On problems on the extremal decomposition. (Russian) Zbl 0608.30026
For the quadratic differential \(Q(z)dz^ 2\), let \(\phi =\phi_ Q\) denote the union of all critical trajectories in the closed complex plane \({\bar {\mathbb{C}}}\), and set \({\hat \phi}=int {\bar \phi}\). If \({\hat \phi}=\emptyset\) (this is the only case considered), then \({\bar \phi}\) divides \({\bar {\mathbb{C}}}\) into finitely many domains \(D_ i\). Set \(D_ Q=\{D_ i\}\). If the only singularities of \(Q(z)dz^ 2\) are simple poles, then \(D_ Q\) has certain extremal properties. If, in addition, to the simple poles, \(Q(z)dz^ 2\) has poles of order 2, then Q(z) has a certain known representation.
For quadratic differentials with poles only of first and second order, the author considers extremal properties of the moduli of the \(D_ i\) and related quantities. The results extend and generalize results by G. V. Kuz’mina [Tr. Mat. Inst. Steklova 139, 240 p. (1980; Zbl 0482.30015)], A. Yu. Solynin [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 144, 136-145 (1985; Zbl 0597.32021)], and the author. The details are too complicated to present here.
For quadratic differentials with poles only of first and second order, the author considers extremal properties of the moduli of the \(D_ i\) and related quantities. The results extend and generalize results by G. V. Kuz’mina [Tr. Mat. Inst. Steklova 139, 240 p. (1980; Zbl 0482.30015)], A. Yu. Solynin [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 144, 136-145 (1985; Zbl 0597.32021)], and the author. The details are too complicated to present here.
Reviewer: Renate McLaughlin
MSC:
30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |
30C55 | General theory of univalent and multivalent functions of one complex variable |