# Meromorphic quadratic differentials and measured foliations on a Riemann surface

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## Abstract

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential *q* if we prescribe, in addition, the principal parts of \(\sqrt{q}\) at the poles. This generalizes a theorem of Hubbard and Masur for holomorphic quadratic differentials. The proof analyzes infinite-energy harmonic maps from the Riemann surface to \(\mathbb {R}\)-trees of infinite co-diameter, with prescribed behavior at the poles.

## Notes

### Acknowledgements

Both authors gratefully appreciate support by NSF Grants DMS-1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR network) as well as the hospitality of MSRI (Berkeley), where some of this work was initiated. The second author acknowledges support of NSF DMS-1564374. The first author thanks the hospitality and support of the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95) during May–June 2016. The first author wishes to acknowledge that the research leading to these results was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Union Framework Programme (FP7/2007-2013) under grant agreement no. 612534, project MODULI—Indo European Collaboration on Moduli Spaces. Finally, both authors are grateful for the referee’s careful reading and insightful comments.

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