# 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.

## References

- 1.Daniele, A., Liu, L., Athanase, P., Su, W.: The horofunction compactification of the arc metric on Teichmüller space, arXiv:1411.6208
- 2.Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S., Moeller, M.: Compactification of strata of abelian differentials,
*preprint*, arXiv:1604.08834 - 3.Bridgeland, T., Smith, I.: Quadratic differentials as stability conditions. Publ. Math. Inst. Hautes Études Sci.
**121**, 155–278 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Culler, M., Morgan, J.W.: Group actions on \({\bf R}\)-trees. Proc. London Math. Soc. (3)
**55**(3), 571–604 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Daskalopoulos, G.D., Dostoglou, S., Wentworth, R.A.: On the Morgan-Shalen compactification of the \({\rm SL}(2,{\bf C})\) character varieties of surface groups. Duke Math. J.
**101**(2), 189–207 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Daskalopoulos, G., Wentworth, R.: Harmonic maps and Teichmüller theory, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, (2007), pp. 33–109Google Scholar
- 7.Evans, L.C.: Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI (2010)Google Scholar
- 8.Fathi, A., Laudenbach, F., Poénaru, V.: Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ (2012) Translated from the 1979 French original by Djun M. Kim and Dan MargalitGoogle Scholar
- 9.Farb, B., Wolf, M.: Harmonic splittings of surfaces. Topology
**40**(6), 1395–1414 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Gupta, S.: Asymptoticity of grafting and Teichmüller rays II. Geometriae Dedicata
**176**, 185–213 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Gupta, S., Wolf, M.: Quadratic differentials, half-plane structures, and harmonic maps to trees. Comment. Math. Helv.
**91**(2), 317–356 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Gupta, S., Wolf, M.: Meromorphic quadratic differentials with complex residues and spiralling foliations. In the tradition of Ahlfors-Bers. VII, Contemp. Math., vol. 696, Am. Math. Soc., Providence, RI, pp. 153–181 (2017)Google Scholar
- 13.Hubbard, J., Masur, H.: Quadratic differentials and foliations, Acta Math.
**142**(3–4) (1979)Google Scholar - 14.Huang, A.: Harmonic maps of punctured surfaces to the hyperbolic plane, arXiv:1605.07715
- 15.Jost, J.: Harmonic maps between surfaces, Lecture Notes in Mathematics, vol. 1062. Springer, Berlin (1984)Google Scholar
- 16.Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom.
**1**(3–4), 561–659 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom.
**1**(3-4), (1993)Google Scholar - 18.Korevaar, N.J., Schoen, R.M.: Global existence theorems for harmonic maps to non-locally compact spaces. Commun. Anal. Geom.
**5**(2), 333–387 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Mese, C.: Uniqueness theorems for harmonic maps into metric spaces. Commun. Contemp. Math.
**4**(4), 725–750 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Mulase, M., Penkava, M.: Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over \(\overline{{\bf Q}}\). Asian J. Math.
**2**(4) (1998), Mikio Sato: a great Japanese mathematician of the twentieth centuryGoogle Scholar - 21.Schoen, R.M.: Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., vol. 2, pp. 321–358Google Scholar
- 22.Skora, R.K.: Splittings of surfaces. J. Am. Math. Soc.
**9**(2), 605–616 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Strebel, K.: Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, (1984)Google Scholar
- 24.Thurston, D.: On geometric intersection of curves in surfaces, Unpublished article available at the author’s webpageGoogle Scholar
- 25.Wolf, M.: The Teichmüller theory of harmonic maps. J. Differ. Geom.
**29**(2), 449–479 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Wolf, M.: Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space. J. Differ. Geom.
**33**(2), 487–539 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Wolf, M.: Harmonic maps from surfaces to \({\bf R}\)-trees. Math. Z.
**218**(4), 577–593 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Wolf, M.: On realizing measured foliations via quadratic differentials of harmonic maps to \({\bf R}\)-trees. J. Anal. Math.
**68**, 107–120 (1996)MathSciNetCrossRefzbMATHGoogle Scholar