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 a...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.展开更多
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 o...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.展开更多
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...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.展开更多
Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius tran...Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.展开更多
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 Poincar...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.展开更多
A new kind of subspaces of the universal Teichmüller space is introduced. Some characterizations of the subspaces are given in terms of univalent functions, Beltrami coefficients and quasisymmetric homeomorphisms...A new kind of subspaces of the universal Teichmüller space is introduced. Some characterizations of the subspaces are given in terms of univalent functions, Beltrami coefficients and quasisymmetric homeomorphisms of the boundary of the unit disc.展开更多
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...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.展开更多
This paper focuses on Teichmüller curves in the space of two-genus double covers of flat tori,identifying all of them, counting them with respect to their triangular areas, formulating the numbers of their cusps,...This paper focuses on Teichmüller curves in the space of two-genus double covers of flat tori,identifying all of them, counting them with respect to their triangular areas, formulating the numbers of their cusps, and characterizing the ones without a simple cusp. Some applications are also discussed.展开更多
The purpose of this paper is to give a relatively elementary and direct proof of the Delta Inequality, which plays a very important role in the study of the extremal problem of quasiconformal mappings.
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 ray...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.展开更多
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 proj...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.展开更多
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.
Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the pu...Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod$ under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mode.展开更多
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.
Let T0（Δ） be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ（Δ） be the unit ball of the space Q（Δ） of all integrable holomorphic quad...Let T0（Δ） be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ（Δ） be the unit ball of the space Q（Δ） of all integrable holomorphic quadratic differentials on the unit disk Δ and Q0（Δ） be defined as Q0（Δ） = {？ ∈ SQ（Δ） ： there exists a k ∈（0, 1） such that [kˉ？ ｜？｜] ∈ T0（Δ）}. In this paper, we show that Q0（Δ） is dense in SQ（Δ）.展开更多
A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 0(h) (indicated by the change in module of the quadrilaterals with vertices o...A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 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 forK 0 (m) (h) =K 1(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK 0 (m) (h)<K 1(h), wherek 1(h) is the maximal dilatation ofh.展开更多
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 Teichmülle...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 .展开更多
文摘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.
基金supported by the National Natural Science Foundation of China(11371045)
文摘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.
基金supported by National Natural Science Foundation of China (Grant No.11901241).
文摘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.
基金supported by the National Science Foundationsupported by a collaboration grant from the Simons Foundation(Grant No.523341)PSC-CUNY awards and a grant from NSFC(Grant No.11571122)。
文摘Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.
文摘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.
文摘A new kind of subspaces of the universal Teichmüller space is introduced. Some characterizations of the subspaces are given in terms of univalent functions, Beltrami coefficients and quasisymmetric homeomorphisms of the boundary of the unit disc.
基金supported by the National Science Foundation of USA (Grant No. DMS1747905)collaboration grant from the Simons Foundation (Grant No. 523341)+1 种基金the Professional Staff Congress of the City University of New York Award (Grant No. PSC-CUNY 66806-00 44)National Natural Science Foundation of China (Grant No. 11571122)
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 11401167).supported by National Natural Science Foundation of China (Grant No. 11371035).supported by National Natural Science Foundation of China (Grant Nos. 11701039 and 11371035)Research and Innovation Program of Beijing University of Posts and Telecommunications for Youth (Grant No. 2017RC18)
文摘This paper focuses on Teichmüller curves in the space of two-genus double covers of flat tori,identifying all of them, counting them with respect to their triangular areas, formulating the numbers of their cusps, and characterizing the ones without a simple cusp. Some applications are also discussed.
基金supported by the National Natural Science Foundation of China(10971008 and 11371045)
文摘The purpose of this paper is to give a relatively elementary and direct proof of the Delta Inequality, which plays a very important role in the study of the extremal problem of quasiconformal mappings.
基金Supported by National Natural Science Foundation of China(Grant Nos.11871085,11371045)。
文摘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.
基金supported by the Program for New Century Excellent Talents in University(Grant No.06-0504)National Natural Science Foundation of China (Grant No.10771153)
文摘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.
文摘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.
文摘Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod$ under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mode.
文摘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.
基金Supported by National Natural Science Foundation of China(Grant No.11371035)
文摘Let T0（Δ） be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ（Δ） be the unit ball of the space Q（Δ） of all integrable holomorphic quadratic differentials on the unit disk Δ and Q0（Δ） be defined as Q0（Δ） = {？ ∈ SQ（Δ） ： there exists a k ∈（0, 1） such that [kˉ？ ｜？｜] ∈ T0（Δ）}. In this paper, we show that Q0（Δ） is dense in SQ（Δ）.
文摘A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 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 forK 0 (m) (h) =K 1(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK 0 (m) (h)<K 1(h), wherek 1(h) is the maximal dilatation ofh.
文摘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 .
