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Weil-Petersson Metric on the Universal Teichmüller Space
Leon A. Takhtajan, SUNY at Stony Brook, NY, and Lee-Peng Teo, Kuala Lumpur, Malaysia
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Memoirs of the American Mathematical Society
2006; 119 pp; softcover
Volume: 183
ISBN-10: 0-8218-3936-5
ISBN-13: 978-0-8218-3936-2
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
Order Code: MEMO/183/861

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 Weil-Petersson metric on $$T(1)$$ by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that $$T(1)$$ is a Kähler-Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita 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 finite-dimensional 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 pre-Bers embeddings by proving that the Grunsky operators $$B_{1}$$ and $$B_{4}$$, associated with the points in $$T_{0}(1)$$ via conformal welding, are Hilbert-Schmidt. We define a "universal Liouville action" -- a real-valued function $${\mathbf S}_{1}$$ on $$T_{0}(1)$$, and prove that it is a Kähler potential of the Weil-Petersson 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 quasi-circle, 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 Segal-Wilson 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 e-prints: Weil-Petersson metric on the universal Teichmuller space I. Curvature properties and Chern forms, arXiv:math.CV/0312172 (2003), and Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping, arXiv:math.CV/0406408 (2004).

Table of Contents

• Introduction
• Curvature Properties and Chern Forms
• Kähler Potential and Period Mapping
• Appendix A. The Hilbert Manifold Structure of $$\mathcal{T}_{0}(1)$$
• Appendix B. The Period Mapping $$\hat{\mathcal{P}}$$
• Bibliography
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