Advertisement

Painlevé III: A Case Study in the Geometry of Meromorphic Connections

  • Martin A. Guest
  • Claus Hertling

Part of the Lecture Notes in Mathematics book series (LNM, volume 2198)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Martin A. Guest, Claus Hertling
    Pages 1-20
  3. Martin A. Guest, Claus Hertling
    Pages 21-32
  4. Martin A. Guest, Claus Hertling
    Pages 33-36
  5. Martin A. Guest, Claus Hertling
    Pages 37-41
  6. Martin A. Guest, Claus Hertling
    Pages 43-47
  7. Martin A. Guest, Claus Hertling
    Pages 49-57
  8. Martin A. Guest, Claus Hertling
    Pages 59-70
  9. Martin A. Guest, Claus Hertling
    Pages 71-85
  10. Martin A. Guest, Claus Hertling
    Pages 87-92
  11. Martin A. Guest, Claus Hertling
    Pages 93-104
  12. Martin A. Guest, Claus Hertling
    Pages 105-114
  13. Martin A. Guest, Claus Hertling
    Pages 115-126
  14. Martin A. Guest, Claus Hertling
    Pages 127-143
  15. Martin A. Guest, Claus Hertling
    Pages 151-159
  16. Martin A. Guest, Claus Hertling
    Pages 161-170
  17. Martin A. Guest, Claus Hertling
    Pages 171-179
  18. Martin A. Guest, Claus Hertling
    Pages 181-196
  19. Back Matter
    Pages 197-204

About this book

Introduction

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.

Keywords

Painlevé III movable poles isomonodromic connections Riemann-Hilbert map monodromy data TERP-structures

Authors and affiliations

  • Martin A. Guest
    • 1
  • Claus Hertling
    • 2
  1. 1.Department of Mathematics, Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  2. 2.Lehrstuhl für Mathematik VIUniversität MannheimMannheimGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-66526-9
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-66525-2
  • Online ISBN 978-3-319-66526-9
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site
Industry Sectors
Aerospace