ENNEPER TYPE IMMERSIONS 5
treated in Darboux [4] (see also Eisenhart [7]). In this case one
wants to solve the hyperbolic sine-Gordon equation. The treatment
in both cases is similar.
Finally, H. Dobriner [5] studied the case of Enneper type
surfaces of constant negative Gauss curvature K = -1. His
treatment is based on the book of Enneper [8] and expresses the
immersions using theta functions. His work is discussed in
Darboux [4]. The analysis in the present work follows the
approach in Darboux's treatise.
If a minimal surface in R has one family of planar curvature
lines then the same must be true of the other family. Besides the
catenoid and Enneper's minimal surface there is a one-parameter
family of such surfaces called Bonnet's surfaces. These are
described in Eisenhart [7]. In some sense these surfaces serve to
connect the catenoid to Enneper*s minimal surface. Finally, in
Section V we shall find that surfaces of revolution are the only
Joachimsthal type minimal surface in hyperbolic space, a somewhat
surprising result.
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