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Theorems on the extremal decomposition in a family of systems of domains of various types. (English. Russian original) Zbl 0940.30018

J. Math. Sci., New York 95, No. 3, 2221-2239 (1999); translation from Zap. Nauchn. Semin. POMI 237, 74-104 (1997).
The study of extremal decompositions was initiated by the reviewer as a method to prove the existence and properties of extremal metrics for the module of multiple curve families [J. A. Jenkins, Ann. Math., II. Ser. 66, 440-453 (1957; Zbl 0082.06301); Tôhoku Math. J., II. Ser. 45, No. 2, 249-257 (1993; Zbl 0780.30019)] considered independently of the extremal metric problem it becomes the study of extremal decompositions of finite Riemann surfaces into various types of domains. The basic case in the fundamental theorem deals with decompositions into doubly-connected domains. The reviewer indicated certain extensions allowing reduced modules for simply-connected domains and quadrangles associated with border components.
In the present paper, the authors continue their study of extremal decompositions. E. G. Emel’yanov [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 154, 76-89 (1986; Zbl 0608.30026)] introduced a concept similar to reduced module for biangles (bigons). The authors consider the case of extremal decompositions where the competing domains include triangles whose behavior at assigned points is asymptotically like that of logarithmic spirals with given slopes.

MSC:

30F30 Differentials on Riemann surfaces
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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[1] J. A. Jenkins, ”On the existence of certain general metrics,”Ann. Math.,66, 440–453 (1957). · Zbl 0082.06301
[2] K. Strebel,Quadratic Differentials (Ergebn. Math. ihrer Grenzgeb., 3. F., Bd 5), Springer, Berlin (1984).
[3] H. Renelt, ”Konstruktion gewisser quadratischer Differentiale mit Hilfe von Dirichletintegralen,”Math. Nachr.,73, 125–142 (1970). · Zbl 0374.30017
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[10] G. V. Kuz’mina, ”On the existence of quadratic differentials with poles of high orders,”Zap. Nauchn. Semin. POMI,212, 129–138 (1994). · Zbl 0863.30032
[11] G. V. Kuz’mina, ”The module problem for the family of curve classes in an annulus,”Zap. Nauchn. Semin. LOMI,144, 115–127 (1985).
[12] I. Kra,Automorphic Forms and Kleinian Groups, Reading, Massachusetts (1972). · Zbl 0253.30015
[13] E. G. Emel’yanov, ”The dependence of the functional in a problem of extremal decomposition on the parameters,” PDMI Preprint 16/1997 (1997).
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