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On Liouville equations, accessory parameters, and the geometry of Teichmüller spaces for Riemann surfaces of genus 0. (Russian) Zbl 0642.32010
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

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F10 Compact Riemann surfaces and uniformization
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