Surfaces of constant mean curvature in 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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1209429-2

MathSciNet review:
1209429

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

**[1]**M. F. Atiyah,*Magnetic monopoles on hyperbolic space*, Vector Bundles on Algebraic Varieties, Oxford Univ. Press, London, 1987, pp. 95-148. MR**893593 (88i:32045)****[2]**M. F. Atiyah and R. S. Ward,*Instantons and algebraic geometry*, Comm. Math. Phys.**55**(1977), 117-124. MR**0494098 (58:13029)****[3]**A. I. Bobenko,*All constant mean curvature tori in**in terms of theta functions*, Math. Ann.**290**(1991), 209-245. MR**1109632 (92h:53072)****[4]**R. L. Bryant,*Surfaces of mean curvature one in hyperbolic space*, Théorie des Variétés Minimal et Applications, Asterisque**154-155**(1987), 321-347. MR**955072****[5]**P. Griffiths and J. Harris,*Principles of algebraic geometry*, Wiley-Interscience, New York, 1978. MR**507725 (80b:14001)****[6]**R. Hartshorne,*Algebraic geometry*, Springer-Verlag, New York, 1977. MR**0463157 (57:3116)****[7]**N. J. Hitchin,*Complex manifolds and Einstein's equations*, Twistor Geometry and Non-Linear Systems, Lecture Notes in Math., vol. 970, Springer-Verlag, New York, 1980.**[8]**-,*Monopoles and geodesies*, Comm. Math. Phys.**83**(1982), 579-602. MR**649818 (84i:53071)****[9]**G. Kerbaugh,*Surfaces of constant mean curvature*1*in hyperbolic space*, Thesis, S.U.N.Y Stony Brook, 1985.**[10]**N. J. Korevaar, R. Kusner, W. H. Meeks III, and B. Solomon,*Constant mean curvature surfaces in hyperbolic space*, Amer. J. Math.**114**(1992), 1-43. MR**1147718 (92k:53116)****[11]**W. T. Shaw,*Twistors, minimal surfaces and strings*, Classical Quantum Gravity**2**(1985), L113-L119. MR**815170 (87e:32046)****[12]**A. J. Small,*Minimal surfaces in**and algebraic curves*, Differential Geom. Appl.**2**(1992), 369-384. MR**1243536 (94h:53014)****[13]**-,*Minimal surfaces in**and the Klein correspondence*(in preparation).**[14]**-,*Null curves in self-dual Einstein manifolds and Einstein-Weyl spaces*(in preparation).**[15]**-,*Monopole charge, null curves and jumping lines*(in preparation).

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1209429-2

Article copyright:
© Copyright 1994
American Mathematical Society