Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems de Frederic Hélein

Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems: Lectures in Mathematics. ETH Zürich

Frederic Hélein

R. Moser

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.

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Cuprins

1 Introduction: Surfaces with prescribed mean curvature.- 2 From minimal surfaces and CMC surfaces to harmonic maps.- 2.1 Minimal surfaces.- 2.2 Constant mean curvature surfaces.- 3 Variational point of view and Noether’s theorem.- 4 Working with the Hopf differential.- 4.1 Appendix.- 5 The Gauss-Codazzi condition.- 5.1 Appendix.- 6 Elementary twistor theory for harmonic maps.- 6.1 Appendix.- 7 Harmonic maps as an integrable system.- 7.1 Maps into spheres.- 7.2 Generalizations.- 7.3 A new setting: loop groups.- 7.4 Examples.- 8 Construction of finite type solutions.- 8.1 Preliminary: the Iwasawa decomposition (for)..- 8.2 Application to loop Lie algebras.- 8.3 The algorithm.- 8.4 Some further properties of finite type solutions.- 9 Constant mean curvature tori are of finite type.- 9.1 The result.- 9.2 Appendix.- 10 Wente tori.- 10.1 CMC surfaces with planar curvature lines.- 10.2 A system of commuting ordinary equations.- 10.3 Recovering a finite type solution.- 10.4 Spectral curves.- 11 Weierstrass type representations.- 11.1 Loop groups decompositions.- 11.2 Solutions in terms of holomorphic data.- 11.3 Meromorphic potentials.- 11.4 Generalizations.