Problems on Extremal Decomposition of the Riemann Sphere. II

Abstract

In the present paper, we solve some problems on the maximum of the weighted sum

$$\sum\limits_{\user1{k = }1}^\user1{n} {\alpha _\user1{k}^2 } M(D_\user1{k} ,\user1{a}_\user1{k} )$$

(M(D_k,a_k) denotes the reduced module of the domain D_k with respect to the point a_k∈ D_k in the family of all nonoverlapping simple connected domains D_k, a_k ∈ D_k, k=1,... ,n, where the points a₁,... ,a_n are free parameters satisfying certain geometric conditions. The proofs involve a version of the method of extremal metric, which reveals a certain symmetry of the extremal system of the points a₁,... ,a_n. The problem on the maximum of the conformal invariant

$$2\pi \sum\limits_{k = 1}^5 {M(D_\user1{k} ,\user1{b}_\user1{k} )} - \frac{1}{2}\sum\limits_{1 \leqslant \user1{b}_\user1{k} < \user1{b}_\user1{l} \leqslant 5} {\log \left| {\user1{b}_\user1{k} - \user1{b}_\user1{i} } \right|} $$

for all systems of points b₁,... ,b_s is also considered. In the case where the systems {b₁,... ,b₅} are symmetric with respect to a certain circle, the problem was solved earlier. A theorem formulated in the author's previous work asserts that the maximum of invariant (*) for all system of points {b₁,... ,b₅} is attained in a certain well-defined case. In the present work, it is shown that the proof of this theorem contains mistake. A possible proof of the theorem is outlined. Bibliography: 10 titles.