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Quadratic differentials, measured foliations, and metric graphs on punctured surfaces. (English) Zbl 1469.30093

Summary: 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.

MSC:

30F30 Differentials on Riemann surfaces
30F60 Teichmüller theory for Riemann surfaces
57R30 Foliations in differential topology; geometric theory
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