# Weil-Petersson metric on the universal Teichmüller space

### About this Title

**Leon A. Takhtajan** and **Lee-Peng Teo**

Publication: Memoirs of the American Mathematical Society

Publication Year
2006: Volume 183, Number 861

ISBNs: 978-0-8218-3936-2 (print); 978-1-4704-0465-9 (online)

DOI: http://dx.doi.org/10.1090/memo/0861

MathSciNet review: 2251887

MSC: Primary 32G15; Secondary 30C55, 30F60, 58B20

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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

**Chapters**

- Introduction
- 1. Curvature properties and Chern forms
- 2. Kähler potential and period mapping