Mean Curvature 1 Surfaces of Revolution in Hyperbolic 3-Space EG-Models Home

image catenoid_minus0pt4_Preview.gif
image catenoid_minus0pt1_Preview.gif
image catenoid_plus0pt05_Preview.gif
image catenoid_plus3pt0_Preview.gif
Electronic Geometry Model No. 2001.01.048


Wayne Rossman, Masaaki Umehara, and Kotaro Yamada


We show four members of the 1 parameter family of catenoid cousins, which are mean curvature 1 surfaces of revolution in hyperbolic 3-space. Only one of two congruent pieces of each surface is shown. (Hyperbolic 3-space is shown here using the Poincare model.)

Robert Bryant, in [1], found a representation for mean curvature 1 surfaces in hyperbolic 3-space. This representation is similar to the Weierstrass representation for minimal surfaces in Euclidean 3-space, in that it also produces surfaces from a meromorphic function and a holomorphic 1-form on a Riemann surface.

Using this representation, Bryant explicitly described all mean curvature 1 surfaces of revolution, which he called catenoid cousins. There is a one parameter family of these surfaces, depending on a parameter mu (see Bryant's work) that is strictly between -1/2 and 0, or is any positive real. When mu is negative, the surface is embedded. When mu is positive, the surface is not embedded. As mu converges to zero, the surfaces converge to two horopheres that are tangent at one point. (Horospheres also have mean curvature 1.)

The four surfaces shown here have the following values for mu, in order: -0.4, -0.1, +0.05, +3.0.

A more detailed description of these surfaces can be found in the LaTeX, postscript, pdf and Mathematica files included in this model.

Model produced with: JavaView v.2.00.a2

Keywordsconstant mean curvature surface; surface of revolution; catenoid; hyperbolic 3-space
MSC-2000 Classification53A10 (53A35,53A42)
Zentralblatt No.05264873


  1. Robert Bryant: Surfaces of mean curvature one in hyperbolic space, Asterisque 154-155 (1987), 321--347.
  2. Masaaki Umehara and Kotaro Yamada: Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space, Annals of Mathematics 137 (1993), 611--638.


Submission information

Submitted: Tue Jan 23 17:56:59 CET 2001.
Revised: Fri Jan 11 09:50:40 GMT 2002.
Accepted: Wed Feb 20 12:44:06 CET 2002.

Authors' Addresses

Wayne Rossman
Kobe University
Mathematics Department
Faculty of Science
Rokko, Kobe 657-8501
Masaaki Umehara
Hiroshima University
Mathematics Department
Faculty of Science
Higashi-Hiroshima 739-8526
Kotaro Yamada
Kyushu University 36, 6-10-1
Faculty of Mathematics
Hakozaki, Higashi-ku, Fukuoka 812-8185