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2018 | OriginalPaper | Chapter

Complex Differential Equations and Geometric Structures on Curves

Author : Adolfo Guillot

Published in: Geometrical Themes Inspired by the N-body Problem

Publisher: Springer International Publishing

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Abstract

These are the notes of a series of lectures on ordinary differential equations in the complex domain delivered at the “Seventh Minimeeting in Differential Geometry” at CIMAT, in Guanajuato, Mexico, in 2015. We use geometric structures on curves as a setting to present some historical results of the theory and as a tool for a better understanding of some classical equations.

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next chapter Blow-Up, Homotopy and Existence for Periodic Solutions of the Planar Three-Body Problem

With the case (2, 2, ∞) meaning μ ₁₂ = 2 and μ ₂₁ = 2, corresponding to a degenerate triangle with two vertices with right internal angles and a third vertex at infinity (with internal angle zero).

L.V. Ahlfors, Complex Analysis, 3rd edn. (McGraw-Hill Book, New York, 1978)

Á. Álvarez-Parrilla, J. Muciño Raymundo, Dynamics of singular complex analytic vector fields with essential singularities I. Conform. Geom. Dyn. 21, 126–224 (2017)MathSciNetCrossRefMATH

A. Artigue, Órbitas periódicas en billares triangulares. Miscelánea Mat. 59, 19–40 (2014)

C. Briot, J.-C. Bouquet, Recherches sur les fonctions doublement périodiques. C. R. Hebdomadaires des Scéances de l’Académie des Sciences 40(7), 342–344 (1855)

G. Cairns, R.W. Sharpe, On the inversive differential geometry of plane curves. Enseign. Math. (2) 36(1–2), 175–196 (1990)

F. Calogero, The neatest many-body problem amenable to exact treatments (a “goldfish”?). Advances in nonlinear mathematics and science. Physica D 152/153, 78–84 (2001)

F. Calogero, J.-P. Françoise, Periodic solutions of a many-rotator problem in the plane. Inverse Probl. 17(4), 871–878 (2001). Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)

C. Camacho, A. Lins Neto, P. Sad, Minimal sets of foliations on complex projective spaces. Inst. Hautes Études Sci. Publ. Math. 68, 187–203 (1988–1989)

J. Chazy, Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Acta Math. 34(1), 317–385 (1911)MathSciNetCrossRefMATH

10.

H.S.M. Coxeter, Inversive distance. Ann. Mat. Pura Appl. (4) 71, 73–83 (1966)

11.

P. de la Harpe, Topics in Geometric Group Theory. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 2000)

12.

H.P. de Saint-Gervais, Uniformisation des surfaces de Riemann (ENS Éditions, Lyon, 2010)MATH

13.

B. Deroin, C. Dupont, Topology and dynamics of laminations in surfaces of general type. J. Am. Math. Soc. 29(2), 495–535 (2016)MathSciNetCrossRefMATH

14.

L.E. Dickson, Theory of linear groups in an arbitrary field. Trans. Am. Math. Soc. 2(4), 363–394 (1901)MathSciNetCrossRefMATH

15.

R.H. Fox, R.B. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2(1), 147–150 (1936)MathSciNetCrossRefMATH

16.

E. Ghys, J.-C. Rebelo, Singularités des flots holomorphes. II. Ann. Inst. Fourier (Grenoble) 47(4), 1117–1174 (1997)

17.

J.J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, 2nd edn. (Birkhäuser Boston, Boston, MA, 2000)MATH

18.

A. Guillot, The Painlevé property for quasihomogenous systems and a many-body problem in the plane. Commun. Math. Phys. 256(1), 181–194 (2005)CrossRefMATH

19.

A. Guillot, Semicompleteness of homogeneous quadratic vector fields. Ann. Inst. Fourier (Grenoble) 56(5), 1583–1615 (2006)

20.

A. Guillot, Sur les équations d’Halphen et les actions de SL₂(C). Publ. Math. Inst. Hautes Études Sci. 105, 221–294 (2007)MathSciNetCrossRefMATH

21.

A. Guillot, Some generalizations of Halphen’s equations. Osaka J. Math. 48(4), 1085–1094 (2011)MathSciNetMATH

22.

A. Guillot, The geometry of Chazy’s homogeneous third-order differential equations. Funkcial. Ekvac. 55(1), 67–87 (2012)MathSciNetCrossRefMATH

23.

A. Guillot, Meromorphic vector fields with single-valued solutions on complex surfaces. arXiv:1603.02288. Preprint (2016)

24.

A. Guillot, J. Rebelo, Semicomplete meromorphic vector fields on complex surfaces. J. Reine Angew. Math. 667, 27–65 (2012)MathSciNetMATH

25.

G.-H. Halphen, Sur certains systèmes d’équations différentielles. Comptes Rendus Hebdomadaires de l’Académie des Sciences XCII 24, 1404–1406 (1881)

26.

E. Hille, Ordinary Differential Equations in the Complex Domain (Dover Publications, Mineola, NY, 1997). Reprint of the 1976 original

27.

W.P. Hooper, Periodic billiard paths in right triangles are unstable. Geom. Dedicata 125, 39–46 (2007)MathSciNetCrossRefMATH

28.

Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations. Graduate Studies in Mathematics, vol. 86 (American Mathematical Society, Providence, RI, 2008)

29.

E.L. Ince, Ordinary Differential Equations (Dover Publications, New York, 1944)MATH

30.

K. Iwasawa, Über die Einfachheit der speziellen projektiven Gruppen. Proc. Imp. Acad. Tokyo 17, 57–59 (1941)MathSciNetCrossRefMATH

31.

F. Loray, J.C. Rebelo, Minimal, rigid foliations by curves on \(\mathbb {C}\mathbb {P}^n\). J. Eur. Math. Soc. 5(2), 147–201 (2003)

32.

W. Magnus, Noneuclidean Tesselations and Their Groups. Pure and Applied Mathematics, vol. 61 (Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, 1974)

33.

B. Maskit, On Poincaré’s theorem for fundamental polygons. Adv. Math. 7, 219–230 (1971)MathSciNetCrossRefMATH

34.

Z. Nehari, The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc. 55, 545–551 (1949)MathSciNetCrossRefMATH

35.

Y. Ohyama, Differential equations for modular forms of level three. Funkcial. Ekvac. 44(3), 377–389 (2001)MathSciNetMATH

36.

H. Poincaré, Théorie des groupes fuchsiens. Acta Math. 1(1), 1–62 (1882)MathSciNetCrossRefMATH

37.

S. Ramanujan, On certain arithmetical functions. Trans. Camb. Philos. Soc. 22(9), 159–184 (1916)

38.

J.C. Rebelo, Singularités des flots holomorphes. Ann. Inst. Fourier (Grenoble) 46(2), 411–428 (1996)

39.

J.C. Rebelo, H. Reis, Local theory of holomorphic foliations and vector fields. arXiv:1101.4309. Preprint (2011)

40.

J.C. Rebelo, H. Reis, Uniformizing complex ODEs and applications. Rev. Mat. Iberoam. 30(3), 799–874 (2014)MathSciNetCrossRefMATH

41.

R.E. Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Exp. Math. 18(2), 137–171 (2009)

42.

W.P. Thurston, in Three-Dimensional Geometry and Topology. Vol. 1, ed. by S. Levy. Princeton Mathematical Series, vol. 35 (Princeton University Press, Princeton, NJ, 1997)

43.

F. Valdez, Billiards in polygons and homogeneous foliations on \(\mathbb {C}^2\). Ergodic Theory Dynam. Syst. 29(1), 255–271 (2009)

44.

F. Valdez, Billares poligonales y curvas de Teichmüller. Universo.math 1(2) (2014). http://universo.math.org.mx/2014-2/Billares/billares.html

45.

Y.B. Vorobets, G.A. Gal’perin, A.M. Stepin, Periodic billiard trajectories in polygons: generation mechanisms. Uspekhi Mat. Nauk 47(3(285)), 9–74, 207 (1992)

46.

A.N. Zemljakov, A.B. Katok, Topological transitivity of billiards in polygons. Mat. Zametki 18(2), 291–300 (1975)MathSciNet

Title: Complex Differential Equations and Geometric Structures on Curves
Author: Adolfo Guillot
Publisher: Springer International Publishing
Book: Geometrical Themes Inspired by the N-body Problem
Print ISBN: 978-3-319-71427-1

Electronic ISBN: 978-3-319-71428-8

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-71428-8_1

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