Weil-Petersson metric on the universal Teichmüller space
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Bibliographic Information
Weil-Petersson metric on the universal Teichmüller space
（Memoirs of the American Mathematical Society, no. 861）
American Mathematical Society, c2006
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"September 2006, volume 183, number 861 (first of 4 numbers)"
Includes bibliographical references (p. 117-119)
Description and Table of Contents
Description
In this memoir, we prove that the universal Teichmuller 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 Kahler-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 Teichmuller curve fibration over the universal Teichmuller space.As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmuller spaces from the formulas for the universal Teichmuller 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 Kahler 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 Kahler 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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