On uniformization of Riemann surfaces and the Weil-Petersson metric on Teichmüller and Schottky spaces

A potential is constructed for the Weil–Petersson metric on the Teichmüller space $T_g$ of marked Riemann surfaces of genus $g>1$ in terms of the density of the Poincaré metric on the region of discontinuity of the corresponding normalized marked Schottky group. It is proved that the difference between the projective connections corresponding to the Fuchsian uniformization and the Schottky uniformization for a marked Riemann surface of genus $g>1$ is the $\partial$-derivative of this potential, and the Weil–Petersson symplectic form on Teichmüller space is the $\overline\partial$-derivative of the Fuchsian projective connection. The results establish how the accessory parameters of the Fuchsian uniformization and the Schottky uniformization of a Riemann surface are connected with the geometries of Teichmüller space and Schottky space.
Bibliography: 31 titles.