Surfaces of constant mean curvature $1$ in $\textbf {H}^ 3$ and algebraic curves on a quadric

Abstract:

We show that there exists a natural correspondence between holomorphic curves in $\mathbb {P}{\text {SL}}(2,\mathbb {C})$ that are null with respect to the Cartan-Killing metric, and holomorphic curves on ${\mathbb {P}_1} \times {\mathbb {P}_1}$. This correspondence derives from classical osculation duality between curves in ${\mathbb {P}_3}$ and its dual, $\mathbb {P}_3^ \ast$. 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.