We show a mean curvature 1 Enneper cousin in hyperbolic 3-space. (Hyperbolic 3-space is shown here using the Poincare model.) This surface is isometric to the minimal Enneper surface in Euclidean 3-space. The second surface we show is the dual surface to the Enneper cousin shown here. Only one of four congruent pieces, with the end cut away, of each surface is shown.
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 the Enneper cousins, and there is a one parameter family of these surfaces, depending on a positive real parameter mu (see Bryant's work). The value of mu chosen for the surface here is mu=1.4.
The hyperbolic Gauss map (described in  and ) of the Enneper cousin has an essential singularity at the end. Since the hyperbolic Gauss map is the map which takes each point on the surface to the asymptotic class of the normal geodesic starting at that point and oriented in the mean curvature vector's direction, this essential singularity is reflected in the fact that the end wraps around infinitely many times as it approaches the sphere at infinity.
The dual of a surface with lift F (as in ) is the surface whose lift is the inverse of F. Note that the dual of the Enneper cousin has a periodic series of bulges moving out toward the end of the surface. The end in this picture approaches the south pole in the sphere at infinity.
More detailed information about these surfaces can be found in the LaTeX and postscript and pdf and Mathematica files included in this model. (One of the included Mathematica files is a program for drawing general mean curvature 1 surfaces in hyperbolic 3-space, based on general Weierstrass data. Also included is a jvx file marking the boundary of the Poincare model for the hyperbolic 3-space.)
Model produced with: JavaView v.2.00.a2
|Keywords||constant mean curvature surface; minimal surface; Enneper's surface; hyperbolic 3-space|
|MSC-2000 Classification||53A10 (53A35,53A42)|
Submitted: Tue Jan 23 17:56:59 CET 2001.
Revised: Tue Feb 12 09:25:29 CET 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