Surfaces of constant mean curvature 1 in 𝐻³ and algebraic curves on a quadric

Abstract

We show that there exists a natural correspondence between holomorphic curves in P SL ( 2 , C ) \mathbb {P}{\text {SL}}(2,\mathbb {C}) that are null with respect to the Cartan-Killing metric, and holomorphic curves on P 1 × P 1 {\mathbb {P}_1} \times {\mathbb {P}_1} . This correspondence derives from classical osculation duality between curves in P 3 {\mathbb {P}_3} and its dual, P 3 ∗ \mathbb {P}_3^ \ast . Thus, via Bryant’s correspondence, surfaces of constant mean curvature 1 in the 3-dimensional hyperbolic space of curvature − 1 - 1 , are studied in terms of complex geometry: in particular, ’Weierstrass representation formulae’ for such surfaces are derived.