Abstract
A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole singularities. In a neighborhood of a pole, such a foliation comprises foliated strips and half-planes, and its leaf space determines a metric graph. We introduce the notion of an asymptotic direction at each pole and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue of a theorem of Gardiner–Masur for meromorphic quadratic differentials. We also prove an analogue of the Hubbard–Masur theorem; namely, for a fixed punctured Riemann surface there exists a meromorphic quadratic differential with any prescribed horizontal foliation, and such a differential is unique provided we prescribe the singular flat geometry at the poles.
Citation
Kealey Dias. Subhojoy Gupta. Maria Trnkova. "Quadratic differentials, measured foliations, and metric graphs on punctured surfaces." Illinois J. Math. 65 (2) 417 - 454, June 2021. https://doi.org/10.1215/00192082-8827639
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