On mixed problems for extremal decompositions (Q2710404)

The study of problems for extremal decompositions of Riemann surfaces was initiated in a paper by the author [Ann. Math., II. Ser. 66, 440-453 (1957; Zbl 0082.06301)] where a problem of this type was introduced to obtain the solution of a module problem for multiple curve families on a finite Riemann surface \(R\). For a free family of homotopy classes \({\mathcal H}_j\), \(j=1, \dots, L\), on \({\mathcal R}\) and admissible doubly-connected domains \(D_j\) associated with \({\mathcal H}_j\), \(j=1,\dots,L\), the problem was the following. \({\mathcal P}(a_1, \dots,a_L)\). Let \(a_j\), \(j=1,\dots,L\), be non-negative numbers not all zero. For a family of admissible domains \(D_j\) of module \(M_j\), \(j=1, \dots,L\), find the least upper bound of \(\sum^L_{j=1} a^2_jM_j\). \textit{H. Renelt} [Math. Nachr. 73, 125-142 (1976; Zbl 0374.30017)] considered the plane case of the following problem \(R(b_1,\dots, b_L)\). Let \(b_j\), \(j=1, \dots,L\), be positive numbers. For an admissible family of domains \(D_j\) of module \(M_j\), \(j= 1, \dots,L\), find the greatest lower bound for \(\sum^L_{j=1} b^2_jM_j^{-1}\). In a later paper [\textit{Tôhoku} Math. J. 43, No. 2, 249-257 (1993; Zbl 0780.30019)] \textit{J. A. Jenkins} gave a proof of the results of the previous paper using only the method of the extremal metric and in particular showed that the solution of the problem \(R(b_1,\dots, b_L)\) is an easy consequence of that of \({\mathcal P}(a_1,\dots,a_L)\). In the present paper the author shows that both of these problems are special cases of the following problem \(X(a_1,\dots, a_N, b_{N+1}, \dots,b_L)\). Let \(a_j\), \(j=1,\dots,N\), and \(b_j\), \(j=N+1, \dots,L\), be non-negative numbers not all zero, with any \(b_j\) which actually occurs positive, \(0\leq N\leq L\). For an admissible family of domains \(D_j\) of module \(M_j\), \(j=1,\dots,L\), find the least upper bound of \(\sum^N_{j=1} a^2_jM_j\to \sum^L_{j=N+1} b^2_jM_j^{-1}\). The author shows that the solution of this problem can be derived from the solution of the problem \({\mathcal P}(a_1, \dots, a_L)\) and gives the appropriate equality statement.