Painleve Equations in the Differential Geometry of Surfaces: 1753 (Lecture Notes in Mathematics, 1753) - Softcover

TU Berlin, Alexander I. Bobenko; Eitner, Ulrich

Softcover

ISBN 10: 3540414142 ISBN 13: 9783540414148

Publisher: Springer, 2000

Synopsis

Since the time of surfaces -+ in differential Gauss, parametrized (x, y) P(x, y) have been described a frame attached to the moving geometry through TI(x, y) surface. One introduces the Gauss- which linear dif- Weingarten equations are , ferential equations = U = TIX T1, VT', !PY (1. for the and their condition frame, compatibility - = V + [U, V] 0, UY (1.2) which the Gauss-Codazzi For surfaces in three-dim- represents equations . a sional Euclidean the frame T1 lies in the usually or space, group SO(3) SU(2). On the other a of a non-linear in the form hand, representation equation (1.2) is the of the of of starting point theory integrable equations (theory solitons), which in mathematical in the 1960's appeared physics [NMPZ, AbS, CD, FT, More the differential for the coefficients of AbC]. exactly, partial equation (1.2) the matrices U and V is considered to be if these matrices can be integrable , extended to U V non-trivially a one-parameter family (x, y, A), (x, y, A) satisfying - = + U(A)y V(A). [U(A), V(A)] 0, (1-3) so that the differential is and original partial equation preserved.' . Usually U(A) V are rational functions of the which is called the (A) parameter A, spectral param- In soliton the eter is called the Lax . theory, representation (1.3) representation the Zakharov-Shabat or representation [ZS].

About this title

Synopsis

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.

Edition details

• Publisher: Springer
• Publication date: 2000
• Language: English
• ISBN 10: 3540414142
• ISBN 13: 9783540414148
• Binding: Paperback
• Number of pages: 124