurfaces in Space Forms, Annals of Global Analysis and Geometry 29 (2006), 157-185. MR 2251001 (2007e:53068)
  • [TW2] K. Tenenblat and Q. Wang: New constant mean curvature surfaces in the hyperbolic space. Illinois Math J. 53 (2009), 135-162. MR 2584939
  • [UY] M. Umehara and K. Yamada: Complete Surfaces of Constant Mean Curvature 1 in the Hyperbolic 3-space. Annals of Mathematics (2) 137 (1993), 611-638. MR 1217349 (94c:53015)

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Additional Information

M. Lemes
Affiliation: Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil

P. Roitman
Affiliation: Departmento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

K. Tenenblat
Affiliation: Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

R. Tribuzy
Affiliation: Departamento de Matemática, Universidade Federal do Amazonas, Manaus, AM, Brazil

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

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