ENNEPER TYPE IMMERSIONS 9

We now impose the condition that the immersion be of Enneper

type. Specifically we demand that the lines of curvature

corresponding to the parameter lines u = constant be spherical.

Theorem 2.2: Suppose x(u,v) is a conformal cmc immersion

3

into M (c) of Enneper type parameterized by its lines of curvature

curvature such that the lines of curvature in the v-direction are

spherical and with fundamental forms (2.1). Besides satisfying

the Gauss equation

2(0

-2C0

(2.16) Aw + Ae - Be = 0

a) For H = 1/2 in R3, A = B = 1/4

b) For H = 0 i n R 3 , A = 0, B = 1

c) For H = 0 i n H 3 , A = -1, B = 1

there are functions a(u), /?(u) such that

(2.17) 2co = a(u)eW + ft(u)e~°d .

u

Proof: To be discussed in Sections III and IV.

We now want to solve the system (2. 16) (2. 17) . A key step is

the following result.

Theorem 2.3: Let oo(u,v) be a solution to the system

(2.16 - 17)

*

20)

„ -20)

Aco + Ae - Be = 0

2w = a(u)e^ + /?(u)e~ .

u

If OJ T£ 0 then the functions (a(u), /?(u)} are solutions to

v

th e system

(2.18)

a " = aa - 2a2/? - 2A/?

ft" = aft - 2a/?2 - 2Ba