June 2021 Quadratic differentials, measured foliations, and metric graphs on punctured surfaces
Kealey Dias, Subhojoy Gupta, Maria Trnkova
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Illinois J. Math. 65(2): 417-454 (June 2021). DOI: 10.1215/00192082-8827639


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.


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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


Received: 24 June 2020; Revised: 20 August 2020; Published: June 2021
First available in Project Euclid: 16 December 2020

Digital Object Identifier: 10.1215/00192082-8827639

Primary: 30F30
Secondary: 30F60 , 57R30

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign


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Vol.65 • No. 2 • June 2021
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