Painlevé Equations in the Differential Geometry of Surfaces

Alexander I. Bobenko, Ulrich Eitner (eds.)3540414142, 9783540414148

This book brings together two different branches of mathematics: the theory of Painlevé and the theory of surfaces. Self-contained introductions to both these fields are presented. It is shown how some classical problems in surface theory can be solved using the modern theory of Painlevé equations. In particular, an essential part of the book is devoted to Bonnet surfaces, i.e. to surfaces possessing families of isometries preserving the mean curvature function. A global classification of Bonnet surfaces is given using both ingredients of the theory of Painlevé equations: the theory of isomonodromic deformation and the Painlevé property. The book is illustrated by plots of surfaces. It is intended to be used by mathematicians and graduate students interested in differential geometry and Painlevé equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.

---

Table of contents :
1. Introduction….Pages 1-5
2. Basics on Painlevé Equations and Quaternionic Description of Surfaces….Pages 7-20
3. Bonnet Surfaces in Euclidean Three-space….Pages 21-64
4. Bonnet Surfaces in S 3 and H 3 and Surfaces with Harmonic Inverse Mean Curvature….Pages 65-88
5. Surfaces with Constant Curvature….Pages 89-108
6. Appendices….Pages 109-112