Quadratic differentials and foliations

  title={Quadratic differentials and foliations},
  author={John H. Hubbard and Howard A. Masur},
  journal={Acta Mathematica},
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 studied… 
Measured foliations and the minimal norm property for quadratic differentials
The objective is to prove two important inequalities of Riemann surface theory and to present them in such a way that one can see their close relationship. The first inequality is the minimal norm
Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms
A quadratic differential on a Riemann surface M determines certain " topological" data: the genus o f M; the orders o f zeros and poles; and the orientability o f the horizontal foliation. In this
The Schwarzian derivative and measured laminations on Riemann surfaces
A holomorphic quadratic differential on a hyperbolic Rie- mann surface has an associated measured foliation, which can be straight- ened to yield a measured geodesic lamination. On the other hand, a
Asymptotic behavior of quadratic differentials
ii Summary Every closed oriented surface S of genus g ≥ 2 can be endowed with a non-unique hyperbolic metric. By the celebrated Uniformization Theorem, hyperbolic and complex structures on S are in
Meromorphic quadratic differentials with complex residues and spiralling foliations
A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by
Topology of Generic Hamiltonian Foliations on Riemann Surfaces
The topology of generic Hamiltonian dynamical systems given by the real parts of generic holomorphic 1-forms on Riemann surfaces is studied. Our approach is based on the notion of transversal
Complex Projective Structures , Grafting , and Teichmüller Theory
We study the space P(S) of marked complex projective (CP1) structures on a compact surface in terms of Teichmüller theory and hyperbolic geometry. In particular, we show that the structure of this
Dynamics in the moduli space of Abelian differentials
We announce the proof of the Zorich–Kontsevich conjecture: the nontrivial Lyapunov exponents of the Teichmüller flow on (any connected component of a stratum of) the moduli space of Abelian
On realizing measured foliations via quadratic differentials of harmonic maps toR-trees
AbstractWe give a brief, elementary and analytic proof of the theorem of Hubbard and Masur [HM] (see also [K], [G]) that every class of measured foliations on a compact Riemann surfaceR of genusg can
A central feature of the study of Kleinian groups is the interplay between 3-dimensional hyperbolic geometry and conformal dynamics
Thurston's ending lamination conjecture states that a hyperbolic manifold is uniquely determined by a collection of Riemann surfaces and geodesic laminations that describe the asymptotic geometry of


On the density of strebel differentials
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 holomorphic
On the existence and uniqueness of Strebel differentials
Let X be a compact Riemann surface of genus g > 1, and q GH°(X> £2® ) be a holomorphic quadratic form on X. A tangent vector £ G Tx X is called horizontal if (q, £ ® £> > 0. The horizontal vectors
On Quadratic Differentials and Extremal Quasi-Conformal Mappings*
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 an
Über quadratische Differentiale mit geschlossenen Trajektorien und extremale quasikonforme Abbildungen
O. Teichmullers Beweis [9] seines Satzes uber die Struktur der extremalen quasikonformen Abbildungen geschlossener Riemannscher Flachen beruht auf einer Kontinuitatsmethode, wie sie schon seiner
On quasiconformal mappings
A fibre bundle description of Teichmüller theory
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