## Illinois Journal of Mathematics

### New constant mean curvature surfaces in the hyperbolic space

#### Abstract

Applying Ribaucour transformations, we construct two new 3-parameter families of complete surfaces, immersed in $H^3$, with constant mean curvature 1 and infinitely many embedded horosphere type ends. Each surface of the first family is locally associated to an Enneper cousin. It has one irregular end and infinitely many regular ends asymptotic to horospheres. The surfaces of the second family are locally associated to a catenoid cousin. Each surface of this family has infinitely many embedded horosphere type ends and one regular end with infinite total curvature.

#### Article information

Source
Illinois J. Math., Volume 53, Number 1 (2009), 135-161.

Dates
First available in Project Euclid: 22 January 2010

https://projecteuclid.org/euclid.ijm/1264170843

Digital Object Identifier
doi:10.1215/ijm/1264170843

Mathematical Reviews number (MathSciNet)
MR2584939

Zentralblatt MATH identifier
1194.53051

#### Citation

Tenenblat, K.; Wang, Q. New constant mean curvature surfaces in the hyperbolic space. Illinois J. Math. 53 (2009), no. 1, 135--161. doi:10.1215/ijm/1264170843. https://projecteuclid.org/euclid.ijm/1264170843

#### References

• L. Bianchi, Le transformazioni di Ribaucour dei sistemi $\mathrm{n}^{pli}$ ortogonali e il teorema generale di permutabilitá, Annali di Matematica (3) 27 (1918), 183–253 e (3), 28 (1919), 187–233.
• L. Bianchi, Lezioni de Geometria Differnziale Vol II, Bologna Nicola Zanichelli Editore, 1927.
• A. I. Bobenko, All constant mean curvature tori in $\mathbb{R}^3, S^3, H^3$ in terms of theta functions, Math. Ann. 290 (1991), 209–245.
• R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Théorie des variétés minimales et applications (Palaiseau, 1983–1984). Astérisque 154155 (1987), 321–347.
• A. V. Corro, W. Ferreira and K. Tenenblat, Minimal surfaces obtained by Ribaucour transformations, Geom. Dedicata 96 (2003), 117–150.
• 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.
• J. Dorfmeister and G. Haak, Investigation and application of the dressing action on surfaces of constant mean curvature, Q. J. Math. 51 (2000), 57–73.
• 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.
• A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13–72.
• K. Groß e-Brauckmann, New surfaces of constant mean curvature, Math. Zeit. 214 (1993), 527–565.
• N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three spaces, Ann. Math. 131 (1990), 239–330.
• N. Kapouleas, Complete constant mean curvature surfaces constructed by using Wenti tori, Invent. Math. 119 (1995), 443–518.
• H. Karcher, The triply periodic minimal surfaces of A. Scheon and their constant mean curvature companions, Manu. Math. 64 (1989), 291–357.
• H. Karcher, Hyperbolic surfaces of constant mean curvature one with compact fundamental domains, Clay Mathematics Proceedings 2 (2005), 311–323.
• M. Kilian, I. McIntosh and N. Schmitt, New constant mean curvature surfaces, Experiment Math. 9 (2000), 595–611.
• B. Lawson, Complete minimal surfaces in $S^3$, Ann. Math. 92 (1970), 335–374.
• M. V. Lemes and K. Tenenblat, On Ribaucour transformations and minimal surfaces, Mat. Contemp. 29 (2005), 13–40.
• 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.
• W. H. Meeks III, A. Ros, H. Rosenberg, The global theory of minimal surfaces in flat spaces, Lecture Notes In Mathematics, vol. 1775, Springer, Berlin, 2002.
• U. Pinkall and I. Sterling, On classification of constant mean curvature tori, Ann. Math. 130 (1989), 407–451.
• 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.
• H. Sievert, Über die Zentraflächen der Enneperschen Flächen konstanten krümmungsmasses, Diss. Tübingen, 1886.
• 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.
• K. Tenenblat, Transformations of manifolds and applications to differential equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 93, Longman, Harlow, 1998.
• K. Tenenblat and Q. Wang, Ribaucour transformations for hypersurfaces in space forms, Ann. Global Anal. Geom. 29 (2006), 157–185.
• M. Umehara and K. Yamada, Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space, Ann. of Math. 137 (1993), 611–638.
• M. Umehara and K. Yamada, Surfaces of constant mean curvature $c$ in $H^{3}(-c^{2})$ with prescribed hyperbolic Gauss map, Math. Ann. 304 (1996), 203–224.