On the equivalence of extremal Teichmller mapping

Let T(△) and B(△) be the Teichmuller space and the infinitesimal Teichmuller space of the unit disk △ respectively. In this paper, we show that [ν]B(△) being an infinitesimal Strebel point does not imply that [ν]T(△) is a Strebel point, vice versa. As an application of our results, problems proposed by Yao are solved.     

Is there always an extremal Teichmüller mapping?

Given a quasisymmetric self-homeomorphismh of the unit circleS ¹, letQ(h) be the set of all quasiconformal mappings with the boundary correspondenceh. In [1], it was shown that there existsh for which no extremal extension inQ(h) as a Teichmüller mapping is possible. This disproved some conjectures of long standing. In the example constructed there, the boundary correspondence has a single extremal quasiconformal extension. We show that even when there are infinitely many extremal extensions of the boundary values, it may still happen that none of the extensions is a Teichmüller mapping. An infinitesimal version of this result is also obtained. The research was supported by the China Postdoctoral Science Foundation and by the National Natural Science Foundation of China (Grant No. 10401036).     

On Bloch functions and the Contraction of Teichmüller metrics

In this note, we consider the properties of Bloch functions derived by Beltrami coefficient. A sufficient condition for extremal quasiconformal mapping with nonexistence of degenerating sequence is obtained. As a result, we consider the contraction or preserved of Teichmüller metrics for the related Beltrami lines under the projection mapping π     

On periodic Teichmüller disks of modular transformations

We study the periodic Teichmüller disks of modular transformations. Especially, we prove that a parabolic modular transformation has either no periodic Teichmullüller disk or infinitely many periodic Teichmüller disks which can be chosen to cover infinitely many arithmetic Teichmüller curves in the Riemann moduli space M _g . Some related topics are also discussed.     

On angles in Teichmüller spaces

We discuss the existence of the angle between two curves in Teichmüller spaces and show that, in any infinite dimensional Teichmüller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length while its three angles are equal to any given three possibly different numbers from 0 to $\pi $ . This implies that the sum of three angles of a geodesic triangle may be equal to any given number from 0 to $3\pi $ in an infinite dimensional Teichmüller space.     

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space

The aim of this paper is to develop the theory of a compactification of Teichmüller space given by F. Gardiner and H. Masur, which we call the Gardiner–Masur compactification of the Teichmüller space. We first develop the general theory of the Gardiner–Masur compactification. Secondly, we will investigate the asymptotic behaviors of Teichmüller geodesic rays under the Gardiner–Masur embedding. In particular, we will observe that the projective class of a rational measured foliation G can not be an accumulation point of every Teichmüller geodesic ray under the Gardiner–Masur embedding, when the support of G consists of at least two simple closed curves. Dedicated to Professor Yoichi Imayoshi on the occasion of his 60th birthday.     

On the quantum Teichmüller invariants of fibred cusped 3-manifolds

This paper defines the pressure metric on the Moduli space of Margulis spacetimes without cusps and shows that it is positive definite on the constant entropy sections. It also demonstrates an identity regarding the variation of the cross-ratios.     

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.     

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.     

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Γ such that ℍ/Γ is a hyperbolic Riemann surface, the Teichmüller curve V(Γ) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Γ) onto V(Γ) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmüller curves is deduced, which generalizes a classical result that the Teichmüller curve V(Γ) depends only on the type of Γ and not on the orders of the elliptic elements of Γ when ℍ/Γ is a compact hyperbolic Riemann surface.     

Fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements

We determine the biholomorphic fiber preserving isomorphisms of fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements. We show that except in some special cases a biholomorphic fiber preserving isomorphism between two Bers fiber spaces is always an allowable mapping. We find that the situation is different for Teichmüller curves, showing that in general there are some other biholomorphic fiber preserving isomorphisms between Teichmüller curves besides the allowable mappings. Research supported by the National Natural Science Foundation of China.     

Strebel differentials and Hamilton sequences

Let T(S) be a Teichmüller space of a hyperbolic Riemann surface S, viewed as a set of Teichmüller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any extremal Beltrami differential μ₀ at a given point τ of T(S), there is a Hamilton sequence for μ₀ formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.     

Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces

A quasisymmetric homeomorphism of the unit circle S¹ is called integrably asymptotic affine if it admits a quasiconformal extension into the unit disk so that its complex dilatation is square integrable in the Poincaré metric on the unit disk. Let QS_* (s¹) be the space of such maps. Here we give some characterizations and properties of maps in QS_* (S¹). We also show that QS_* (S₁)/M?b (S¹) is the completion of Diff(S¹)/M?b(S₁) in the Weil-Petersson metric.     

Geodesic discs in teichmüller space

LetT(S) be the Teichmüller space of a Riemann surfaceS. By definition, a geodesic disc inT(S) is the image of an isometric embedding of the Poincaré disc intoT(S). It is shown in this paper that for any non-Strebel pointτ ∈ T(S), there are infinitely many geodesic discs containing [0] and τ.     

Conjugate points at infinity in infinite dimensional teichmüller spaces

We show in this article that there exists conjugate points at infinity in infinite dimensional Teichmüller spaces.     

A new coordinate of teichmüller space

By using the theory of quadratic differentials, we give a new coordinate to the Teichmüller space as well as the trajectory structures of a special class of Jenkins-Strebel quadratic differentials.     

Koebe Problems and Teichmüller Theory

In this paper we study the deformation space of certain Kleinian groups. As a result, we give a new proof of the finite Koebe theorem on Riemann surfaces from a viewpoint of Teichmüller theory. This work is supported by Post–Doctoral Foundation of China