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.