Extremal properties of quadratic differentials with trajectories which are asymptotically similar to logarithmic spirals

One considers the module problem for a family $\mathcal{H}$ of homotopy classes $H_i$ of curves on the $z$-sphere $\bar{ \mathbb{C} }$, where some of the classes $H_i$ consist of curves which in the neighborhoods of the distinguished points on $\bar{ \mathbb{C} }$ behave asymptotically similar to logarithmic spirals. The connection of the indicated extremal metric problem with the problem on the extremal partitioning of $\bar{ \mathbb{C} }$ is established. This paper complements a previous theorem of the author (Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., Vol. 154, pp. 110–129, 1986).