On existence of quasi-Strebel structures for meromorphic $k$-differentials

Abstract

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order $k\ge 4$ satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive $k$-differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine–Stieltjes theory of linear differential equations with polynomial coefficients.