Zograf, P. G.; Takhtadzhyan, L. A. On Liouville equations, accessory parameters, and the geometry of Teichmüller spaces for Riemann surfaces of genus 0. (Russian) Zbl 0642.32010 Mat. Sb., N. Ser. 132(174), No. 2, 147-166 (1987). The authors prove the results they had announced in Funct. Anal. Appl. 19, 219-220 (1985); translation from Funkts. Anal. Prilozh. 19, No.3, 67- 68 (1985; Zbl 0612.32018) and thoroughly explain the occuring concepts. Let X be a Riemann surface of type (0,n), \(n\geq 3\), i.e. without restriction \(X={\bar {\mathbb{C}}}\setminus \{w_ 1,...,w_ n\}\) with \(w_{n-2}=0\), \(w_{n-1}=1\), \(w_ n=\infty\), and \((w_ 1,...,w_{n- 3})\in W:=\{v\in ({\mathbb{C}}\setminus \{0,1\})^{n-3}:v_ i\neq v_ j\) for all \(i\neq j\}\). The accessory parameters \(c_ 1,...,c_ n\in {\mathbb{C}}\) of X are functions of \((w_ 1,...,w_{n-3})\in W\); \(c_{n-2}\), \(c_{n-1}\), \(c_ n\) are easy to compute from the \(w_ j\) and \(c_ j\) (j\(\leq n-3).\) The solution \(\phi\) of the Liouville equation \(\phi_{w\bar w}=e^{\phi}\) on X is the extremal of a functional S on the set of all smooth functions on X with a certain boundary behaviour (S is called the action integral of the Liouville equation). S(\(\phi)\) is a function on W and it is shown \[ c_ j=-\frac{1}{2\pi}\frac{\partial S(\phi)}{\partial w_ j}\quad for\quad j\leq n-3. \] It is further proven that S(\(\phi)\) is a potential function of the Weil Petersson metric on W, i.e. \[ <\frac{\partial}{\partial w_ i},\frac{\partial}{\partial w_ j}> = - \frac{\partial^ 2S}{\partial w_ i\partial \bar w_ j}, \] and hence induces a potential of the Weil Petersson metric on the Teichmüller space of Riemann surfaces of type (0,n). In the last section a relation between the accessory parameters and the Eichler integrals is developped. Reviewer: K.Wolffhardt Cited in 4 ReviewsCited in 11 Documents MSC: 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F10 Compact Riemann surfaces and uniformization Keywords:Schottky space; accessory parameters; Weil Petersson metric; Teichmüller space of Riemann surfaces Citations:Zbl 0612.32018 PDFBibTeX XMLCite \textit{P. G. Zograf} and \textit{L. A. Takhtadzhyan}, Mat. Sb., Nov. Ser. 132(174), No. 2, 147--166 (1987; Zbl 0642.32010) Full Text: EuDML