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

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

2003-12-08

arxiv.org/abs/math/0312172v3

Mathematics

Complex Variables

70 pages, appendix A added

Scientific paper

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.

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