num_pages: 119
description: 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 u
language: English
title: Weil-Peterson Metric on the Universal Teichmüller Space
titlecomplete: Weil-Peterson Metric on the Universal Teichmüller Space
ratings_count: 1
publisher: American Mathematical Society