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 Teichmɒller space T₀ of marked Riemann surfaces of genus 0 with n 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.