Teichmuller Theory in Riemannian Geometry: Based on Lecture Notes by Jochen Denzler

by Anthony J. Tromba

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.- 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.- Indexes.- Index of Notation.- A Chart of the Maps Used.- Index of Key Words.

224 pages, Paperback

First published April 1, 1992