Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping
arXiv:math/0406408
Abstract
We study the Hilbert manifold structure on -- the connected component of the identity of the Hilbert manifold T(1). We characterize points on in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators and , associated with the points in via conformal welding, are Hilbert-Schmidt. We define a ``universal Liouville action'' -- a real-valued function $\SSS_{1}$ on , and prove that it is a Kähler potential of the Weil-Petersson metric on . We also prove that $\SSS_{1}$ is 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 $\hat{\cP}: T(1)\to\cB(\ell^{2})$ of T(1) into the Banach space of bounded operators on the Hilbert space , prove that $\hat{\cP}$ is a holomorphic mapping of Banach manifolds, and show that $\hat{\cP}$ coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of $\hat{\cP}$ to is an inclusion of into the Segal-Wilson universal Grassmannian, which is a holomorphic mapping of Hilbert manifolds. We also prove that the image of the topological group of symmetric homeomorphisms of under the mapping $\hat{\cP}$ consists of compact operators on .
59 pages, Part II for math.CV/0312172