The problems on extremal decomposition in spaces of Riemann surfaces

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.