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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
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Order Code: MEMO/183/861
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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 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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