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Quadratic differentials, measured foliations, and metric graphs on punctured surfaces

Dias, K and Gupta, S and Trnkova, M (2021) Quadratic differentials, measured foliations, and metric graphs on punctured surfaces. In: Illinois Journal of Mathematics, 65 (2). pp. 417-454.

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Official URL: https://doi.org/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 com-patible pair of measured foliations uniquely determines a complex structure and a mero-morphic 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.

Item Type: Journal Article
Publication: Illinois Journal of Mathematics
Publisher: Duke University Press
Additional Information: The copyright for this article belongs to the Authors.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 01 Jun 2023 10:15
Last Modified: 01 Jun 2023 10:15
URI: https://eprints.iisc.ac.in/id/eprint/81740

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