Painlevé III: A Case Study in the Geometry of Meromorphic Connections (Lecture Notes in Mathematics Book 2198)

The purpose of this monograph is two-fold: it introduces a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation of type PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1 with meromorphic connections. This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears widely in geometry and physics. It is used here as a very concrete and classical illustration of the modern theory of vector bundles with meromorphic connections. Complex multi-valued solutions on C\* are the natural context for most of the monograph, but in the last four chapters real solutions on R>0 (with or without singularities) are addressed. These provide examples of variations of TERP structures, which are related to __tt∗__ geometry and harmonic bundles. As an application, a new global picture o0 is given. Front Matter ....Pages i-xii Introduction (Martin A. Guest, Claus Hertling)....Pages 1-20 The Riemann-Hilbert Correspondence for P3D6 Bundles (Martin A. Guest, Claus Hertling)....Pages 21-32 (Ir)Reducibility (Martin A. Guest, Claus Hertling)....Pages 33-36 Isomonodromic Families (Martin A. Guest, Claus Hertling)....Pages 37-41 Useful Formulae: Three 2 × 2 Matrices (Martin A. Guest, Claus Hertling)....Pages 43-47 P3D6-TEP Bundles (Martin A. Guest, Claus Hertling)....Pages 49-57 P3D6-TEJPA Bundles and Moduli Spaces of Their Monodromy Tuples (Martin A. Guest, Claus Hertling)....Pages 59-70 Normal Forms of P3D6-TEJPA Bundles and Their Moduli Spaces (Martin A. Guest, Claus Hertling)....Pages 71-85 Generalities on the Painlevé Equations (Martin A. Guest, Claus Hertling)....Pages 87-92 Solutions of the Painlevé Equation PIII(0, 0, 4, −4) (Martin A. Guest, Claus Hertling)....Pages 93-104 Comparison with the Setting of Its, Novokshenov, and Niles (Martin A. Guest, Claus Hertling)....Pages 105-114 Asymptotics of All Solutions Near 0 (Martin A. Guest, Claus Hertling)....Pages 115-126 Rank 2 TEPA Bundles with a Logarithmic Pole (Martin A. Guest, Claus Hertling)....Pages 127-143 Symmetries of the Universal Family of Solutions of PIII(0, 0, 4, −4) (Martin A. Guest, Claus Hertling)....Pages 145-149 Three Families of Solutions on \(\mathbb{R}_{>0}\) (Martin A. Guest, Claus Hertling)....Pages 151-159 TERP Structures and P3D6-TEP Bundles (Martin A. Guest, Claus Hertling)....Pages 161-170 Orbits of TERP Structures and Mixed TERP Structures (Martin A. Guest, Claus Hertling)....Pages 171-179 Real Solutions of PIII(0, 0, 4, −4) on \(\mathbb{R}_{>0}\) (Martin A. Guest, Claus Hertling)....Pages 181-196 Back Matter ....Pages 197-204 The purpose of this monograph is two-fold: it introduces a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation of type PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1 with meromorphic connections. This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears widely in geometry and physics. It is used here as a very concrete and classical illustration of the modern theory of vector bundles with meromorphic connections. Complex multi-valued solutions on C* are the natural context for most of the monograph, but in the last four chapters real solutions on R>0 (with or without singularities) are addressed. These provide examples of variations of TERP structures, which are related to tt∗ geometry and harmonic bundles. As an application, a new global picture of0 is given