In this paper we carry out the construction of constant mean
curvature {cmc) immersed surfaces in R {or more generally the
spaces M (c) of constant curvature c) which satisfy the following
geometric condition. One family of curvature lines of the
immersed surface is spherical; that is, each curvature line of the
family is to lie on some sphere. Immersed surfaces satisfying
this condition have been the subject of considerable study by
classical differential geometers. An extensive discussion may be
found in the treatise of G. Darboux [4], see also L.P.Eisenhart
[7], and there was a book on the subject by A. Enneper [8] in
1880. For this reason we shall call any surface satisfying this
geometric condition a surface of Enneper type.
Let x(u,v) be a cmc immersion from an open set in R into
M (c). Suppose that the immersion is conformal and that the
preimage of the lines of curvature are straight lines parallel
to the coordinate axes. If we write the first fundamental form
2 2
(1.1) ds = e (du + dv )
then, as will be developed in Section II, the Gauss equation will
(1.2) Aoo + Ae - Be = 0.
Here B is a positive constant while A = H + c where H is the mean
curvature of the immersion. The character of the solution is
strongly influenced by the sign of A so we shall consider each of
3 3
these possibilities: H = 1/2 in R , minimal surfaces H = 0 in R ,
3 3
and minimal surfaces H = 0 in hyperbolic 3-space M (-1) = H . If
the immersion is of Enneper type then the following auxilliary
condition must be satisfied
(1.3) 2co = a(u)eW + fi(u)e~°*
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