Quantization of the universal Teichmüller space

In the first part of the paper, we describe the Kähler geometry of the universal Teichmüller space, which can be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The universal Teichmüller space contains classical Teichmüller spaces T(G), where G is a Fuchsian group, as complex submanifolds. The quotient Diff₊(S ¹)/Möb(S ¹) of the diffeomorphism group of the unit circle modulo Möbius transformations can be considered as a “smooth” part of the universal Teichmüller space. In the second part we describe how to quantize Diff₊(S ¹)/Möb(S ¹) by embedding it in an infinite-dimensional Siegel disc. This quantization method does not apply to the whole universal Teichmüller space. However, this space can be quantized using the “quantized calculus” of A. Connes and D. Sullivan.     

On a relation between the universal Teichmüller space and the Grunsky operator

Being a Strebel point gives a sufficient condition for that the extremal Beltrami coefficient is uniquely determined in a Teichmiiller equivalence class. We consider how Strebel points are characterized. In this paper, we will give a new characterization of Strebel points in a certain subset of the universal Teichmfiller space by a property of the Grunsky operator.     

Hyperbolic metrics on universal Teichmüller space and extremal problems

The paper consists of two parts, both related to the complex geometry of the universal Teichmüller space. We reprove that all contractible invariant metrics on this space coincide and apply this important fact to solving the general extremal problems for univalent functions with quasiconformal extensions.     

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space II

In this paper, we study the asymptotic behavior of Teichmüller geodesic rays in the Gardiner–Masur compactification. We will observe that any Teichmüller geodesic ray converges in the Gardiner–Masur compactification. Therefore, we get a mapping from the space of projective measured foliations to the Gardiner–Masur boundary by assigning the limits of associated Teichmüller rays. We will show that this mapping is injective but is neither surjective nor continuous. We also discuss the set of points where this mapping is bicontinuous.     

A quantum Teichmüller space

We explicitly describe a noncommutative deformation of the *-algebra of functions on the Teichmüller space of Riemann surfaces with holes that is equivariant with respect to the action of the mapping class group. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 3, pp. 511–528, September, 1999.     

Teichmüller contraction in the Teichmüller space of a closed set in the sphere

We generalize the principle of Teichmüller contraction and deduce the Hamilton-Krushkaĺ condition for extremal quasiconformal mappings in the Teichmüller space of a closed set in the Riemann sphere.     

Bounded combinatorics and the Lipschitz metric on Teichmüller space

Considering the Teichmüller space of a surface equipped with Thurston’s Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are stable. Our main tool is to show that one can get a good estimate for the Lipschitz distance by considering the length ratio of finitely many curves.     

Some characterizations of the integrable Teichmüller space

In this paper, we prove that the Bers projection of the integrable Teichmller space is holomorphic. By using the Douady-Earle extension, we obtain some characterizations of the integrable Teichmller space as well as the p-integrable asymptotic affine homeomorphism.     

Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative \({\epsilon}\)-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and \({\epsilon}\). Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.     

Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative e{epsilon}-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and e{epsilon}. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.     

The Velling-Kirillov metric on the universal Teichmüller curve

We extend Velling's approach and prove that the second variation of the spherical areas of a family of domains defines a Hermitian metric on the universal Teichmüller curve, whose pull-back to Diff +(S ¹)/S ¹ coincides with the Kirillov metric. We call this Hermitian metric the Velling-Kirillov metric. We show that the vertical integration of the square of the symplectic form of the Velling-Kirillov metric on the universal Teichmüller curve is the symplectic form that defines the Weil-Petersson metric on the universal Teichmüller space. Restricted to a finite dimensional Teichmüller space, the vertical integration of the corresponding form on the Teichmüller curve is also the symplectic form that defines the Weil-Petersson metric on the Teichmüller space.     

Iterated grafting and holonomy lifts of Teichmüller space

Let X be a closed hyperbolic surface and λ, η be weighted geodesic multicurves which are short on X. We show that the iterated grafting along λ and η is close in the Teichmüller metric to grafting along a single multicurve which can be given explicitly in terms of λ and η. Using this result, we study the holonomy lifts gr _λ ρ _X,λ of Teichmüller geodesics ρ _X,λ for integral laminations λ and show that all of them have bounded Teichmüller distance to the geodesic ρ _X,λ. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences gr _nλ X and show that they converge geometrically to a punctured surface.     

Stability of quasi-geodesics in Teichmüller space

Let S be a surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2. We show that a Teichmüller quasi-geodesic in the thick part of Teichmüller space for S is contained in a bounded neighborhood of a geodesic if and only if it induces a quasi-geodesic in the curve graph.     

Dynamics on Teichmüller spaces and self-covering of Riemann surfaces

A non-injective holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmüller space. We investigate the dynamics of such self-embeddings by applying our structure theorem of self-covering of Riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the Teichmüller space. We also extend our observation to quasiregular self-covers of Riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self-cover.     

Stars at infinity in Teichmüller space

Geometriae Dedicata - We investigate a metric structure on the Thurston boundary of  Teichmüller space. To do this, we develop tools in sup metrics and apply Minsky’s...     

Local rigidity of the Teichmüller space with the Thurston metric

We show that every R-linear surjective isometry between the cotangent spaces to the Teichmüller space equipped with the Thurston norm is induced by some isometry between the underlying hyperbolic surfaces.This is an analogue of Royden’s theorem concerning the Teichmüller norm.     

Thurston’s pullback map on the augmented Teichmüller space and applications

Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σ _f of a finite-dimensional Teichmüller space. We prove that this map extends continuously to the augmented Teichmüller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston’s pullback map near invariant strata of the boundary of the augmented Teichmüller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston’s pullback map. Our approach also yields new proofs of Thurston’s theorem and Pilgrim’s Canonical Obstruction theorem.     

Convergence of earthquake and horocycle paths to the boundary of Teichmüller space

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into the space of geodesic currents, we prove that a horocycle path in Teichmüller space, which is induced by a quadratic differential whose vertical measured foliation is unique ergodic, converges to the Gardiner-Masur boundary and to the Thurston boundary.