For the TZ metric on the moduli space k to view the MathML source">M0,n of n -pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space k to view the MathML source">Sg,n as holomorphic fibration k to view the MathML source">Sg,n→Sg over the Schottky space k to view the MathML source">Sg of compact Riemann surfaces of genus g, where the fibers are configuration spaces of n points. For the tautological line bundles k to view the MathML source">Li over k to view the MathML source">Sg,n, we define Hermitian metrics k to view the MathML source">hi in terms of Fourier coefficients of a covering map J of the Schottky domain. We define the regularized classical Liouville action S and show that k to view the MathML source">exp{S/π} is a Hermitian metric in the line bundle over k to view the MathML source">Sg,n. We explicitly compute the Chern forms of these Hermitian line bundles
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We prove that a smooth real-valued function on k to view the MathML source">Sg,n, a potential for this special difference of WP and TZ metrics, coincides with the renormalized hyperbolic volume of a corresponding Schottky 3-manifold. We extend these results to the quasi-Fuchsian groups of type k to view the MathML source">(g,n).