Weil-Petersson metric on the universal Teichmüller space

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 Hubert 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 T₀(1) and characterize points on T₀(1) in terms of Bers and pre-Bers embeddings by proving that the Grunsky operators B₁ and B₄, associated with the points in T₀(1) via conformal welding, are Hilbert-Schmidt. We define a "universal Liouville action" - a real-valued function S₁ on T ₀(1), and prove that it is a Kähler potential of the Weil-Petersson metric on T₀(1). We also prove that S₁ is - 1/12π 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 ℘̂: T(1) → ℬ(ℓ²) of T(1) into the Banach space of bounded operators on the Hilbert space ℓ², prove that ℘̂ is a holomorphic mapping of Banach manifolds, and show that ℘̂ coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of ℘̂ to T₀(1) is an inclusion of T₀(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 S¹ under the mapping ℘̂ consists of compact operators on ℓ². 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).