The problems on extremal decomposition in spaces of Riemann surfaces

E. G. Emel'yanov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 84-104

Résumé

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface $\Re$ with a prescribed set $P\subset \Re$ of distinguished points to the case of the Teichmüller space $T_\Re'$ of Riemann surfaces $\widehat{\Re}$ corresponding to $\Re$ under quasiconformal homeomorphisms $f$. For the functional $\mathscr M$ of our problem on extremal decomposition of a surface $\widehat{\Re}$, we consider a function $\mathscr M^*(x)$ expressing the dependence of the extremal value of $\mathscr M$ on a point $x\in T_{\Re'}$ . Differentiation formulas for the function $\mathscr M^*(x)$ are derived. These formulas are different and depend on the genus $g$ of the surface $\mathscr M$. The case where the function $\mathscr M^*(x)$ is pluriharmonic is considered.