Level functions of quadratic differentials, signed measures, and the Strebel property

In this paper, motivated by the classical notion of a Strebel qua- dratic differential on a compact Riemann surface without boundary, we in- troduce several classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess some properties of Strebel differentials and appear in applications. We discuss the relation between gradient differen- tials and special signed measures supported on their set of critical trajectories. We provide a characterisation of gradient differentials for which there exists a positive measure in the latter class.