  In this paper, the focus will be on both the existence and non-existence
respectively of finite critical trajectories and recurrent trajectories of a
quadratic differential on the Riemann sphere. We show the connection between
these two items. More precisely, we collect some criterions for the
non-existence of recurrent trajectories. The first criterion is in the same
vein of Jenkins Three-pole Theorem, while the second one is in relation with
the so-called Level function of quadratic differentials.

Classical Analysis and ODEs