Painlev III: A Case Study in the Geometry of Meromorphic Connections [Paperback]

The purpose of this monograph is two-fold:? it introduces a conceptual language for the geometrical objects underlying Painlev? equations,? and it offers new results on a particular Painlev? 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.?

The purpose of this monograph is two-fold:? it introduces a conceptual language for the geometrical objects underlying Painlev? equations,? and it offers new results on a particular Painlev? 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 natl33