ENNEPER TYPE IMMERSIONS 3
remarkable paper by U. Pinkall and I. Sterling [11]. These
solutions fall into classes indexed by the positive integers. The
solutions constructed here lie in the class N = 2. Solutions of
finite type are also amenable to the techniques of soliton theory.
This aspect has recently been carried out by A.I. Bobenko [2].
In Section III we also look at some specific interesting
examples. For any integer n 2 one can construct an immersed
cylinder with two embedded ends which are asymptotically round
cylinders and which has n sphere-like lobes attached in the
middle. It is symmetric about a plane perpendicular to the axis
passing through these lobes. This striking surface seems to have
been first observed by Pinkall and Sterling and a computer picture
of it appears in their paper [11]. Instead of embedded ends which
are round cylinders one can also construct surfaces whose embedded
ends are asymptotic to Delaunay unduloids of any type. Again
these cmc immersions will have n-lobes in the middle. These
immersions now come in one-parameter families. Also some of the
complicated quasi-periodic examples illustrated in [11] seem to be
of Enneper type.
One finds many possibilities of immersed tori of Enneper
type. In fact, one expects that they should fall into
one-parameter families, and one verifies that this is true at
least in certain cases.
3
In Section IV we look at the case of minimal surfaces in R .
This problem was solved by H. Dobriner [6] in 1887 exhibiting the
possible minimal surfaces using theta functions. For the sake of
completeness I rederive the results here. The Gauss equation is
now the classical Liouville equation. The solution co(u,v)
constructed here will have the same general features as for the
case H = 1/2 except that oo(u,v) will have scattered in the
plane points where the solution becomes positively infinite.
These correspond to flat ends for the immersed minimal surface.
In particular, we look for the Weierstrass representation of these
surfaces which is expressed explicitly in terms of the Weierstrass
P-function. The surfaces depicted here resemble a catenoid,
perhaps covered infinitely often, from which a number of flat ends
have been extruded.
In Section V we consider the case of minimal immersions into
3 3
hyperbolic 3-space M (-1) = H . This leads to the Gauss equation
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