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 Zap. Nauchn. Sem. POMI, 2002, Volume 286, Pages 126–147 (Mi znsl1572)

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 $D_k$, $a_k\in D_k$, $k=1,…,n$, where the points $a_1,…,a_n$, 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 $a_1,…,a_n$. The problem on the maximum of the conformal invariant
$$2\pi\sum^5_{k=1}M(D_k,b_k)-\frac12\sum_{1\le b_k<b_l<5}\log|b_k-b_l| \tag{*}$$
for all systems of points $b_1,…,b_s$ is also considered. In the case where the systems $\{b_1,…,b_5\}$ 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 $\{b_1,…,b_5\}$ 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.

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English version:
Journal of Mathematical Sciences (New York), 2004, 122:6, 3654–3666

Bibliographic databases:

UDC: 517.54
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

Citation in format AMSBIB
\Bibitem{Kuz02} \by G.~V.~Kuz'mina \paper Problems on extremal decomposition of the Riemann sphere.~II \inbook Analytical theory of numbers and theory of functions. Part~18 \serial Zap. Nauchn. Sem. POMI \yr 2002 \vol 286 \pages 126--147 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl1572} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1937373} \zmath{https://zbmath.org/?q=an:1086.30027} \transl \jour J. Math. Sci. (N. Y.) \yr 2004 \vol 122 \issue 6 \pages 3654--3666 \crossref{https://doi.org/10.1023/B:JOTH.0000035241.35530.6f} 

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This publication is cited in the following articles:
1. G. V. Kuz'mina, “The method of extremal metric in extremal decomposition problems with free parameters”, J. Math. Sci. (N. Y.), 129:3 (2005), 3843–3851
2. G. V. Kuz'mina, “Problems of extremal decomposition of the Riemann sphere. III”, J. Math. Sci. (N. Y.), 133:6 (2006), 1676–1685
3. V. N. Dubinin, “Capacities of condensers, generalizations of Grötzsch Lemmas, and symmetrization”, J. Math. Sci. (N. Y.), 143:3 (2007), 3053–3068
4. J. Math. Sci. (N. Y.), 222:5 (2017), 645–689
5. G. V. Kuz'mina, “On an extremal metric approach to extremal decomposition problems”, J. Math. Sci. (N. Y.), 225:6 (2017), 980–990
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