## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 52, Number 4 (2000), 877-898.

### Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space

Reiko AIYAMA and Kazuo AKUTAGAWA

#### Abstract

Our primary object of this paper is to give a representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space of curvature -1 in terms of their normal Gauss maps. For CMC (constant mean curvature) surfaces, we derive another representation formula in terms of their adjusted Gauss maps. These formulas are hyperbolic versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. As an application, we give a construction of complete simply connected CMC $$ surfaces embedded in the hyperbolic 3-space.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 52, Number 4 (2000), 877-898.

**Dates**

First available in Project Euclid: 10 June 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1213107115

**Digital Object Identifier**

doi:10.2969/jmsj/05240877

**Mathematical Reviews number (MathSciNet)**

MR1774634

**Zentralblatt MATH identifier**

0995.53047

**Subjects**

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 58E20: Harmonic maps [See also 53C43], etc.

**Keywords**

Surfaces in the hyperbolic 3-space normal Gauss maps harmonic maps representation formula

#### Citation

AIYAMA, Reiko; AKUTAGAWA, Kazuo. Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space. J. Math. Soc. Japan 52 (2000), no. 4, 877--898. doi:10.2969/jmsj/05240877. https://projecteuclid.org/euclid.jmsj/1213107115