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

Author: A. J. Small
Journal: Proc. Amer. Math. Soc. 122 (1994), 1211-1220
MSC: Primary 53A10; Secondary 14H10, 81R25
MathSciNet review: 1209429
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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.

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