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Item Details
| Title:
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WEIL-PETERSSON METRIC ON THE UNIVERSAL TEICHMULLER SPACE
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| By: |
A. Takhtajan-Leon, Lee-Peng Teo |
| Format: |
Paperback |
| List price:
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£64.95 |
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We currently do not stock this item, please contact the publisher directly for
further information.
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| ISBN 10: |
0821839365 |
| ISBN 13: |
9780821839362 |
| Publisher: |
AMERICAN MATHEMATICAL SOCIETY |
| Pub. date: |
15 August, 2006 |
| Series: |
Memoirs of the American Mathematical Society No. 183 |
| Pages: |
119 |
| Description: |
Proves 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. This book defines Weil-Petersson metric on $T(1)$ by Hilbert space inner products on tangent spaces. |
| Synopsis: |
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). |
| Publication: |
US |
| Imprint: |
American Mathematical Society |
| Returns: |
Returnable |
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