Weil-petersson Metric on the Universal Teichmüller Space Ii. Kähler Potential and Period Mapping

We study the Hilbert manifold structure on T0(1) — the connected component of the identity of the Hilbert manifold T (1). We characterize points on T0(1) in terms of Bers and pre-Bers embeddings, and prove 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 T0(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 P̂ : T (1) → B(l) of T (1) into the Banach space of bounded operators on the Hilbert space l, prove that P̂ is a holomorphic mapping of Banach manifolds, and show that P̂ coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of P̂ to T0(1) is an inclusion of T0(1) into the SegalWilson 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 under the mapping P̂ consists of compact operators on l.