Some properties of the moduli of families of curves

Abstract

Let A={a₁,...,a_n} and B={b₁,...,b_m} be systems of distinct points in$\bar C$, let χ be a family of homotopic classes H_i,i=1,..., j+m, of closed Jordan curves on

, where the classes H_j+l, l=1,...,m, consist of curves that are homotopic to a point curve in b_ℓ. Let α=α₁,..., α_j+m be a system of positive numbers and letU be the modulus of the extremal-metric problem for the family χ and the system α. In this paper we investigate the dependence of the modulusU=U(α,A,B) on the parameters α_i and on the disposition of the points a_k and b_ℓ. One shows thatU is a smooth function of the indicated arguments and one obtains expressions for the derivatives$\frac{\partial }{{\partial \alpha _i }}$ U,$\frac{\partial }{{\partial \bar a_\kappa }}$ U, and$\frac{\partial }{{\partial \bar b_\ell }}$ U. One gives some applications of these results.