Surfaces of constant mean curvature in and algebraic curves on a quadric
Author:
A. J. Small
Journal:
Proc. Amer. Math. Soc. 122 (1994), 12111220
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 that are null with respect to the CartanKilling 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 3dimensional hyperbolic space of curvature , are studied in terms of complex geometry: in particular, 'Weierstrass representation formulae' for such surfaces are derived.
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, Minimal surfaces in and the Klein correspondence (in preparation).
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, Null curves in selfdual Einstein manifolds and EinsteinWeyl spaces (in preparation).
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, Monopole charge, null curves and jumping lines (in preparation).
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 M. F. Atiyah, Magnetic monopoles on hyperbolic space, Vector Bundles on Algebraic Varieties, Oxford Univ. Press, London, 1987, pp. 95148. MR 893593 (88i:32045)
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 M. F. Atiyah and R. S. Ward, Instantons and algebraic geometry, Comm. Math. Phys. 55 (1977), 117124. MR 0494098 (58:13029)
 [3]
 A. I. Bobenko, All constant mean curvature tori in in terms of theta functions, Math. Ann. 290 (1991), 209245. MR 1109632 (92h:53072)
 [4]
 R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Théorie des Variétés Minimal et Applications, Asterisque 154155 (1987), 321347. MR 955072
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 P. Griffiths and J. Harris, Principles of algebraic geometry, WileyInterscience, New York, 1978. MR 507725 (80b:14001)
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 R. Hartshorne, Algebraic geometry, SpringerVerlag, New York, 1977. MR 0463157 (57:3116)
 [7]
 N. J. Hitchin, Complex manifolds and Einstein's equations, Twistor Geometry and NonLinear Systems, Lecture Notes in Math., vol. 970, SpringerVerlag, New York, 1980.
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 , Monopoles and geodesies, Comm. Math. Phys. 83 (1982), 579602. MR 649818 (84i:53071)
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 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), 143. MR 1147718 (92k:53116)
 [11]
 W. T. Shaw, Twistors, minimal surfaces and strings, Classical Quantum Gravity 2 (1985), L113L119. MR 815170 (87e:32046)
 [12]
 A. J. Small, Minimal surfaces in and algebraic curves, Differential Geom. Appl. 2 (1992), 369384. MR 1243536 (94h:53014)
 [13]
 , Minimal surfaces in and the Klein correspondence (in preparation).
 [14]
 , Null curves in selfdual Einstein manifolds and EinsteinWeyl spaces (in preparation).
 [15]
 , Monopole charge, null curves and jumping lines (in preparation).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412094292
PII:
S 00029939(1994)12094292
Article copyright:
© Copyright 1994 American Mathematical Society
