One considers the modulus problem for a family ~ of homotopic classes {Hi}, in the extended complex plane ~ , of the following types. The classes H i consist of closed Jordan curves, homotopic to appropriate nondegenerate contours or point curves, and also of arcs with endpoints in (distinct or coinciding) distinguished points in ~.One establishes the relation of the indicated extremal metric problem and the problem on the extremal decomposition of ~ in the family ~=~)~ of systems of mutually nonoverlapping domains {Di}, associated with the family ~ of the classes ~I 9 The results of this paper complement a previous theorem of the author [Moduli of families of curves and quadratic differentials.Trudy Mat. Inst.

where the integrals are taken, for example, along linear intervals. Let O,(r darZ~. Q(,~-~ a~,az) dwz-(3) Let ])(a~,az) denote a doubly connected domain of the ~-plane which is symmetric with respect to the circle Iwl=~ and whose inner boundary continuum for a=o is the interval o

No abstract

Let a1, . . . , an be a system of distinct points on the z-sphere C, and let D be a system of all non-overlapping simply-connected domains D1, . . . , Dn on C such that ak ∈ Dk, k = 1, . . . , n. Let M (Dk, ak) be the reduced module of the domain Dk with respect to the point ak ∈ Dk. In the present paper, we solve some problems concerning the maximum of weighted sums of the reduced modules M (Dk, ak) in certain families of systems of domains {Dk} described above, where the systems of points {ak} satisfy prescribed symmetry conditions. In each case, the proof is based on an explicit construction of an admissible metric of the module problem, which is equivalent to the extremal problem under consideration, from known extremal metrics of simpler module problems. Bibliography: 7 titles.

No abstract