Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems

Helein, Frederic

ISBN 10: 3764365765 / ISBN 13: 9783764365769

Published by Birkh?user, 2001

About this Item

This title provides and introduction to harmonic maps between a surface and a symmetric manifold and constant mean curvature surfaces as completely integrable systems. It should help the reader to access the ideas of the theory and to aquire a unified perspective of the subject. Series: Lectures in Mathematics. ETH Zurich. Num Pages: 128 pages, biography. BIC Classification: PBCD; PBM. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly; (UU) Undergraduate. Dimension: 244 x 170 x 6. Weight in Grams: 237. . 2001. 2001st Edition. paperback. . . . .

Seller Inventory # V9783764365769

Bibliographic Details

• Title: Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems
• Publisher: Birkh?user
• Publication Date: 2001
• Language: English
• ISBN 10: 3764365765
• ISBN 13: 9783764365769
• Binding: Soft cover
• Edition: 1st Edition
• Condition: New

About this title

Synopsis

One of the most striking development of the last decades in the study of minimal surfaces, constant mean surfaces and harmonic maps is the discovery that many classical problems in differential geometry - including these examples - are actually integrable systems. This theory grew up mainly after the important discovery of the properties of the Korteweg-de Vries equation in the sixties. After C. Gardner, J. Greene, M. Kruskal et R. Miura [44] showed that this equation could be solved using the inverse scattering method and P. Lax [62] reinterpreted this method by his famous equation, many other deep observations have been made during the seventies, mainly by the Russian and the Japanese schools. In particular this theory was shown to be strongly connected with methods from algebraic geom- etry (S. Novikov, V. B. Matveev, LM. Krichever...), loop techniques (M. Adler, B. Kostant, W. W. Symes, M. J. Ablowitz ...) and Grassmannian manifolds in Hilbert spaces (M. Sato ...). Approximatively during the same period, the twist or theory of R. Penrose, built independentely, was applied successfully by R. Penrose and R. S. Ward for constructing self-dual Yang-Mills connections and four-dimensional self-dual manifolds using complex geometry methods. Then in the eighties it became clear that all these methods share the same roots and that other instances of integrable systems should exist, in particular in differential ge- ometry. This led K.

Synopsis

This book intends to give an introduction to harmonic maps between a surface and a symmetric manifold and constant mean curvature surfaces as completely integrable systems. The presentation is accessible to undergraduate and graduate students in mathematics but will also be useful to researchers. It is among the first textbooks about integrable systems, their interplay with harmonic maps and the use of loop groups, and it presents the theory, for the first time, from the point of view of a differential geometer. The most important results are exposed with complete proofs (except for the last two chapters, which require a minimal knowledge from the reader). Some proofs have been completely rewritten with the objective, in particular, to clarify the relation between finite mean curvature tori, Wente tori and the loop group approach - an aspect largely neglected in the literature. The book helps the reader to access the ideas of the theory and to acquire a unified perspective of the subject.