# Quadratic differentials and foliations

@article{Hubbard1979QuadraticDA, title={Quadratic differentials and foliations}, author={John H. Hubbard and Howard A. Masur}, journal={Acta Mathematica}, year={1979}, volume={142}, pages={221-274} }

This paper concerns the interplay between the complex structure of a Riemann surface and the essentially Euclidean geometry induced by a quadratic differential. One aspect of this geometry is the " trajectory structure" of a quadratic differential which has long played a central role in Teichmfiller theory starting with Teichmiiller's proof of the existence and uniqueness of extremal maps. Ahlfors and Bers later gave proofs of that result. In other contexts, Jenkins and Strebel have studiedâ€¦Â

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## References

SHOWING 1-10 OF 20 REFERENCES

On the density of strebel differentials

- Mathematics
- 1975

w 1. Statement of the Main Result Let X be a compact Riemann surface of genus g > 2 and denote by O the sheaf of germs of holomorphic differential 1-forms on X. Let q~H~ ~| be a nonzero holomorphicâ€¦

On the existence and uniqueness of Strebel differentials

- Mathematics
- 1976

Let X be a compact Riemann surface of genus g > 1, and q GHÂ°(X> ÂŁ2Â® ) be a holomorphic quadratic form on X. A tangent vector ÂŁ G Tx X is called horizontal if (q, ÂŁ Â® ÂŁ> > 0. The horizontal vectorsâ€¦

On Quadratic Differentials and Extremal Quasi-Conformal Mappings*

- Mathematics
- 1975

1. Extremal quasi-conformal mappings and Teichmueller mappings. A regular quasi-conformal mapping of a plane domain G onto a domain G' (more generally of a Riemann surface R onto a surface R') is anâ€¦

Ăśber quadratische Differentiale mit geschlossenen Trajektorien und extremale quasikonforme Abbildungen

- 1966

O. Teichmullers Beweis [9] seines Satzes uber die Struktur der extremalen quasikonformen Abbildungen geschlossener Riemannscher Flachen beruht auf einer Kontinuitatsmethode, wie sie schon seinerâ€¦

The Asymptotic Geometry of Teichmiiller Space

- The Asymptotic Geometry of Teichmiiller Space
- 1978

Stir les sections analytiques de la courbe universelle de Teichmiiller

- Mere. Amer. Math. Soc
- 1976

Lacuna for hyperbolic differential operators II

- Acta Math
- 1973