On Liouville's Equation, Accessory Parameters, and the Geometry of Teichmüller Space for Riemann Surfaces of Genus 0

For the Weil-Petersson metric on the Teichmuller space of marked Riemann surfaces of genus 0 with punctures, a potential is constructed in terms of the density of the hyperbolic metric on the corresponding surface (i.e., in terms of a solution of Liouville's equation). It is shown that this potential is a generating function of the accessory parameters of the Fuchsian uniformization of the corresponding Riemann surface. Also, a connection is established between the accessory parameters and the Eichler integrals of Fuchsian groups.Bibliography: 18 titles.