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
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
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
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
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+n>1.Let T(X)be the Teichmller space of X.By constructing a certain subset E of T(X),w...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+n>1.Let T(X)be the Teichmller space of X.By constructing a certain subset E of T(X),we show that the convex hull of E with respect to the Teichmller metric,the Carathodory metric and the Weil-Petersson metric is not in any thick part of the Teichmler space,respectively.This implies that convex hulls of thick part of Teichmller space with respect to these metrics are not always in thick part of Teichmller space,as well as the facts that thick part of Teichmller space is not always convex with respect to these metrics.展开更多
We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associ...We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.展开更多
For a Riemann surface X of conformally finite type （g, n）, let dT, dL and dpi （i = 1, 2） be the Teichmuller metric, the length spectrum metric and Thurston＇s pseudometrics on the Teichmutler space T（X）, respect...For a Riemann surface X of conformally finite type （g, n）, let dT, dL and dpi （i = 1, 2） be the Teichmuller metric, the length spectrum metric and Thurston＇s pseudometrics on the Teichmutler space T（X）, respectively. The authors get a description of the Teichmiiller distance in terms of the Jenkins-Strebel differential lengths of simple closed curves. Using this result, by relatively short arguments, some comparisons between dT and dL, dpi （i = 1, 2） on Tε（X） and T（X） are obtained, respectively. These comparisons improve a corresponding result of Li a little. As applications, the authors first get an alternative proof of the topological equivalence of dT to any one of dL, dp1 and dp2 on T（X）. Second, a new proof of the completeness of the length spectrum metric from the viewpoint of Finsler geometry is given. Third, a simple proof of the following result of Liu-Papadopoulos is given： a sequence goes to infinity in T（X） with respect to dT if and only if it goes to infinity with respect to dL （as well as dpi （i = 1, 2））.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
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 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 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 National Natural Science Foundation of China(Grant Nos.11271378,11071179 and 10871211)
文摘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+n>1.Let T(X)be the Teichmller space of X.By constructing a certain subset E of T(X),we show that the convex hull of E with respect to the Teichmller metric,the Carathodory metric and the Weil-Petersson metric is not in any thick part of the Teichmler space,respectively.This implies that convex hulls of thick part of Teichmller space with respect to these metrics are not always in thick part of Teichmller space,as well as the facts that thick part of Teichmller space is not always convex with respect to these metrics.
基金This work was supported by National Science Foundation of USA(Grant No.DMS-1747905)the Simons Foundation(Grant No.523341)+1 种基金Professional Sta Congress of the City University of New York Enhanced Award(Grant No.62777-0050)National Natural Science Foundation of China(Grant No.11571122).
文摘We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.
基金supported by the National Natural Science Foundation of China (No. 10871211)
文摘For a Riemann surface X of conformally finite type （g, n）, let dT, dL and dpi （i = 1, 2） be the Teichmuller metric, the length spectrum metric and Thurston＇s pseudometrics on the Teichmutler space T（X）, respectively. The authors get a description of the Teichmiiller distance in terms of the Jenkins-Strebel differential lengths of simple closed curves. Using this result, by relatively short arguments, some comparisons between dT and dL, dpi （i = 1, 2） on Tε（X） and T（X） are obtained, respectively. These comparisons improve a corresponding result of Li a little. As applications, the authors first get an alternative proof of the topological equivalence of dT to any one of dL, dp1 and dp2 on T（X）. Second, a new proof of the completeness of the length spectrum metric from the viewpoint of Finsler geometry is given. Third, a simple proof of the following result of Liu-Papadopoulos is given： a sequence goes to infinity in T（X） with respect to dT if and only if it goes to infinity with respect to dL （as well as dpi （i = 1, 2））.
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）.
