Extremal decompositions of a Riemann surface and quasiconformal mappings of a special form. II

Let $\mathfrak R$ be a finite Riemann surface. For a quadratic differential on $\mathfrak R$ associated with a certain problem on extremal decomposition of $\mathfrak R$ into $n$ domians, a parametric family of quasiconformal mappings $f_{\mathbf K}\colon\mathfrak R\to\mathfrak R_{\mathbf K}$, $\mathbf K=(k_1,\dots,k_s)$, $k_i\to\mathbb C$, is defined. These mappings map the domians of the extremal decomposition of $\mathfrak R$ onto the domians of the extremal decomposition of $\mathfrak R_{\mathbf K}$. This allows one to study the functional dependence of the problem on the parameters.