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
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
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
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
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
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
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.
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
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
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
LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS 1=?Δ. Denote byV h [z 0] the variability set ofz 0 with respect toh∈T. The following ...LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS 1=?Δ. Denote byV h [z 0] the variability set ofz 0 with respect toh∈T. The following is proved: Leth 0 be a point ofT. Suppose thatμ 0 is an arbitrarily given extremal Beltrami differential ofh 0 andf 0: μ→μ is a quasiconformal mapping with the Beltrami coefficientμ 0 andf 01s=h 0. Then there are a sequenceh n of points inT and a sequencew n of points in Δ withh n ∈(Δ?V h [z 0]) andw n →f 0(z 0) andh n →h 0 andn∞ such that the point shift differentials determined byh n asw n form a Hamilton sequence ofμ 0.展开更多
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
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.
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.
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).
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.
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.
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.
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.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19531060)the Doctoral Education Program Foundation of China
文摘LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS 1=?Δ. Denote byV h [z 0] the variability set ofz 0 with respect toh∈T. The following is proved: Leth 0 be a point ofT. Suppose thatμ 0 is an arbitrarily given extremal Beltrami differential ofh 0 andf 0: μ→μ is a quasiconformal mapping with the Beltrami coefficientμ 0 andf 01s=h 0. Then there are a sequenceh n of points inT and a sequencew n of points in Δ withh n ∈(Δ?V h [z 0]) andw n →f 0(z 0) andh n →h 0 andn∞ such that the point shift differentials determined byh n asw n form a Hamilton sequence ofμ 0.
