Synopses & Reviews
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.
Description
Includes bibliographical references (p. [113]-117) and index.
Table of Contents
1. Introduction
2. Basics of Painlevé Equations and Quaternionic Description of Surfaces
2.1. Painlevé Property and Painlevé Equations
2.2. Isomonodromic Deformations 2.3. Conformally Parametrized Surfaces 2.4. Quaternionic Description of Surfaces 3. Bonnet Surfaces in Euclidean three-space 3.1. Definition of Bonnet Surfaces and Simplest Properties 3.2. Local Theory away from Critical Points 3.3. Local Theory at Critical Points 3.4. Bonnet Surfaces via Painlev Transcendents 3.5. Global Properties of Bonnet Surfaces 3.6. Examples of Bonnet Surfaces 3.7. Schlesinger Transformations for Bonnet Surfaces 4. Bonnet Surfaces in S and H and Surfaces with Harmonic Inverse Mean Curvature 4.1. Surfaces in S3 and H3 4.2. Definition and Simplest Properties 4.3. Bonnet Surfaces in S3 and H3 away from Critical Points 4.4. Local Theory of Bonnet Surfaces in S and H at Critical Points 4.5. Bonnet Surfaces in S3 and H3 in Terms of Painlev Transcendents 4.6. Global Properties of Bonnet Surfaces in Space Forms 4.7. Surfaces with Harmonic Inverse Mean Curvature 4.8. Bonnet Pairs of HIMC Surfaces 4.9. HIMC Bonnet Pairs in Painlev Transcendents 4.10. Examples of HIMC Surfaces 5. Surfaces with Constant Curvature 5.1. Surfaces with Constant Negative Gaussian Curvature and Two Straight Asymptotic Lines 5.2. Smyth Surfaces 5.3. Affine Spheres with Affine Straight Lines 6. Appendices