Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems

Inhaltsverzeichnis

• 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.