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 ,
which we state as a theorem.
Theorem 2.1: Let x = F(w), w = u + iv be a conformal cmc
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 -
. 2 ,
a) ds = e du + dv )
b) -(dx-d£) = Ldu2 + 2Mdudv + Ndv2.
The Codazzi-Mainardi equations are equivalent to
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
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
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
k + k = 2H we find