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) =fL 2 N 1 - (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
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