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

Recurrent Trajectories and Finite Critical Trajectories of Quadratic Differentials on the Riemann Sphere