Quadratic differentials and signed measures

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