Some properties of the moduli of families of curves

E. G. Emel'yanov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 6, Tome 144 (1985), pp. 72-82

Résumé

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.