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

Boris Shapiro; Guillaume Tahar

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arXiv · 2002.10280

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

Boris Shapiro·Guillaume Tahar

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Abstract

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces 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 exceeding 2 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.

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Boris Shapiro, Guillaume Tahar. 2020-02-24. On existence of quasi-Strebel structures for meromorphic k-differentials. https://arxiv.org/abs/2002.10280

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