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.     

Maximal surfaces and the universal Teichmüller space

We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdS _n+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS₃, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.     

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.     

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.     

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.     

A binary infinitesimal form of the Teichmüller metric

Let S be a hyperbolic Riemann surface of finite analytic type. We give a binary infinitesimal form of the Teichmüllermetric on the Teichmüller space T(S) of S by use of the fundamental Reich-Strebel inequalities. As an application, we reduce the Teichmüller disk problem to the infinitesimal setting. Moreover, we define the “angle” between geodesic rays in T(S) and give an explicit formula.     

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.     

Exponential mixing for the Teichmüller flow

We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the $SL(2,\mathbb{R})We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of H?lder observables. A geometric consequence is that the action in the moduli space has a spectral gap.     

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.     

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.     

The Grunsky function and Carathéodory metric of Teichmüller spaces

A new proof of the equality of all non-expanding invariant metrics on the universal Teichmüller space is given. The arguments extend to Teichmüller space of the punctured disc.     

Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space

Geometriae Dedicata - We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with...     

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.     

Teichmüller Spaces and Function Spaces

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The Yang–Mills flow near the boundary of Teichmüller space

We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of representations of a once punctured surface. This foliation universalizes over Teichmüller space and is equivariant with respect to the action of the mapping class group. It is shown how to extend the foliation as a singular foliation over the augmented boundary of Teichmüller space obtained by adding nodal Riemann surfaces. Continuity of this extension is the main result of the paper. Received May 18, 1998 / Revised August 30, 1999 / Published online July 20, 2000     

Teichmüller spaces and holomorphic motions

The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmüller space of a closed subset of the Reimann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the “Harmonic γ-Lemma” of Bers and Royden. We also study some other applications.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

A uniqueness property for the quantization of Teichmüller Spaces

Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space of a punctured surface S, and an exponential version of this construction was developed by Bonahon and Liu. The construction of crucially depends on certain coordinate change isomorphisms between the Chekhov–Fock algebras associated to different ideal triangulations of S. We show that these coordinate change isomorphisms are essentially unique, once we require them to satisfy a certain number of natural conditions.