Contents
Introduction 1
Chapter 1. Curvature Properties and Chern Forms 7
1. The universal Teichmiiller space 7
1.1. Teichmiiller theory 7
1.1.1. Main definitions 7
1.1.2. The group structure 8
1.1.3. The Bers embedding 9
1.1.4. The complex structure 9
1.1.5. The universal Teichmiiller curve 10
1.2. Homogeneous spaces of HomeoQS(Sx) 11
1.2.1. Conformal welding 11
1.2.2. The horizontal and vertical subspaces 14
1.2.3. The isomorphisms of the tangent spaces 15
1.3. Teichmiiller spaces and Teichmiiller curves of Fuchsian groups 17
1.4. Resolvent kernel 18
1.5. Variational formulas 19
2. T(l) as a Hilbert manifold 22
2.1. Hilbert space structure on tangent spaces 22
2.2. The
L2-estimates
27
2.3. The Hilbert manifold structure of T(l) 29
2.4. Integral manifolds of the distribution TT 31
3. TQ (1) as a topological group 32
4. Velling-Kirillov and Weil-Petersson metrics 36
4.1. Velling-Kirillov metric on the universal Teichmiiller curve 36
4.2. Weil-Petersson metric on the universal Teichmiiller space 37
5. Characteristic forms of the universal Teichmiiller curve 38
5.1. The form ci(V) as Velling-Kirillov symplectic form 39
5.2. The Chern form ci(V) and the resolvent kernel 41
5.3. Mumford-Morita-Miller characteristic forms 43
6. First and second variations of the hyperbolic metric 44
6.1. The first variation 44
6.2. The second variation 44
7. Riemann curvature tensor 46
7.1. The first variation of the Weil-Petersson metric 46
7.2. The second variation of the Weil-Petersson metric 50
7.3. Ricci and sectional curvatures 54
8. Finite-dimensional Teichmiiller spaces 56
Chapter 2. Kahler Potential and Period Mapping 65
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