Teichmuller theory in Riemannian geometry

Includes index

Includes bibliography (p. 205-213) and references

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 C The 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

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.