Problems on extremal decomposition of the Riemann sphere.~II

G. V. Kuz'mina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 126-147

Résumé

In the present paper, we solve some problems on the maximum of the weighted sum $$ \sum^n_{k=1}\alpha^2_kM(D_k, a_k) $$ ($M(D_k, a_k)$ denote the reduced module of the domian $D_k$ with respect to the point $a_k\in D_k$) in the family of all nonoverlapping simple connected domians $D_k$, $a_k\in D_k$, $k=1,\dots,n$, where the points $a_1,\dots,a_n$, 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 $a_1,\dots,a_n$. The problem on the maximum of the conformal invariant \begin{equation} 2\pi\sum^5_{k=1}M(D_k,b_k)-\frac12\sum_{1\le b_k5}\log|b_k-b_l| \tag{*} \end{equation} for all systems of points $b_1,\dots,b_s$ is also considered. In the case where the systems $\{b_1,\dots,b_5\}$ 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 $\{b_1,\dots,b_5\}$ 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.