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_k<b_l<5}\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.