The Geometry of the Moduli Space of Riemann Surfaces

We wish to describe how the hyperbolic geometry of a Riemann surface of genus g y g > 2, leads to a symplectic geometry on Tg, the genus g Teichmüller space, and ~Mg, the moduli space of genus g stable curves. The symplectic structure has three elements: the Weil-Petersson Kahler form, the FenchelNielsen vector fields t+, and the geodesic length functions I*. Weil introduced a Kahler metric for Tg based on the Petersson product for automorphic forms; the metric is invariant under the covering of Tg onto Mg, the classical moduli space of Riemann surfaces [1]. To a geodesic a on a marked Riemann surface R with hyperbolic metric is associated the Fenchel-Nielsen vector field ta on Tg; ta is the infinitesimal generator of the flow given by the Fenchel-Nielsen (fractional twist) deformation for a [7]. The infinitesimal generators of Thurston's earthquake flows form the completion (in the compact-open topology) of the Fenchel-Nielsen vector fields [6]. A basic invariant of the geodesic a on a marked surface R is its length la(R)', the exterior derivative dla is a 1-form on Tg. We have the following formulas for the Weil-Petersson Kahler form u [8].