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Zapiski Nauchnykh Seminarov POMI, 2002, Volume 286, Pages 126–147
(Mi znsl1572)
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This article is cited in 5 scientific papers (total in 5 papers)
Problems on extremal decomposition of the Riemann sphere. II
G. V. Kuz'mina St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
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.
Received: 25.12.2001 Revised: 25.03.2002
Citation:
G. V. Kuz'mina, “Problems on extremal decomposition of the Riemann sphere. II”, Analytical theory of numbers and theory of functions. Part 18, Zap. Nauchn. Sem. POMI, 286, POMI, St. Petersburg, 2002, 126–147; J. Math. Sci. (N. Y.), 122:6 (2004), 3654–3666
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https://www.mathnet.ru/eng/znsl1572 https://www.mathnet.ru/eng/znsl/v286/p126
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