I. INTRODUCTION

In this paper we carry out the construction of constant mean

3

curvature {cmc) immersed surfaces in R {or more generally the

3

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.

2

Let x(u,v) be a cmc immersion from an open set in R into

3

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

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2 2

(1.1) ds = e (du + dv )

then, as will be developed in Section II, the Gauss equation will

be

(1.2) Aoo + Ae - Be = 0.

2

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~°*

u

1