Theorems on the extremal decomposition in the family of systems of domains of various types

E. G. Emel'yanov ; G. V. Kuz'mina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 14, Tome 237 (1997), pp. 74-104

Résumé

A problem on extremal decomposition in a family $\mathscr D$ of systems of domains of various types on a finite Riemann surface $\mathfrak R$ is studied. In contrast to the known cases, the family $\mathscr D$ contains a system of bigons whose boundary arcs are asymptotically similar to logarithmic spirals with arbitrarily given slopes in neighborhoods of their vertices. The main result of this work is a full description of the extremal system of domains in the family $\mathscr D$ in terms of the associated quadratic differential $Q(z)dz^2$ which is uniquely determined by a number of conditions. This differential has poles of the second order at distinguished points on $\mathfrak R$ with prescribed initial terms in the expansions of $Q(z)$ with respect to local parameters representing these poles.