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

TL;DR

This paper introduces new classes of quadratic differentials inspired by Strebel differentials, explores their properties, and characterizes those associated with positive measures, advancing understanding of their geometric and measure-theoretic aspects.

Contribution

It defines non-chaotic, gradient, and positive gradient quadratic differentials, linking them to signed measures and characterizing gradient differentials with positive measures.

Findings

Introduction of new classes of quadratic differentials.

Relationship established between gradient differentials and signed measures.

Characterization of gradient differentials with positive measures.

Abstract

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.