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Saddle towers and minimal k-noids in ℍ2 × ℝ

Published online by Cambridge University Press:  21 June 2011

Filippo Morabito
Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain
M. Magdalena Rodríguez
Departamento de Geometría y Topología, Universidad de Granada, Fuentenueva s/n, 18071, Granada, Spain (


Given k ≥ 2, we construct a (2k − 2)-parameter family of properly embedded minimal surfaces in ℍ2 × ℝ invariant by a vertical translation T, called saddle towers, which have total intrinsic curvature 4π(1 − k), genus zero and 2k vertical Scherk-type ends in the quotient by T. Each of those examples is obtained from the conjugate graph of a Jenkins–Serrin graph over a convex polygonal domain with 2k edges of the same (finite) length. As limits of saddle towers, we obtain properly embedded minimal surfaces, called minimal k-noids, which are symmetric with respect to a horizontal slice (in fact they are vertical bi-graphs) and have total intrinsic curvature 4π(1 − k), genus zero and k vertical planar ends.

Research Article
Copyright © Cambridge University Press 2012

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1.Collin, P. and Rosenberg, H., Construction of harmonic diffeomorphisms and minimal graphs, Annals Math. 172(3) (2010), 18791906.CrossRefGoogle Scholar
2.Daniel, B., Isometric immersions into × ℝ and ℍn × ℝ and applications to minimal surfaces, Trans. Am. Math. Soc. 361(12) (2009), 62556282.CrossRefGoogle Scholar
3.Hauswirth, L. and Rosenberg, H., Minimal surfaces of finite total curvature in ℍ2 × ℝ, Mat. Contemp. 31 (2006), 6580.Google Scholar
4.Hauswirth, L., Earp, R. Sa and Toubiana, E., Associate and conjugate minimal immersions in ℍ2 × ℝ, Tohoku Math. J. 60(2) (2008), 267286.CrossRefGoogle Scholar
5.Jenkins, H. and Serrin, J., Variational problems of minimal surface type II: boundary value problems for the minimal surfaces equation, Arch. Ration. Mech. Analysis 21 (1966), 321342.CrossRefGoogle Scholar
6.Karcher, H., Embedded minimal surfaces derived from Scherk examples, Manuscr. Math. 62 (1988), 83114.CrossRefGoogle Scholar
7.Karcher, H., Construction of minimal surfaces, in Surveys in geometry, pp. 196 (University of Tokyo, 1989).Google Scholar
8.Karcher, H., Introduction to conjugate Plateau constructions, in Global theory of minimal surfaces, Clay Mathematics Proceedings, Volume 2, pp. 137161 (American Mathematical Society, Providence, RI, 2005).Google Scholar
9.Mazet, L., Rodríguez, M. M. and Traizet, M., Saddle towers with infinitely many ends, Indiana Univ. Math. J. 56(6) (2007), 28212838.CrossRefGoogle Scholar
10.Mazet, L., Rodríguez, M. M. and Rosenberg, H., The Dirichlet problem for the minimal surface equation with possible infinite boundary data over domains in a Riemannian surface, Proc. Lond. Math. Soc., in press.Google Scholar
11.Meeks, W. H. III, Pérez, J. and Ros, A., Properly embedded minimal planar domains, preprint (available at Scholar
12.Nelli, B. and Rosenberg, H., Minimal surfaces in ℍ2 × ℝ, Bull. Braz. Math. Soc. 33 (2002), 263292.CrossRefGoogle Scholar
13.Pérez, J. and Traizet, M., The classification of singly periodic minimal surfaces with genus zero and Scherk type ends, Trans. Am. Math. Soc. 359(3) (2007), 965990.CrossRefGoogle Scholar
14.Pyo, J., Singly-periodic minimal surfaces in ℍ2 × ℝ, preprint.Google Scholar
15.Pyo, J., New complete embedded minimal surfaces in ℍ2 × ℝ, Annals Glob. Analysis Geom., in press.Google Scholar
16.Scherk, H. F., Bemerkungen Über die kleinste Fläche innerhalb gegebener Grenzen, J. Reine Angew. Math. 13 (1935), 185208.Google Scholar