Abstract
This chapter is an introduction to immersions of surfaces in hyperbolic space with constant mean curvature equal to one (CMC 1 immersions in H3). The approach follows Bryant’s paper (Bryant, Asterisque 154–155, 12, 1987, 321–347, 353, 1988), which replaces the hyperboloid model of H3 by the set of all 2 × 2 hermitian matrices with determinant one and positive trace. This model is acted upon isometrically by SL(2, C), the universal cover of the group of all isometries of hyperbolic space. The method of moving frames is applied to the study of immersed surfaces in this homogeneous space. Departing from Bryant’s approach, we use frames adapted to a given complex coordinate to great advantage. A null immersion from a Riemann surface into SL(2, C) projects to a CMC 1 immersion into hyperbolic space. The null immersions are analogous to minimal curves of the Weierstrass representation of minimal immersions into Euclidean space. A solution of these equations leads to a more complicated monodromy problem, which is described in detail. The chapter ends with some examples.
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