Surfaces of constant mean curvature $1$ in $\textbf {H}^ 3$ and algebraic curves on a quadric
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- by A. J. Small
- Proc. Amer. Math. Soc. 122 (1994), 1211-1220
- DOI: https://doi.org/10.1090/S0002-9939-1994-1209429-2
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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.References
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Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 1211-1220
- MSC: Primary 53A10; Secondary 14H10, 81R25
- DOI: https://doi.org/10.1090/S0002-9939-1994-1209429-2
- MathSciNet review: 1209429