THE REAL HYPER-ELLIPTIC SUBSPACES OF TEICHMüLLER SPACE AND MODULI SPACE

Although the existence and uniqueness of Strebel differentials are proved by Jenkins and Strebel, the specific constructions of Strebel differentials are difficult. Two special kinds of special Strebel differentials are constructed in [5, 6]. The Strebel rays [3] and the eventually distance minimizing rays [9] are important in Teichmüller spaces and moduli spaces, respectively. Motivated by the study of [3, 9], two special kinds of Strebel rays in the real hyper-elliptic subspace of Teichmüller space and two special kinds of EDM rays in the real hyper-elliptic subspace of moduli space are studied in this article.

万有Teichmller空间与Loewner链

提要通过采用构造Loewner链的方法,得到了单叶函数的充分条件,同时,结合万有Teichm(u|¨)ller空间理论,利用Loewner链构造了单叶函数的拟共形扩张表达式,并且得到了一些拟圆区域的单叶性内径的下界估计不等式.

THE EVENTUALLY DISTANCE MINIMIZING RAYS IN MODULI SPACES

The eventually distance minimizing ray（EDM ray） in moduli spaces of the Riemann surfaces of analytic finite type with 3 g ＋ n-3 〉 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmu¨ller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.

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.

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 Geometry of Infinite Dimensional Teichmüller Spaces

We prove that in any infinite dimensional Teichmüller space,there exists a minimal geodesic lying in two distinct geodesic disks.

Isometric Embeddings of Subsets of Boundaries of Teichmüller Spaces of Compact Hyperbolic Riemann Surfaces

It is known that every finitely unbranched holomorphic covering π:S→S of a compact Riemann surface S with genus g≥2 induces an isometric embedding Φπ:Teich(S)→Teich(S).By the mutual relations between Strebel rays in Teich(S)and their embeddings in Teich(S),we show that the 1 st-strata space of the augmented Teichmüller space Teich(S)can be embedded in the augmented Teichmüller space Teich(S)isometrically.Furthermore,we show that Φπ induces an isometric embedding from the set Teich(S)B(∞)consisting of Busemann points in the horofunction boundary of Teich(S)into Teich(S)B(∞)with the detour metric.

Teichmüller spaces for pointed Fuchsian groups

Let T(G) be the Teichmüller space of a Fuchsian group G and T(G) be the pointed Teichmüller space of a corresponding pointed Fuchsian group G.We will discuss the existence of holomorphic sections of the projection from the space M(G) of Beltrami coefficients for G to T(G) and of that from T(G) to T(G) as well.We will also study the biholomorphic isomorphisms between two pointed Teichmüller spaces.

A function model for the Teichmüller space of a closed hyperbolic Riemann surface

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface.Then we introduce a new metric on the Teichmüller space by using the maximum norm on the function space.We prove that the identity map from the Teichmüller space equipped with the Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Moreover, we prove that the inverse of the identity, i.e., the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the Teichmüller metric, is continuous(but not uniformly). Therefore, the topology induced by the new metric is the same as the topology induced by the Teichmüller metric on the Teichmüller space.Finally, we give a remark about the pressure metric on the function model and the Weil-Petersson metric on the Teichmüller space.

Geodesic discs in Teichmüller space

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

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.

Koebe Problems and Teichmiiller 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.

A problem in extremal quasiconformal extensions

A constant K (m) 0(h) is introduced for every quasisymmetric mapping h of the unit circle and every integer m ≥4 which contains the constant K 0(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established for K (m) 0(h)=K 1(h) . It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property that K (m) 0(h)<K 1(h) , where K 1(h) is the maximal dilatation of h .

On the boundary value problem for harmonic maps of the Poincaré disc

The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfyingwhere ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with normsIt is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .

On Strebel points

Any Beltrami coefficient μ on a hyperbolic Riemann surface X of infinite type represents a point [μ]T in the Teichmüller space T (X) and a point [μ]B in the tangent space of T (X) at the base point as well. The paper deals with the problem of determining whether that [μ]T is a Strebel point is equivalent to that [μ]B is an infinitesimal Strebel point.

Non-Strebel points and variability sets

This paper studies the subset of the non-Strebel points in the universal Teichmüller space T. Let z0 ∈△ be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curveγ . [0, 1] → T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈(0, 1], the variability set Vγ(t)[zo] of γ(t) has non-empty interior, i.e. Vγ(t) [z0] ≠φ.

Local boundary dilatation of quasiconformal maps

By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary the same as the usual ones in the case of Jordan domains. Then we show that the answer to the problem of F. Gardiner and N. Lakic is affirmative according to this new definition. Comparing with the usual definitions, the problem of F. Gardiner and N. Lakic is partly answered and the results of G. Cui and Y. Qi is generalized.

On the Compactness of Grunsky Differential Operators

Grunsky operators play an important role in classical geometric function theory and in the study of Teichmüller spaces.The Grunsky map is known to be holomorphic on the universal Teichmüller space.In this paper the authors deal with the compactness of a Grunsky differential operator.They will give upper and lower estimates of the essential norm of a Grunsky differential operator and discuss when a Grunsky differential operator is a p-Schatten class operator.

Strebel differentials and Hamilton sequences

Let T(S) be a Teichmaller space of a hyperbolic Riemann surface S, viewed as a set of Teichmaller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any ex-tremal Beltrami differential μo at a given point T of T( S), there is a Hamilton sequence forμo 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.