| Summary: | In this memoir, we prove that the universal Teichmuller 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 Hilbert 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 Kahler-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 Teichmuller curve fibration over the universal Teichmuller space.As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmuller spaces from the formulas for the universal Teichmuller space. We study in detail the Hilbert manifold structure on $T_{{0}}(1)$ and characterize points on $T_{{0}}(1)$ in terms of Bers and pre-Bers embeddings by proving 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 ${{\mathbf S}}_{{1}}$ on $T_{{0}}(1)$, and prove that it is a Kahler potential of the Weil-Petersson metric on $T_{{0}}(1)$.We also prove that ${{\mathbf S}}_{{1}}$ is $-\tfrac{{1}}{{12\pi}}$ 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 $\hat{{\mathcal{{P}}}}: T(1)\rightarrow\mathcal{{B}}(\ell^{{2}})$ of $T(1)$ into the Banach space of bounded operators on the Hilbert space $\ell^{{2}}$, prove that $\hat{{\mathcal{{P}}}}$ is a holomorphic mapping of Banach manifolds, and show that $\hat{{\mathcal{{P}}}}$ coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan.We prove that the restriction of $\hat{{\mathcal{{P}}}}$ 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 $\hat{{\mathcal{{P}}}}$ consists of compact operators on $\ell^{{2}}$. 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).
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