# Weil-Petersson Metric on the Universal Teichmüller Space

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*Leon A. Takhtajan; Lee-Peng Teo*

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

# Table of Contents

## Weil-Petersson Metric on the Universal Teichmuller Space

- Contents v6 free
- Introduction 110 free
- Chapter 1. Curvature Properties and Chern Forms 716 free
- 1. The universal Teichmuller space 716
- 2. T(1) as a Hilbert manifold 2231
- 3. T[sub(0)] (1) as a topological group 3241
- 4. Velling- Kirillov and Weil-Petersson metrics 3645
- 5. Characteristic forms of the universal Teichmüller curve 3847
- 6. First and second variations of the hyperbolic metric 4453
- 7. Riemann curvature tensor 4655
- 8. Finite-dimensional Teichmüller spaces 5665

- Chapter 2. Kähler Potential and Period Mapping 6574
- Appendix A. The Hilbert Manifold Structure of T[sub(0)](1) 105114
- Appendix B. The Period Mapping P 109118
- Bibliography 117126