The model of the universal Teichmuller union of infinite disconnected components. between different components is 0, and the every other component is 2. space by the derivatives of logarithm is the In this paper, it i...The model of the universal Teichmuller union of infinite disconnected components. between different components is 0, and the every other component is 2. space by the derivatives of logarithm is the In this paper, it is proved that the distance distance from the center of a component to展开更多
We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties...We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore,the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.展开更多
In this paper,the inner radius of univalency of hyperbolic domains by pre-Schwarzian derivative is studied,and some general formulas for the lower bound of the inner radius are established. As their applications,the l...In this paper,the inner radius of univalency of hyperbolic domains by pre-Schwarzian derivative is studied,and some general formulas for the lower bound of the inner radius are established. As their applications,the lower bounds of inner radiuses for angular domains and strongly starlike domains are obtained.展开更多
In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain ...In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.展开更多
It is proved that for any Fuchsian group Г such that H/Г is a hyperbolic Riemann surface, the Teichmuller curve V(Г) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(...It is proved that for any Fuchsian group Г such that H/Г is a hyperbolic Riemann surface, the Teichmuller curve V(Г) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Г) onto V(Г) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmuller curves is deduced, which generalizes a classical result that the Teichmuller curve V(Г) depends only on the type of Г and not on the orders of the elliptic elements of Г when H/Г is a compact hyperbolic Riemann surface.展开更多
We will be mainly concerned with some important fiber spaces over Teichmuller spaces, including the Bers fiber space and Teichmuller curve, establishing an isomorphism theorem between 'punctured' Teichmuller c...We will be mainly concerned with some important fiber spaces over Teichmuller spaces, including the Bers fiber space and Teichmuller curve, establishing an isomorphism theorem between 'punctured' Teichmuller curves and determining the biholomorphic isomorphisms of these fiber spaces.展开更多
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...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.展开更多
Let B denote the set of those points p in an infinite dimensional Teichmiiller space such that a geodesic between the zero point and p is not unique. Some properties of the set B are given.
This paper is to discuss the problem whether a sphere in a Teichmuller space is strictly convex with respect to geodesics. It is shown that the answer to this problem is negative for any Teichmuller space of a Fuchsia...This paper is to discuss the problem whether a sphere in a Teichmuller space is strictly convex with respect to geodesics. It is shown that the answer to this problem is negative for any Teichmuller space of a Fuchsian group of the second kind. For the case where the Fuchsian group is of the first kind, the problem is still open.展开更多
Abstract The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmfiller space AT（D） are studied in this paper. It is proved that if # is asymptotically extremal in [[#]] with h （μ） ...Abstract The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmfiller space AT（D） are studied in this paper. It is proved that if # is asymptotically extremal in [[#]] with h （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）.展开更多
Any covering Y→X of a hyperbolic Riemann surface X determines an inclusion of Teichmuller spaces T(X)→T(Y). This map is shown to be an isometry for the Teichmuller metric iff the covering is amenable, and contractin...Any covering Y→X of a hyperbolic Riemann surface X determines an inclusion of Teichmuller spaces T(X)→T(Y). This map is shown to be an isometry for the Teichmuller metric iff the covering is amenable, and contracting iff for any [μ]εT(X), where is the Poincare series operator. Furthermore the inclusion is not a uniform contraction on T(X).展开更多
Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then...Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.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
The author shows the non-differentiability of the Teichmuller cometric for infinite-dimensional Teichmuller spaces. Let Y be a Riemann surface, the Teichm uller space of Y denoted by T(Y). For[X, f]∈T(Y), where [X. f...The author shows the non-differentiability of the Teichmuller cometric for infinite-dimensional Teichmuller spaces. Let Y be a Riemann surface, the Teichm uller space of Y denoted by T(Y). For[X, f]∈T(Y), where [X. f] is an equivalent class of marked Riemann surfaces (X. f),展开更多
Let S<sub>0</sub> be a compact Riemann surface with genus g,g】1. For any compact Riemannsurface S and any orientation preserved homeomorphism f:S<sub>0</sub>→S, we denote the pair （S,f） a m...Let S<sub>0</sub> be a compact Riemann surface with genus g,g】1. For any compact Riemannsurface S and any orientation preserved homeomorphism f:S<sub>0</sub>→S, we denote the pair （S,f） a marked Riemann surface. Two marked Riemann surfaces （S<sub>1</sub>,f<sub>1</sub>） and （S<sub>2</sub>,f<sub>2</sub>） are equiv-alent if there exists a conformal mapping φ: S<sub>1</sub>→S<sub>2</sub> which is homotopic to f<sub>2</sub> f<sub>1</sub><sup>-1</sup>. De-展开更多
The authors identify the function space which is the tangent space to the integrable Teichmfiller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations ...The authors identify the function space which is the tangent space to the integrable Teichmfiller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.展开更多
The authOrs define the scenery flow of the torus. The flow space is the union of all flat 2- dimensional tori of area 1 with a marked direction (or equivalently the union of all tori with a quadratic differential of n...The authOrs define the scenery flow of the torus. The flow space is the union of all flat 2- dimensional tori of area 1 with a marked direction (or equivalently the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant 3-dimensional subset on which it is the suspension flow of that map. The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of : stacking and rescaling periodic tilings of the plane; symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the S-adic system generated by two substitutions; zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity; the natural extension of two-dimensional continued fractions; induction on exchanges of three intervals; rescaling on pairs of transverse measure foliations on the torus, or the Teichmuller flow on the twice-punctured torus.展开更多
It is proved that, for any elementary torsion free Fuchsian group F, the natural projection from the Teichmiiller curve V（F） to the Teichmiiller space T（F） has no holomorphic section.
Consider the Teichmuller mapping f associated with in the unit disc D and the class of all quasiconformal mappings in D with the boundary values of f. For a special holomorphic function, the present paper gives the ne...Consider the Teichmuller mapping f associated with in the unit disc D and the class of all quasiconformal mappings in D with the boundary values of f. For a special holomorphic function, the present paper gives the necessary and sufficient condition on, such that f is uniquely extremal among the class. Further, for a general holomorphic function, the anthors suggest the models of the best possible growth conditions on, such that f is extremal or uniquely extremal among the class respectively.展开更多
In this paper, we prove that the Bers projection of the integrable Teichmller space is holomorphic. By using the Douady-Earle extension, we obtain some characterizations of the integrable Teichmller space as well as t...In this paper, we prove that the Bers projection of the integrable Teichmller space is holomorphic. By using the Douady-Earle extension, we obtain some characterizations of the integrable Teichmller space as well as the p-integrable asymptotic affine homeomorphism.展开更多
基金Project supported by the National Natural Science Foundation of China (No.10271029).
文摘The model of the universal Teichmuller union of infinite disconnected components. between different components is 0, and the every other component is 2. space by the derivatives of logarithm is the In this paper, it is proved that the distance distance from the center of a component to
文摘We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore,the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10571028).
文摘In this paper,the inner radius of univalency of hyperbolic domains by pre-Schwarzian derivative is studied,and some general formulas for the lower bound of the inner radius are established. As their applications,the lower bounds of inner radiuses for angular domains and strongly starlike domains are obtained.
基金Supported by China Postdoctoral Science Foundation funded project (No. 20080430571)Jiangxi Educa tional Bureau Foundation (No. G JJ08163)
文摘In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.
基金supported by the National Natural Science Foundation of China(Grant No.10231040).
文摘It is proved that for any Fuchsian group Г such that H/Г is a hyperbolic Riemann surface, the Teichmuller curve V(Г) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Г) onto V(Г) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmuller curves is deduced, which generalizes a classical result that the Teichmuller curve V(Г) depends only on the type of Г and not on the orders of the elliptic elements of Г when H/Г is a compact hyperbolic Riemann surface.
基金The authors would like to thank the referee for his many valuable suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 10231040).
文摘We will be mainly concerned with some important fiber spaces over Teichmuller spaces, including the Bers fiber space and Teichmuller curve, establishing an isomorphism theorem between 'punctured' Teichmuller curves and determining the biholomorphic isomorphisms of these fiber spaces.
基金supported by the National Natural Science Foundation of China (Grant No. 10571028)
文摘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.
文摘Let B denote the set of those points p in an infinite dimensional Teichmiiller space such that a geodesic between the zero point and p is not unique. Some properties of the set B are given.
基金Project supported by the National Natural Science Foundation of China anti DPF.
文摘This paper is to discuss the problem whether a sphere in a Teichmuller space is strictly convex with respect to geodesics. It is shown that the answer to this problem is negative for any Teichmuller space of a Fuchsian group of the second kind. For the case where the Fuchsian group is of the first kind, the problem is still open.
基金Supported by National Natural Science Foundation of China(Grant No.11371045)
文摘Abstract The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmfiller space AT（D） are studied in this paper. It is proved that if # is asymptotically extremal in [[#]] with h （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）.
文摘Any covering Y→X of a hyperbolic Riemann surface X determines an inclusion of Teichmuller spaces T(X)→T(Y). This map is shown to be an isometry for the Teichmuller metric iff the covering is amenable, and contracting iff for any [μ]εT(X), where is the Poincare series operator. Furthermore the inclusion is not a uniform contraction on T(X).
文摘Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.
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The author shows the non-differentiability of the Teichmuller cometric for infinite-dimensional Teichmuller spaces. Let Y be a Riemann surface, the Teichm uller space of Y denoted by T(Y). For[X, f]∈T(Y), where [X. f] is an equivalent class of marked Riemann surfaces (X. f),
文摘Let S<sub>0</sub> be a compact Riemann surface with genus g,g】1. For any compact Riemannsurface S and any orientation preserved homeomorphism f:S<sub>0</sub>→S, we denote the pair （S,f） a marked Riemann surface. Two marked Riemann surfaces （S<sub>1</sub>,f<sub>1</sub>） and （S<sub>2</sub>,f<sub>2</sub>） are equiv-alent if there exists a conformal mapping φ: S<sub>1</sub>→S<sub>2</sub> which is homotopic to f<sub>2</sub> f<sub>1</sub><sup>-1</sup>. De-
基金supported by the National Natural Science Foundation of China(Nos.11371268,11171080,11601100,11701459)the Jiangsu Provincial Natural Science Foundation of China(No.BK20141189)the Ph.D Research Startup Foundation of Guizhou Normal University(No.11904-05032130006)
文摘The authors identify the function space which is the tangent space to the integrable Teichmfiller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.
文摘The authOrs define the scenery flow of the torus. The flow space is the union of all flat 2- dimensional tori of area 1 with a marked direction (or equivalently the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant 3-dimensional subset on which it is the suspension flow of that map. The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of : stacking and rescaling periodic tilings of the plane; symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the S-adic system generated by two substitutions; zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity; the natural extension of two-dimensional continued fractions; induction on exchanges of three intervals; rescaling on pairs of transverse measure foliations on the torus, or the Teichmuller flow on the twice-punctured torus.
基金Supported by Program for New Century Excellent Talents in University (Grant No.NCET-06-0504)National Natural Science Foundation of China (Grant No.10771153)
文摘It is proved that, for any elementary torsion free Fuchsian group F, the natural projection from the Teichmiiller curve V（F） to the Teichmiiller space T（F） has no holomorphic section.
文摘Consider the Teichmuller mapping f associated with in the unit disc D and the class of all quasiconformal mappings in D with the boundary values of f. For a special holomorphic function, the present paper gives the necessary and sufficient condition on, such that f is uniquely extremal among the class. Further, for a general holomorphic function, the anthors suggest the models of the best possible growth conditions on, such that f is extremal or uniquely extremal among the class respectively.
基金supported by National Natural Science Foundation of China (Grant No.10831004)
文摘In this paper, we prove that the Bers projection of the integrable Teichmller space is holomorphic. By using the Douady-Earle extension, we obtain some characterizations of the integrable Teichmller space as well as the p-integrable asymptotic affine homeomorphism.
