SURFACES OF CONSTANT MEAN-CURVATURE-1 IN H-3 AND ALGEBRAIC-CURVES ON A QUADRIC

We show that there exists a natural correspondence between holomorphic curves in PSL(2, C) that are null with respect to the Cartan-Killing metric, and holomorphic curves on P-1 x P-1. This correspondence derives from classical osculation duality between curves in P-3 and its dual, P-3*. Thus, via Bryant's correspondence, surfaces of constant mean curvature 1 in the 3-dimensional hyperbolic space of curvature -1, are studied in terms of complex geometry: in particular, 'Weierstrass representation formulae' for such surfaces are derived.