Kuz’mina, G. V. Problems on extremal decomposition of the Riemann sphere. (English. Russian original) Zbl 1071.30022 J. Math. Sci., New York 118, No. 1, 4880-4894 (2003); translation from Zap. Nauchn. Semin. POMI 276, 253-275 (2001). Summary: We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let \({\mathbf a}= \{a_1,\dots, a_n\}\) be a system of distinct points on \(\overline{\mathbb{C}}\) and let \({\mathcal D}({\mathbf a})\) be the family of all systems \(\mathbb{D}= \{D_1,\dots, D_n\}\) of nonoverlapping simply connected domains on \(\overline{\mathbb{C}}\) such that \(a_k\in D_k\), \(k= 1,\dots, n\). Let \[ J({\mathbf a})= \max_{\mathbb{D}\subset{\mathcal D}({\mathbf a}}\,\Biggl\{2\pi \sum^n_{k=1} M(D_k, a_k)- {2\over n-1} \sum_{1\leq k< l\leq n}\log|a_k- a_l|\Biggr\}, \] where \(M(D_k,a_k)\) is the reduced module of the domain \(D_k\) with respect to the point \(a_k\). At present, the problem concerning the value \(\max_{{\mathbf a}} J({\mathbf a})\) was solved completely for \(n= 2, 3,4\). In this work, we continue the previous author’s investigations and consider the case \(n= 5\). In addition, we consider the problem concerning the maximum of the sum \[ \alpha^2\{M(D_0, 0)+ M(D_{n+1},\infty)\}+ \sum^n_{k=1} M(D_k, a_k) \] in the family \({\mathcal D}({\mathbf a})\) introduced above, where \({\mathbf a}= \{0, a_1,\dots, a_n,\infty\}\), \(a_k\), \(k= 1,\dots, n\), are arbitrary points of the circle \(|z|= 1\), and \(\alpha\) is a positive number. We prove that if \(\alpha/n\leq 1/\sqrt{8}\), then the maximum is attained only for systems of equidistant points of the circle \(|z|= 1\). For \(\alpha/n= 1/\sqrt{8}\), this result was obtained earlier by Dubinin who applied the method of symmetrization. It is shown that if \(n\geq 2\), where \(\alpha/n\geq 1/2\) is an even number, then equidistant points of the circle \(|z|= 1\) do not realize the indicated maximum. Cited in 1 ReviewCited in 6 Documents 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 Keywords:extremal metric × Cite Format Result Cite Review PDF Full Text: DOI