Teichmuller Distance is not C2+{varepsilon}

The Teichmüller space of a finite-type surface is considered.It is shown that Teichmüller distance is not C^{2 +} forany > 0. Furthermore, Teichmüller distance is not C^{2+ g} for any gauge function g with . 2000 Mathematics Subject Classification 30F60.     

On the Geometry of Hyperbolic Surfaces with a Conical Singularity

The geometry of compact hyperbolic surfaces with conical singularities isinvestigated. The Teichmüller space of a hyperbolic pair of pants with asingle singularity is studied. A finite number of real parameters whichdetermine the geometry of a hyperbolic torus of genus two with a singlesingularity, is given.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces

In this paper we construct a closed geodesic in any infinite-
dimensional Teichmüller space. The construction also leads to a proof of non-differentiability of the metric in infinite-dimensional Teichmüller spaces, which provides a negative answer to a problem of Goldberg.

     

Moduli spaces of convex projective structures on surfaces

We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for SL₃(R) defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149]. We investigate the cluster structure of this moduli space, and define its quantum version.     

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 equivalence of extremal Teichmüller mapping

Let T(Δ) and B(Δ) be the Teichmüller space and the infinitesimal Teichmüller space of the unit disk Δ respectively. In this paper, we show that [ν] _B(Δ) being an infinitesimal Strebel point does not imply that [ν] _T(Δ) is a Strebel point, vice versa. As an application of our results, problems proposed by Yao are solved. This work was supported by the National Natural Science Foundation of China (Grant No. 10571028)     

The Weil–Petersson Isometry Group

We prove that, for 3g–3+n>1 and (g,n)(1,2), the group of Weil–Petersson isometries of the Teichmüller space T _g,n coincides with the extended mapping class group.     

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     

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced by Rupflin and Topping (2012)  [11], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.     

Remarks on invariant metrics on Teichmüller space

We make some comparisons concerning the induced infinitesimal Kobayashi metric, the induced Siegel metric, the L² Bergman metric, the Teichmüller metric and the Weil-Petersson metric on the Teichmüller space of a compact Riemann surface of genus g?2. As a consequence, among others, we show that the moduli space has finite volume with respect to the L² Bergman metric. This answers a question raised by Nag in 1989.     

The cross-ratio distortion of integrably asymptotic affine homeomorphism of unit circle

Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary condition for a quasisymmetric homeomorphism to belong to QS *(S 1) from the aspect of cross-ratio distortion.     

渐近Teichmüller空间的不唯一性

设AT(△)是单位圆盘△上所有渐近Teichmüller等价类[[μ]]或[[f~μ]]构成的渐近Teichmüller空间.本文证明了对AT(△)内的任意渐近极值的f~μ,总存在一个[[f~μ]]内的渐近极值映射g~v,使边界伸缩商h~*(μ_(fog)~(-1)(g(z)))≠0.同时也获得了AT(△)在基点处的切空间上的类似结果.     

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.     

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.     

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.     

Configuration Spaces of Points on the Circle and Hyperbolic Dehn Fillings, II (Dedicated to Professor Mitsuyoshi Kato on his 60th birthday)/atl>

In our previous paper (Topology 38 (1999), 497–516), we discussed the hyperbolization of the configuration space of n ( 5) marked points with weights in the projective line up to projective transformations. A variation of the weights induces a deformation. It was shown that this correspondence of the set of the weights to the Teichmüller space when n = 5 and to the Dehn filling space when n = 6 is locally one-to-one near the equal weight. In this paper, we establish its global injectivity.     

Isomorphisms of fiber spaces over Teichmüller spaces

We will be mainly concerned with some important fiber spaces over Teichmüller spaces, including the Bers fiber space and Teichmüller curve, establishing an isomorphism theorem between “punctured” Teichmüller curves and determining the biholomorphic isomorphisms of these fiber spaces.     

On the measure with maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials

The Teichmüller flow g _t on the moduli space of Abelian differentials with zeros of given orders on a Riemann surface of a given genus is considered. This flow is known to preserve a finite absolutely continuous measure and is ergodic on every connected component ? of the moduli space. The main result of the paper is that µ/µ(?) is the unique measure with maximal entropy for the restriction of g _t to ?. The proof is based on the symbolic representation of g _t .     

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.