Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping
| dc.creator | Takhtajan, Leon A. | |
| dc.creator | Teo, Lee-Peng | |
| dc.date | 2004-06-21 | |
| dc.date.accessioned | 2026-07-07T05:09:26Z | |
| dc.date.available | 2026-07-07T05:09:26Z | |
| dc.description | We study the Hilbert manifold structure on $T_{0}(1)$ -- the connected component of the identity of the Hilbert manifold T(1). We characterize points on $T_{0}(1)$ in terms of Bers and pre-Bers embeddings, and prove 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 $\SSS_{1}$ 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 $\SSS_{1}$ is $-\tfrac{1}{12π}$ 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 $\hat{\cP}: T(1)\to\cB(\ell^{2})$ of T(1) into the Banach space of bounded operators on the Hilbert space $\ell^{2}$, 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 $T_{0}(1)$ 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 $\hat{\cP}$ consists of compact operators on $\ell^{2}$. | |
| dc.description | 59 pages, Part II for math.CV/0312172 | |
| dc.identifier | https://arxiv.org/abs/math/0406408 | |
| dc.identifier | http://arxiv.org/abs/math/0406408 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71625 | |
| dc.subject | Complex Variables | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Differential Geometry | |
| dc.subject | 30F60 (Primary) 30C55, 32G15, 46E20, 58B20, 58B25 (Secondary) | |
| dc.title | Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping | |
| dc.type | text |