Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping
Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping
复制标题
通用 Teichmuller 空间 II 上的 Weil-Petersson 度量。
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
L. Teo
中科院分区：
文献类型：
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作者：
L. Takhtajan;L. Teo
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 K"{a}hler potential of the Weil-Petersson metric on $T_{0}(1)$. We also prove that $SSS_{1}$ is $- frac{1}{12pi}$ 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 $hat{cP}: T(1) ocB(ell^{2})$ 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 $ell^{2}$.
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 K"{a}hler potential of the Weil-Petersson metric on $T_{0}(1)$. We also prove that $SSS_{1}$ is $- frac{1}{12pi}$ 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 $hat{cP}: T(1) ocB(ell^{2})$ 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 $ell^{2}$.