Some properties of the moduli of families of curves

Let $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_m\}$ be systems of distinct points in $\bar{ \mathbb{C} }$, let $H$ be a family of homotopic classes $H_i$, $i=1,\dots,j+m$, of closed Jordan curves on $\bar{ \mathbb{C} }^\prime=\bar{ \mathbb{C} }\setminus\{A\cup B\}$, where the classes $H_{j+\ell}$, $\ell=1,\dots,m$, consist of curves that are homotopic to a point curve in $b_\ell$. Let $\alpha=\{\alpha_1,\dots,\alpha_{j+m}\}$ be a system of positive numbers and let $M$ be the modulus of the extremal-metric problem for the family $H$ and the system $\alpha$. In this paper we investigate the dependence of the modulus $M=M(\alpha,A,B)$ on the parameters $\alpha_1$ and on the disposition of the points $a_k$ and $b_\ell$. One shows that $M$ is a smooth function of the indicated arguments and one obtains expressions for the derivatives $\frac{\partial}{\partial\alpha_i}M$, $\frac{\partial}{\partial\bar a_k}M$, and $\frac{\partial}{\partial b_\ell}M$. One gives some applications of these results.