Weil-Petersson metric on the universal Teichmuller space I: Curvature properties and Chern forms

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We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We 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.
70 pages, appendix A added

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Complex Variables, High Energy Physics - Theory, Differential Geometry, 30F60 (Primary) 32G15, 46E20, 58B20, 58B25 (Secondary)

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http://salesiana.dossiersoluciones.com/handle/123456789/69517

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0312172
 http://arxiv.org/abs/math/0312172

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arXiv