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
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
(2.16) 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 .
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)
Aco + Ae - Be = 0
2w = a(u)e^ + /?(u)e~ .
If OJ T£ 0 then the functions (a(u), /?(u)} are solutions to
th e system
a " = aa - 2a2/? - 2A/?
ft" = aft - 2a/?2 - 2Ba