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 in hyperbolic spaces*, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 1–33. MR**893593****[2]**M. F. Atiyah and R. S. Ward,*Instantons and algebraic geometry*, Comm. Math. Phys.**55**(1977), no. 2, 117–124. MR**0494098****[3]**A. I. Bobenko,*All constant mean curvature tori in 𝑅³,𝑆³,𝐻³ in terms of theta-functions*, Math. Ann.**290**(1991), no. 2, 209–245. MR**1109632**, https://doi.org/10.1007/BF01459243**[4]**Robert L. Bryant,*Surfaces of mean curvature one in hyperbolic space*, Astérisque**154-155**(1987), 12, 321–347, 353 (1988) (English, with French summary). Théorie des variétés minimales et applications (Palaiseau, 1983–1984). MR**955072****[5]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725****[6]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[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]**N. J. Hitchin,*Monopoles and geodesics*, Comm. Math. Phys.**83**(1982), no. 4, 579–602. MR**649818****[9]**G. Kerbaugh,*Surfaces of constant mean curvature*1*in hyperbolic space*, Thesis, S.U.N.Y Stony Brook, 1985.**[10]**Nicholas J. Korevaar, Rob Kusner, William H. Meeks III, and Bruce Solomon,*Constant mean curvature surfaces in hyperbolic space*, Amer. J. Math.**114**(1992), no. 1, 1–43. MR**1147718**, https://doi.org/10.2307/2374738**[11]**William T. Shaw,*Twistors, minimal surfaces and strings*, Classical Quantum Gravity**2**(1985), no. 6, L113–L119. MR**815170****[12]**A. J. Small,*Minimal surfaces in 𝐑³ and algebraic curves*, Differential Geom. Appl.**2**(1992), no. 4, 369–384. MR**1243536**, https://doi.org/10.1016/0926-2245(92)90003-6**[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