/ 1
x A
( 1 . 7 ) Aoo = e + e
The techniques developed in Section II still apply although now
the separation of variables method becomes more complicated.
Again the solutions will be periodic in the v-direction. Now the
points of singularity which developed for minimal surfaces in R
have expanded into holes. The maximal domain for the solutions to
(1.7) is no longer simply connected. It will be a planar domain
with periodic holes. Also in general the domain will be contained
inside a vertical strip. The lines of curvature in the
v-direction must now lie on spheres, horospheres, or "pseudo-"
spheres. Once again the centers of the spheres all lie on a
geodesic line in H . We also look at some examples. In
particular, one finds examples where the immersion has ends which
are the H counterpart of Delaunay surfaces.
We conclude the introduction with the following historical
remarks. The simplest geometrical condition one can impose on a
cmc surface in R is that it be a surface of revolution. In this
situation one obtains the classical Delaunay surfaces.
A Joachimsthal surface is one for which the lines of
curvature of one family are all planar and such that these planes
all contain a common line, the axis. It follows that the lines of
curvature of the other family are all spherical with the centers of
the spheres lying on the axis. Furthermore, the immersed surface
will intersect these spheres at right angles along these curvature
lines (see Eisenhart [7]).
If x(u,v) is an H = 1/2 immersion in R with planar lines of
curvature in the u-direction then the parallel surface y= x+ t,
where £ is the oriented unit normal vector, is an immersed surface
of constant Gauss curvature K = +1. It is a Joachimsthal surface.
For this reason we call the corresponding cmc immersion a surface
of Joachimsthal type. It is a special case of an Enneper type
surface. These were the surfaces initially studied by the author
in the construction of closed cmc tori in R , [14]. The explicit
representation of these surfaces in terms of elliptic integrals
has been carried out by U. Abresch [1] and also by R. Walter [13].
We should also note that the Joachimsthal surfaces with K = -1 are
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