## Tohoku Mathematical Journal

### Surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends

Zuhuan Yu

#### Abstract

We investigate surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends, and prove that their irregular ends must self-intersect, which answers affirmatively a conjecture of Umehara and Yamada. Moreover we also obtain an explicit representation of a constant mean curvature one surface and a new minimal surface in the Euclidean three-space.

#### Article information

Source
Tohoku Math. J. (2), Volume 53, Number 2 (2001), 305-318.

Dates
First available in Project Euclid: 3 May 2007

https://projecteuclid.org/euclid.tmj/1178207483

Digital Object Identifier
doi:10.2748/tmj/1178207483

Mathematical Reviews number (MathSciNet)
MR1829983

Zentralblatt MATH identifier
1027.53011

#### Citation

Yu, Zuhuan. Surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends. Tohoku Math. J. (2) 53 (2001), no. 2, 305--318. doi:10.2748/tmj/1178207483. https://projecteuclid.org/euclid.tmj/1178207483

#### References

• [1] R. BRYANT, Surfaces of constant mean curvature one in the hyperbolic 3-space, Asterisque 144-145 (1987), 321-347.
• [2] P. COLLIN, L. HAUSWIRTH AND H. ROSENBERG, The geometry of finite topology surfaces properly embed ded in hyperbolic space with constant mean curvature one, (preprint).
• [3] N. KOREVAAR, R. KUSNER, W. H. MEEKS AND B. SOLOMON, Constant mean curvature surfaces in hyper bolic space, Amer. J. Math. 114 (1992), 1-43.
• [4] H. L. TURRITTIN, Convergent solutions of ordinary linear homogeneous differential equations in the neibor hood of an irregular singular point, Acta Math. 93 (1955), 27-65.
• [5] M. UMEHARA AND K. YAM ADA, Complete surfaces of constant mean curvature one in the hyperbolic 3 space, Ann. of Math. (2) 137 (1993), 611-638.
• [6] M. UMEHARA AND K. YAMADA, A duality on CMC-1 surfaces in hyperbolic analogue of the Osserma inequality, Tsukuba J. Math. 21 (1997), 229-237.
• [7] Z. H. Yu, The value distribution of hyperbolic Gauss maps, Proc. Amer. Math. Soc. 125 (1997), 2997-3001