Extremal decompositions of riemann surfaces and quasiconformal mappings of special form

It is shown that the extremal decomposition of a finite Riemann surface ℛ into a system of doubly connected domains may be associated with a family of quasiconformal mappings ℛ → ℛ′, which are similar to the Teichmüller mappings. In the case ℛ = ℂ̄, this construction allows us to prove that the extremal value of the functional in the indicated problem on the extremal decomposition is a pluriharmonic function of the coordinates of the distinguished points on ℂ̄. © 2001 Plenum Publishing Corporation.