Convex hull of set in thick part of Teichmller space

Let X be a non-elementary Riemann surface of type(g,n),where g is the number of genus and n is the number of punctures with 3g-3+n1.Let T(X)be the Teichmller space of X.By constructing a certain subset E of T(X),we show that the convex hull of E with respect to the Teichmller metric,the Carathodory metric and the Weil-Petersson metric is not in any thick part of the Teichmler space,respectively.This implies that convex hulls of thick part of Teichmller space with respect to these metrics are not always in thick part of Teichmller space,as well as the facts that thick part of Teichmller space is not always convex with respect to these metrics.     

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.     

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.     

Hausdorff dimension of quasi-cirles of polygonal mappings and its applications

We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one.Furthermore,we apply this result to the theory of extremal quasiconformal mappings.Let [μ] be a point in the universal Teichmller space such that the Hausdorff dimension of fμ(Δ) is bigger than one.We show that for every kn∈(0,1) and polygonal differentials ψn,n=1,2,...,the sequence {[kn ψn/|ψn|]} cannot converge to [μ] under the Teichmer metric.     

A Metric Space with Transfinite Asymptotic Dimension 2ω + 1*

The authors construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both 2ω + 1, where ω is the smallest infinite ordinal number. Therefore, an example of a metric space with asymptotic property C is obtained.     

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.     

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 Teichm¨uller 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 Teichm¨uller space and two special kinds of EDM rays in the real hyper-elliptic subspace of moduli space are studied in this article.     

On Nonuniqueness of Geodesics and Geodesic Disks in the Universal Asymptotic Teichmüller Space

The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmüller space AT(D) are studied in this paper. It is proved that if μ is asymptotically extremal in [[μ]] with h_ζ~*(μ) h~*(μ) for some point ζ∈ ?D, then there exist infinitely many geodesic segments joining [[0]]and [[μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [[μ]] in AT(D).     

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.     

Stable Teichmller mappings of type (0,4)

Let X = C\{0,1} and X = X\{}. We get a necessary and suficient condition on the position of  in X such that X has stable Teichmller mappings. Furthermore, we can formulate all these stable Teichmller mappings. The main result in this paper partially answers a question posed by Kra.     

A Sufficient Condition for Rigidity in Extremality of Teichmller Equivalence Classes by Schwarzian Derivative

The Strebel point is a Teichm ¨uller equivalence class in the Teichm ¨uller space that has a certain rigidity in the extremality of the maximal dilatation. In this paper,we give a sufficient condition in terms of the Schwarzian derivative for a Teichm ¨uller equivalence class of the universal Teichm ¨uller space under which the class is a Strebel point. As an application, we construct a Teichm ¨uller equivalence class that is a Strebel point and that is not an asymptotically conformal class.     

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.     

一些超空间的非双Lipschitz 齐次性

A metric space(X, d) is called bi-Lipschitz homogeneous if for any points x, y ∈ X,there exists a self-homeomorphism h of X such that both h and h-1are Lipschitz and h(x) = y.Let 2(X,d)denote the family of all non-empty compact subsets of metric space(X, d) with the Hausdorff metric. In 1985, Hohti proved that 2([0,1],d)is not bi-Lipschitz homogeneous, where d is the standard metric on [0, 1]. We extend this result in two aspects. One is that 2([0,1],e)is not bi-Lipschitz homogeneous for an admissible metric e satisfying some conditions. Another is that 2(X,d)is not bi-Lipschitz homogeneous if(X, d) has a nonempty open subspace which is isometric to an open subspace of m-dimensional Euclidean space Rm.     

Permanence of Metric Sparsification Property under Finite Decomposition Complexity

The notions of metric sparsification property and finite decomposition com- plexity are recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper, it is proved that a metric space X has finite decomposition complexity with respect to metric sparsification property if and only if X itself has metric sparsification property. As a consequence, the authors obtain an alterna- tive proof of a very recent result by Guentner, Tessera and Yu that all countable linear groups have the metric sparsification property and hence the operator norm localization property.     

ON THE UNIFORM CONTINUITY OF METRIC PROJECTIONS IN BANACH SPACES

Let M be a convex Chebyshev subset of a uniformly convex and uniformly smoothBanach space.It is proved that the metric projection P_M of X onto M is uniformly continu-ous on every bounded subset of X.Moreover,a global and explicit estimate on the modulus ofcontinuity of the metric projection is obtained.     

拟定曲率空间上的几个定理

The present paper is devoted to determining the metric g for an n-dimension-al (n≥4) Riemannian manifold (M, g) of quasi-constant curvature [1]. By the way, we have identified the space of quasi-constant curvature with the κ-special conformally flat space of K.Yano & B.Y.Chen [8]. Based upon the results so obtained, we have completely determined the canonical metric for such a space to admit the relevant field X as geodesic field, and the geometric structure for (M, g) to be a recurrent space of quasi-constant curvature. Also we have examined the validity of our results just obtained for a 3-dimensional conformally flat space of quasi-constant cvrvature. Besides, we have deduced some global properties of a complete manifold of quasi-constant curvature, which may be useful in applications.     

On the Equi-nuclearity of Roe Algebras of Metric Spaces

The authors define the equi-nuclearity of uniform Roe algebras of a family of metric spaces. For a discrete metric space X with bounded geometry which is covered by a family of subspaces {Xi}∞i=1, if {Cu*(Xi)}∞i=1 are equi-nuclear and under some proper gluing conditions, it is proved that Cu*(X) is nuclear. Furthermore, it is claimed that in general, the coarse Roe algebra C*(X) is not nuclear.     

SOME THEOREMS ON THE SPACES OF QUASI-CONSTANT CURVATURE

The present paper is devoted to determining the metric g for an n-dimensional (n≥4) Riemannian manifold (M, g) of quasi-constant curvature [1]. By the way, we have identified the space of quasi-constant curvature with the k-special conformally flat space of K. Yano & B. Y. Chen [8]. Based upon the results so obtained, we have completely determined the canonical metric for such a space to admit the relevant field X as geodesic field, and the geometric structure for (M, g) to be a recurrent space of quasi-constant curvature. Also we have examined the validity of our results just obtained for a 3-dimensional conformally flat space of quasi-constant cvrvature. Besides, we have deduced some global properties for a complete manifold of quasi-constant curvature, which may be useful in applications.     

Besov Functions and Tangent Space to the Integrable Teichmüller Space

The authors identify the function space which is the tangent space to the integrable Teichm¨uller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.