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.
Author(s): Frederic Hélein, R. Moser
Series: Lectures in Mathematics. ETH Zürich
Edition: 1
Publisher: Birkhäuser Basel
Year: 2001
Language: English
Pages: 124
Preface ......Page 8
1 Introduction: Surfaces with prescribed mean curvature ......Page 10
2 From minimal surfaces and CMC surfaces to harmonic maps ......Page 16
2.1 Minimal surfaces ......Page 17
2.2 Constant mean irvature.surfaces ......Page 19
3 Variational point of view and Noether's theorem ......Page 23
4 Working with the Hopf differential ......Page 35
4.1 Appendix ......Page 40
5 The Gauss-Codazzi condition ......Page 42
5.1 Appendix ......Page 51
6 Elementary twistor theory for harmonic maps ......Page 53
6.1 Appendix ......Page 61
7.1 Maps into spheres ......Page 64
7.2 Generalizations ......Page 69
7.3 A new setting: loop groups ......Page 72
7.4 Examples ......Page 75
8.1 Preliminary: the Iwasawa decomposition (for Rc) ......Page 82
8.2 Application to loop Lie algebras ......Page 84
8.3 The algorithm ......Page 85
8.4 Some further properties of finite type solutions ......Page 88
9.1 The result ......Page 90
9.2 Appendix ......Page 95
10.1 CMC surfaces with planar curvature lines ......Page 108
10.2 A system of commuting ordinary equations ......Page 110
10.3 Recovering a finite type solution ......Page 111
10.4 Spectral curves ......Page 112
11.1 Loop groups decompositions ......Page 113
11.2 Solutions in terms of holomorphic data ......Page 114
11.3 Meromorphic potentials ......Page 116
11.4 Generalizations ......Page 117
Bibliography ......Page 118
