Electronic ISBN:  9781470404659 
Product Code:  MEMO/183/861.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 183; 2006; 119 ppMSC: Primary 30; Secondary 32; 46; 58;
In this memoir, we prove that the universal Teichmüller space \(T(1)\) carries a new structure of a complex Hilbert manifold and show that the connected component of the identity of \(T(1)\) — the Hilbert submanifold \(T_{0}(1)\) — is a topological group. We define a WeilPetersson metric on \(T(1)\) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that \(T(1)\) is a KählerEinstein manifold with negative Ricci and sectional curvatures. We introduce and compute MumfordMillerMorita characteristic forms for the vertical tangent bundle of the universal Teichmüller curve fibration over the universal Teichmüller space. As an application, we derive Wolpert curvature formulas for the finitedimensional Teichmüller spaces from the formulas for the universal Teichmüller space. We study in detail the Hilbert manifold structure on \(T_{0}(1)\) and characterize points on \(T_{0}(1)\) in terms of Bers and preBers embeddings by proving that the Grunsky operators \(B_{1}\) and \(B_{4}\), associated with the points in \(T_{0}(1)\) via conformal welding, are HilbertSchmidt. We define a “universal Liouville action” — a realvalued function \({\mathbf S}_{1}\) on \(T_{0}(1)\), and prove that it is a Kähler potential of the WeilPetersson metric on \(T_{0}(1)\). We also prove that \({\mathbf S}_{1}\) is \(\tfrac{1}{12\pi}\) times the logarithm of the Fredholm determinant of associated quasicircle, which generalizes classical results of Schiffer and Hawley. We define the universal period mapping \(\hat{\mathcal{P}}: T(1)\rightarrow\mathcal{B}(\ell^{2})\) of \(T(1)\) into the Banach space of bounded operators on the Hilbert space \(\ell^{2}\), prove that \(\hat{\mathcal{P}}\) is a holomorphic mapping of Banach manifolds, and show that \(\hat{\mathcal{P}}\) coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of \(\hat{\mathcal{P}}\) to \(T_{0}(1)\) is an inclusion of \(T_{0}(1)\) into the SegalWilson universal Grassmannian, which is a holomorphic mapping of Hilbert manifolds. We also prove that the image of the topological group \(S\) of symmetric homeomorphisms of \(S^{1}\) under the mapping \(\hat{\mathcal{P}}\) consists of compact operators on \(\ell^{2}\).
The results of this memoir were presented in our eprints: WeilPetersson metric on the universal Teichmuller space I. Curvature properties and Chern forms, arXiv:math.CV/0312172 (2003), and WeilPetersson metric on the universal Teichmuller space II. Kahler potential and period mapping, arXiv:math.CV/0406408 (2004). 
Table of Contents

Chapters

Introduction

1. Curvature properties and Chern forms

2. Kähler potential and period mapping


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In this memoir, we prove that the universal Teichmüller space \(T(1)\) carries a new structure of a complex Hilbert manifold and show that the connected component of the identity of \(T(1)\) — the Hilbert submanifold \(T_{0}(1)\) — is a topological group. We define a WeilPetersson metric on \(T(1)\) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that \(T(1)\) is a KählerEinstein manifold with negative Ricci and sectional curvatures. We introduce and compute MumfordMillerMorita characteristic forms for the vertical tangent bundle of the universal Teichmüller curve fibration over the universal Teichmüller space. As an application, we derive Wolpert curvature formulas for the finitedimensional Teichmüller spaces from the formulas for the universal Teichmüller space. We study in detail the Hilbert manifold structure on \(T_{0}(1)\) and characterize points on \(T_{0}(1)\) in terms of Bers and preBers embeddings by proving that the Grunsky operators \(B_{1}\) and \(B_{4}\), associated with the points in \(T_{0}(1)\) via conformal welding, are HilbertSchmidt. We define a “universal Liouville action” — a realvalued function \({\mathbf S}_{1}\) on \(T_{0}(1)\), and prove that it is a Kähler potential of the WeilPetersson metric on \(T_{0}(1)\). We also prove that \({\mathbf S}_{1}\) is \(\tfrac{1}{12\pi}\) times the logarithm of the Fredholm determinant of associated quasicircle, which generalizes classical results of Schiffer and Hawley. We define the universal period mapping \(\hat{\mathcal{P}}: T(1)\rightarrow\mathcal{B}(\ell^{2})\) of \(T(1)\) into the Banach space of bounded operators on the Hilbert space \(\ell^{2}\), prove that \(\hat{\mathcal{P}}\) is a holomorphic mapping of Banach manifolds, and show that \(\hat{\mathcal{P}}\) coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of \(\hat{\mathcal{P}}\) to \(T_{0}(1)\) is an inclusion of \(T_{0}(1)\) into the SegalWilson universal Grassmannian, which is a holomorphic mapping of Hilbert manifolds. We also prove that the image of the topological group \(S\) of symmetric homeomorphisms of \(S^{1}\) under the mapping \(\hat{\mathcal{P}}\) consists of compact operators on \(\ell^{2}\).
The results of this memoir were presented in our eprints: WeilPetersson metric on the universal Teichmuller space I. Curvature properties and Chern forms, arXiv:math.CV/0312172 (2003), and WeilPetersson metric on the universal Teichmuller space II. Kahler potential and period mapping, arXiv:math.CV/0406408 (2004).

Chapters

Introduction

1. Curvature properties and Chern forms

2. Kähler potential and period mapping