Six results on Painleve VI

Abstract

After recalling some of the geometry of the sixth Painleve equation, we will describe how the Okamoto symmetries arise naturally from symmetries of Schlesinger's equations and summarise the classification of the Platonic Painleve six solutions. A key observation is that Painleve VI governs the isomonodromic deformations of certain Fuchsian systems on rank \emph{three} bundles.

16 pages, written for Angers 2004 International Conference on Asymptotic Theories and Painleve Equations (updated, added Remark 7 giving simple formulae for isomonodromic families of rank three systems in terms of PVI solutions and as an example the full family of "Klein connections" are written down)

References in corpus (5)

• Transformations ${RS}_4^2(3)$ of the Ranks $\le 4$ and Algebraic Solutions of the Sixth Painlevé Equation
• Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
• Some explicit solutions to the Riemann-Hilbert problem
• Dessins d'Enfants, Their Deformations and Algebraic the Sixth Painlevé and Gauss Hypergeometric Functions
• Painlevé equations and the middle convolution

Cited by in corpus (7)

• Transformations of Spherical Blocks
• On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System
• Regge and Okamoto symmetries
• Classification of algebraic solutions of irregular Garnier systems
• Solutions algébriques. Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre $2$
• Singular Instantons and Painlevé VI
• Okamoto Transformations relating Equivariant Instanton Bundles via Painleve VI