Extremal decompositions of a Riemann surface and quasiconformal mappings of a special type

E. G. Emel'yanov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 15, Tome 254 (1998), pp. 108-115

Résumé

It is shown that the extremal decomposition of a finite Riemann surface $\mathfrak R$ into a system of doubly connected domains may be associated with a family of quasiconformal mappings $\mathfrak R\to\mathfrak R'$, which are similar to the Teichmüller mappings. In the case $\mathfrak R=\overline{\mathbb C}$, 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 $\overline{\mathbb C}$.