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Problems on extremal decomposition of the Riemann sphere. II. (English. Russian original) Zbl 1086.30027

J. Math. Sci., New York 122, No. 6, 3654-3666 (2004); translation from Zap. Nauchn. Semin. POMI 286, 126-147 (2002).
Summary: [For part I see ibid. 118, No. 1, 4880–4894 (2003; Zbl 1071.30022).]
In the present paper, we solve some problems on the maximum of the weighted sum \[ \sum_{k=1}^n \alpha_k^2 M(D_k,a_k) \] \((M(D_k,a_k)\) denotes the reduced module of the domain \(D_k\) with respect to the point \(a_k\in D_k\)) in the family of all nonoverlapping simple connected domains \(D_k\), \(a_k\in D_k\), \(k=1,\dots,n\), 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 \[ 2\pi \sum_{k=1}^6 M(D_k,b_k)- \tfrac12 \sum_{1\leq b_k< b_l\leq 5} \log|b_k-b_l|, \tag \(*\) \] 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 \(\{b1,\dots,b_5\}\) is attained in a certain well-defined case. In the present work, it is shown that the proof of this theorem contains a mistake. A possible proof of the theorem is outlined.

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions

Citations:

Zbl 1071.30022
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