Osaka Journal of Mathematics

Some generalizations of Halphen's equations

Abstract

Halphen's equations are given by a remarkable polynomial vector field in $\mathbf{C}^{3}$ having only single-valued solutions, defined in domains bounded by a circle or by a line. By generalizing the Lie-theoretic principle behind Halphen's equations and borrowing some facts from the theory of deformations of Fuchsian groups, we exhibit a family of polynomial vector fields in $\mathbf{C}^{3}$ having only single-valued solutions. The solutions of vector fields within this family are defined in domains which had not been previously observed as domains of definition of solutions of polynomial vector fields in $\mathbf{C}^{3}$. For example, we obtain polynomial vector fields having solutions defined in domains that are bounded by a fractal curve.

Article information

Source
Osaka J. Math., Volume 48, Number 4 (2011), 1085-1094.

Dates
First available in Project Euclid: 11 January 2012

https://projecteuclid.org/euclid.ojm/1326291219

Mathematical Reviews number (MathSciNet)
MR2871295

Zentralblatt MATH identifier
1251.32019

Citation

Guillot, Adolfo. Some generalizations of Halphen's equations. Osaka J. Math. 48 (2011), no. 4, 1085--1094. https://projecteuclid.org/euclid.ojm/1326291219

References

• W. Abikoff: Two theorems on totally degenerate Kleinian groups, Amer. J. Math. 98 (1976), 109–118.
• M. Atiyah and N. Hitchin: The Geometry and Dynamics of Magnetic Monopoles, M.B. Porter Lectures, Princeton Univ. Press, Princeton, NJ, 1988.
• L. Bers: On boundaries of Teichmüller spaces and on Kleinian groups, I, Ann. of Math. (2) 91 (1970), 570–600.
• R.D. Canary and E. Taylor: Kleinian groups with small limit sets, Duke Math. J. 73 (1994), 371–381.
• S. Chakravarty, M.J. Ablowitz and P.A. Clarkson: Reductions of self-dual Yang–Mills fields and classical systems, Phys. Rev. Lett. 65 (1990), 1085–1087.
• S. Chakravarty and R. Halburd: First integrals of a generalized Darboux–Halphen system, J. Math. Phys. 44 (2003), 1751–1762.
• G. Darboux: Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux, Ann. Sci. École Norm. Sup. (2) 7 (1878), 101–150.
• \begingroup A. Guillot: Semicompleteness of homogeneous quadratic vector fields, Ann. Inst. Fourier (Grenoble) 56 (2006), 1583–1615. \endgroup
• A. Guillot: Sur les équations d'Halphen et les actions de $\mathrm{SL}_{2}(\mathbf{C})$, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 221–294.
• A. Guillot and J. Rebelo: Semicomplete meromorphic vector fields on complex surfaces, to appear in J. Reine Angew. Math. (2011).
• \begingroup G.-H. Halphen: Sur certains systèmes d'équations différentielles, Comptes Rendus Hebdomadaires de l'Académie des Sciences 92 (1881), 1404–1406. \endgroup
• G.-H. Halphen: Sur un système d'équations différentielles, Comptes Rendus Hebdomadaires de l'Académie des Sciences 92 (1881), 1101–1102.
• E. Hille: Ordinary Differential Equations in the Complex Domain, reprint of the 1976 original, Dover, Mineola, NY, 1997.
• A.J. Maciejewski and J.-M. Strelcyn: On the algebraic non-integrability of the Halphen system, Phys. Lett. A 201 (1995), 161–166.
• B. Maskit: On boundaries of Teichmüller spaces and on Kleinian groups, II, Ann. of Math. (2) 91 (1970), 607–639.
• B. Maskit: Kleinian Groups, Grundlehren der Mathematischen Wissenschaften 287, Springer, Berlin, 1988.
• C. McMullen: Cusps are dense, Ann. of Math. (2) 133 (1991), 217–247.
• Y. Ohyama: Differential relations of theta functions, Osaka J. Math. 32 (1995), 431–450.
• Y. Ohyama: Differential equations for modular forms of level three, Funkcial. Ekvac. 44 (2001), 377–389.
• J.C. Rebelo: Singularités des flots holomorphes, Ann. Inst. Fourier (Grenoble) 46 (1996), 411–428.
• M. Spivak: A Comprehensive Introduction to Differential Geometry, I, second edition, Publish or Perish Inc., Wilmington, Del., 1979.
• W.P. Thurston: Three-Dimensional Geometry and Topology, I, Princeton Univ. Press, Princeton, NJ, 1997.
• V.V. Zudilin: Theta constants and differential equations, Mat. Sb. 191 (2000), 77–122.