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 , 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
|Keywords||constant mean curvature surface; surface of revolution; catenoid; hyperbolic 3-space|
|MSC-2000 Classification||53A10 (53A35,53A42)|
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.
Kobe UniversityMasaaki Umehara
Faculty of Science
Rokko, Kobe 657-8501
Hiroshima UniversityKotaro Yamada
Faculty of Science
Kyushu University 36, 6-10-1
Faculty of Mathematics
Hakozaki, Higashi-ku, Fukuoka 812-8185