II. THE DIFFERENTIAL GEOMETRY

The construction of cmc immersions is based on the following

well-known result due to 0. Bonnet in the Euclidean case [12],

which we state as a theorem.

Theorem 2.1: Let x = F(w), w = u + iv be a conformal cmc

2 3

immersion from an open set in R into M (c), the complete

simply-connected Riemannian 3-manifold of constant curvature c.

Suppose the first and second fundamental forms are given by

, 2 -

2W

. 2 ,

2,

a) ds = e du + dv )

(2.1)

b) -(dx-d£) = Ldu2 + 2Mdudv + Ndv2.

The Codazzi-Mainardi equations are equivalent to

i M

is holomorphic, while the Gauss equation becomes

(2.2) 0(w) =fL2 N1 -

(2.3) Aw + (H2 + c)e2W - \p\2e"2^ = 0

where H is the mean curvature of the immersion. Conversely, given

the pair ($(w), co(u,v)} satisfying (2.2) (2.3) on a simply

connected region Q there is determined a cmc immersed surface

3

x(u,v) from 0 into M (c) with mean curvature H whose fundamental

forms are as above. This immersion is unique to within an

3

isometry of M (c).

A straightforward calculation establishes that

(2.4) |#(w)I = Ik - k je2C°/2

where k , k are the principal curvatures. Given that

l 2

k + k = 2H we find

1 2

6