Teichmüller Theory in Riemannian Geometry: (Lectures in Mathematics. ETH Zürich)

These lecture notes are based on the joint work of the author and Arthur Fischer on Teichmiiller theory undertaken in the years 1980-1986. Since then many of our colleagues have encouraged us to publish our approach to the subject in a concise format, easily accessible to a broad mathematical audience. However, it was the invitation by the faculty of the ETH Ziirich to deliver the ETH N achdiplom-Vorlesungen on this material which provided the opportunity for the author to develop our research papers into a format suitable for mathematicians with a modest background in differential geometry. We also hoped it would provide the basis for a graduate course stressing the application of fundamental ideas in geometry. For this opportunity the author wishes to thank Eduard Zehnder and Jiirgen Moser, acting director and director of the Forschungsinstitut fiir Mathematik at the ETH, Gisbert Wiistholz, responsible for the Nachdiplom Vorlesungen and the entire ETH faculty for their support and warm hospitality. This new approach to Teichmiiller theory presented here was undertaken for two reasons. First, it was clear that the classical approach, using the theory of extremal quasi-conformal mappings (in this approach we completely avoid the use of quasi-conformal maps) was not easily applicable to the theory of minimal surfaces, a field of interest of the author over many years. Second, many other active mathematicians, who at various times needed some Teichmiiller theory, have found the classical approach inaccessible to them.

Table of Contents:
0 Mathematical Preliminaries.- 1 The Manifolds of Teichmüller Theory.- 1.1 The Manifolds A and As.- 1.2 The Riemannian Manifolds M and Ms.- 1.3 The Diffeomorphism Ms /? s ? As.- 1.4 Some Differential Operators and their Adjoints.- 1.5 Proof of Poincaré’s Theorem.- 1.6 The Manifold Ms-1 and the Diffeomorphism with Ms / s.- 2 The Construction of Teichmüller Space.- 2.1 A Rapid Course in Geodesic Theory.- 2.2 The Free Action of D0 on M-1.- 2.3 The Proper Action of D0 on M-1.- 2.4 The Construction of Teichmüller Space.- 2.5 The Principal Bundles of Teichmüller Theory.- 2.6 The Weil-Petersson Metric on T(M).- 3 T(M) is a Cell.- 3.1 Dirichlet’s Energy on Teichmüller Space.- 3.2 The Properness of Dirichlet’s Energy.- 3.3 Teichmüller Space is a Cell.- 3.4 Topological Implications; The Contractibility of D0.- 4 The Complex Structure on Teichmüller Space.- 4.1 Almost Complex Principal Fibre Bundles.- 4.2 Abresch-Fischer Holomorphic Coordinates for A.- 4.3 Abresch-Fischer Holomorphic Coordinates for T(M).- 5 Properties of the Weil-Petersson Metric.- 5.1 The Weil-Petersson Metric is Kähler.- 5.2 The Natural Algebraic Connection on A.- 5.3 Further Properties of the Algebraic Connection and the non-Integrability of the Horizontal Distribution on A.- 5.4 The Curvature of Teichmüller Space with Respect to its Weil-Petersson Metric.- 5.5 An Asymptotic Property of Weil-Petersson Geodesies.- 6 The Pluri-Subharmonicity of Dirichlet’s Energy on T(M); T(M) is a Stein-Manifold.- 6.1 Pluri-Subharmonic Functions and Complex Manifolds.- 6.2 Dirichlet’s Energy is Strictly Pluri-Subharmonic.- 6.3 Wolf’s Form of Dirichlet’s Energy on T(M) is Strictly Weil-Petersson Convex.- 6.4 The Nielsen Realization Problem.- A Proof of Lichnerowicz’ Formula.- B On Harmonic Maps.- CThe Mumford Compactness Theorem.- D Proof of the Collar Lemma.- E The Levi-Form of Dirichlet’s Energy.- F Riemann-Roch and the Dimension of Teichmüller Space.- Indexes.- Index of Notation.- A Chart of the Maps Used.- Index of Key Words.