Weil-petersson Metric on the Universal Teichmüller Space I: Curvature Properties and Chern Forms

We prove that the universal Teichmüller 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 T0(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 Kähler -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 Teichmüller curve fibration over the universal Teichmüller space. As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmüller spaces from the formulas for the universal Teichmüller space.