Extremal decomposition problems in the space of Riemann surfaces (Q1608608)

Let \({\mathfrak R}\) be a finite Riemann surfaces and let \(A\) be a set of distinguished points on \({\mathfrak R}\). Let \(D\) be the family of all systems of nonoverlapping doubly connected domains \(D_1,D_2,\dots, D_n\), on \({\mathfrak R}'={\mathfrak R}\setminus A\). The paper deals with the problem to find the maximal value of the functional \[ {\mathfrak M}= \sum^n_{j=1} \alpha_j M(D_j), \] where \(M\) denotes the module of a doubly connected domain and \(\alpha_j\) are a given collection of positive numbers. This problem is closely related to the problem to find quadratic differentials with closed trajectories and goes back to \textit{H. Grötzsch} [Ber. Verh. Sächs. Akad. Leipzig 83, 254-279 (1931; Zbl 0003.26102)]. This problem was solved by \textit{U. Pirl} [Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg 4, 1225-1252 (1955; Zbl 0066.05804)], \textit{J. A. Jenkins} [Ann. Math. (2) 66, 440-453 (1957; Zbl 0082.06301)] and \textit{K. Strebel} [Festband 70. Geb. Rolf Nevanlinna, Vortr. 2. R. Nevanlinna-Kolloquiums Zürich 1965, 105-127 (1966; Zbl 0156.09002), J. Anal. Math. 19, 373-382 (1967; Zbl 0158.32402), Ann. Acad. Sci. Fenn. Ser. A I 405, 12 (1967)]. Another approach was given by \textit{H. Renelt} [Math. Nachr. 73, 125-142 (1976; Zbl 0374.30017)]. Recently, \textit{J. A. Jenkins} [Indiana Univ. Math. J. 49, No. 3, 891-896 (2000; Zbl 0969.30024)] has a more general extremal problem solved and a similar more general extremal decomposition problem was also solved by the author and \textit{G. V. Kuz'mina} [Zap. Nauchn. Semin. POMI 237, 74-104 (1997; Zbl 0940.30018)]. The aim of the paper under review is to extend such kind of extremal problems to the Teichmüller space \(T_{{\mathfrak R}'}\). Differentiation formulas are given for the function of extremal values \({\mathfrak M}^*\) depending on \(x\in T_{{\mathfrak R}'}\). In the case considered by Jenkins it is shown that \({\mathfrak M<}^*\) is a pluriharmonic function in a neighborhood of the initial point \(x_0\) and the quadratic differentials \(Q(x)\) describing the extremal decomposition are holomorphic functions on \(x\in T_{{\mathfrak R}'}\).