The Liouville current of holomorphic quadratic differential metrics

Abstract In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in \cite{bankovic2018marked} that for a fixed closed surface, there is a injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain an interesting result that almost all geodesics of a quadratic differential metric will be dense in the surface, and no other flat metrics have this property, which gives another judgement for whether a metric comes from a holomorphic quadratic differential. Mathematics Subject Classification (2020) 51H25 · 53C99