Complete description
This book brings together two different branches of mathematics: the theory of Painleve 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 Painleve 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 Painleve equations: the theory of isomonodromic deformation and the Painleve 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 Painleve equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.
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General info
Publisher & Imprint:
Springer-Verlag Berlin and Heidelberg GmbH & Co. K
City:
Berlin
Pages:
124
More info:
height 234 mm
width 156 mm
weight 191 gr
thickness 6 mm
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Age recommended:
College/higher education
Subject Indexing & Classification
Dewey:(DC21) 515.352
Library of Congress Subject: 00066065 Surfaces
Departments:
Differential equations;
Record updated at:
03 May, 2013
time:
01:07
Summary
Painleve Equations in the Differential Geometry of Surfaces
1. Introduction 2. Basics of Painleve Equations and Quaternionic Description of Surfaces 2.1. Painleve Property and Painleve 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
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