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# Canonical Metrics on the Moduli Space of Riemann Surfaces II

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*in*Journal of differential geometry 69(1) · October 2004

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DOI: 10.4310/jdg/1121540343 · Source: arXiv

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In this paper we continue our study on the canonical metrics on the Teichm\"uller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carath\'eodory metric to the K\"ahler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the perturbed Ricci metric has negative upper and lower bounds, and it also has bounded geometry. Then we study in detail the boundary behaviors of the K\"ahler-Einstein metric and prove that it has bounded geometry, and all of the covariant derivatives of its curvature are uniformly bounded on the Teichm\"uller space. As an application of our detailed understanding of these metrics, we prove that the logarithmic cotangent bundle of the moduli space is stable in the sense of Mumford.

- ... Sun and S.-T. Yau [84, 85, 86, 87] showed the goodness of the Hermitian metrics on the logarithmic tangent bundle on M g which are induced by the Ricci and the perturbed Ricci metrics on M g . They also showed that the Ricci metric on M g extends naturally to the divisor D g := M DM g \ M g and coincides with the Ricci metric on each component of D g . ...... They also showed that the Ricci metric on M g extends naturally to the divisor D g := M DM g \ M g and coincides with the Ricci metric on each component of D g . Liu, Sun and Yau [85] showed that the existence of Kähler-Einstein metric on M g is related to the stability of the logarithmic cotangent bundle over M DM g . Let E be a holomorphic vector bundle over a complex manifold X of dimension n. ...... Liu, Sun and Yau [85] proved the following. Theorem B.1. ...Article
- Mar 2017

In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the Andr{\'e}-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms, Siegel-Jacobi spaces, stable Jacobi forms and the Schottky problem. - ... More precisely, we want to understand the relationships among all of the known canonical complete metrics introduced in history on the moduli and the Teichmüller spaces, and more importantly to introduce new complete Kähler metrics with good curvature properties: the Ricci metric and the perturbed Ricci metric. Through a detailed study we proved that these new metrics have very good curvature properties and very nice Poincaré-type asymptotic behaviors [10], [11]. In particular we proved that the perturbed Ricci metric has bounded negative Ricci and holomorphic sectional curvature and has bounded geometry. ...... We have obtained a series of results. In [10] and [11] we have proved that all of these known complete metrics are actually equivalent, as consequences we proved two old conjectures of Yau about the equivalence between the Kähler-Einstein metric and the Teichmüller metric and also its equivalence with the Bergman metric. In both [24] and [14] which were both written in early 80s, Yau raised various questions about the Kähler-Einstein metric on the Teichmüller space. ...... It is based on the lecture delivered by the first author in the First International Conference of Several Complex Variables held in the Capital Normal University in August 23-28, 2004. All of the main results mentioned here are contained in [10] and [11] which the interested reader may read for details. They have been circulated for a while. ...Article
- Dec 2005
- SCI CHINA SER A

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kähler metrics were introduced on the moduli space and Teichmüller 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 Kähler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincaré type growth. Furthermore, the Kähler-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. - ... Keywords: squeezing function, extremal map, holomorphic homogeneous regular domain. 319 was introduced [ Liu et al. 2004;Liu et al. 2005]. Holomorphic homogeneous regular domains are generalizations of Teichmüller spaces, and they admit some nice geometric and analytic properties [ Liu et al. 2004; Liu et al. 2005;Yeung 2009]. ...... 319 was introduced [ Liu et al. 2004;Liu et al. 2005]. Holomorphic homogeneous regular domains are generalizations of Teichmüller spaces, and they admit some nice geometric and analytic properties [ Liu et al. 2004; Liu et al. 2005;Yeung 2009]. ...... Motivated by the above works, especially [ Liu et al. 2004; Liu et al. 2005], we introduce the notion of squeezing functions defined on general bounded domains as follows: ...This paper introduces the notion of squeezing functions on bounded domains and studies some of their properties. The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved. Holomorphic homogeneous regular domains introduced by Liu, Sun and Yau are exactly domains whose squeezing functions have positive lower bounds. Completeness of certain intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular domains are proved by alternative method. In the dimension one case, we get a neat description of boundary behavior of squeezing functions of finitely connected planar domains. This leads to necessary and sufficient conditions for a finitely connected planar domain to be a holomorphic homogeneous regular domain. Consequently, we can recover some important results in complex analysis. For annuli, we obtain some interesting properties of their squeezing functions. Finally, we present some examples of bounded domains whose squeezing functions can be given explicitly.
- ... For example, the Teichmüller metric is a complete Finsler metric. The McMullen metric, Ricci metric, and perturbed Ricci metric have bounded geometry [16, 17, 18] . The Weil- Petersson metric is Kähler [1] and incomplete [5, 26]. ...... There are also some other metrics on T g like the Bergman metric, Caratheodory metric, Kähler-Einstein metric , Kobayashi metric, and so on. In [16, 17], the authors showed that some metrics listed above are comparable. In this paper we focus on the Weil-Petersson case. ...... In [15] Liu-Sun-Yau also used Wolpert's curvature formula to show that Teich(S) has dual Nakano negative curvature, which says that the complex curvature operator on the dual tangent bundle is positive in some sense. For some other related problems one can refer to [3, 10, 11, 16, 17, 23, 28, 30]. Let X ∈ Teich(S). ...Fix a number g > 1, let S be a close surface of genus g, and let Teich(S) be the Teichmuoller space of S endowed with the Weil- Petersson metric. In this paper we show that the Riemannian sectional curvature operator of Teich(S) is non-positive definite. As an application we show that any twist harmonic map from rank-one hyperbolic spaces HQ,m = Sp(m, 1)/Sp(m) · Sp(1) or HO,2 = F-20 4 /SO(9) into Teich(S) is a constant.
- ... The Asymptotic Poincaré metric, Induced Bergman metric, Kähler-Einstein metric, McMullen metric, Ricci metric, and perturbed Ricci metric are complete and Kähler. In [10, 11, 13, 21], the authors showed that the metrics listed above except the Weil-Petersson metric are equivalent. It is shown that the perturbed Ricci metric [10, 11] has pinched negative Ricci curvature. ...... In [10, 11, 13, 21], the authors showed that the metrics listed above except the Weil-Petersson metric are equivalent. It is shown that the perturbed Ricci metric [10, 11] has pinched negative Ricci curvature. So does the scalar curvature of the perturbed Ricci metric. ...... As introduced before there is a list of canonical metrics which are equivalent (or quasi-isometric) to the Teichmüller metric ||·|| T (see [10, 11, 13, 21]). Our second aim is the following uniform upper bound on the infimum of the scalar curvature of any Hermitian metric on a given class. ...In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which is equivalent to the Teichm\"uller metric, on any finite cover of the moduli space is negative provided that the scalar curvature is bounded from below.
- ... In [7,8] and [11], the concept called holomorphic-homogeneous-regular and equivalently the uniformly-squeezing, respectively, for complex manifolds has been introduced. This concept was essential for estimation of several invariant metrics. ...... Furthermore, the squeezing constantˆσconstantˆ constantˆσ Ω for Ω is defined by ˆ σ Ω := inf p∈Ω σ Ω (p). Definition 1.1 (Liu-Sun-Yau [7,8]; Yeung [11]). A complex manifold Ω is called holomorphic homogeneous regular (HHR), or equivalently uniformly squeezing (USq), ifˆσifˆ ifˆσ Ω > 0. ...ArticleFull-text available
- Jun 2013

We describe the boundary behaviors of the squeezing functions for all bounded convex domains in $\mathbb{C}^n$ and bounded domains with a $C^2$ strongly convex boundary point. - ... These manifolds possess many important geometric properties (e.g. all classical metrics on them are equivalent) [19,20] and have also been studied by several authors (see, for example, [6,7,8,14,24]) in the case of complex domains. In particular, it has been shown in [24] that a holomorphic homogeneous regular bounded domain D in C n must be pseudoconvex and all strongly convex domains in C n are holomorphic homogeneous regular. ...... We now extend the concept of a finite dimensional HHR manifold introduced in [19,20] ...Preprint
- Aug 2018

We extend the concept of a finite dimensional {\it holomorphic homogeneous regular} (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing function of all HHR bounded symmetric domains, including the two exceptional domains. We also show that uniformly elliptic domains in Hilbert spaces are HHR. - ... Motivated by the method in [15], the second author in [40] showed that the Teich(S g ) has negative semi-definite Riemannian curvature operator . One can also see [9, 16, 17, 18, 29, 37, 39] for other aspects of the curvature of Teich(S g ). ...We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichm\"uller space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius. We also show that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to $-1$.
- ... The above comparison theorem [18] [19] [20] [21] are in this sense. Recently, Kefeng Liu, Xiaofeng Sun and Shing-Tung Yau study the equivalence between the classical metrics on Teichmüller spaces and moduli spaces[33] [34] [35]. They proved that on Teichmüller spaces and moduli spaces the four classical metrics ω B (D), ω C (D), ω K (D), ω EK (D) are equivalent. ...Article
- Jan 2006

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, we prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the Bergman metric is equivalent to the Einstein-K\"ahler metric. That means the Yau's conjecture is true on Cartan-Hartogs domain. - ... The research field of metric completions of Teichmüller spaces is vast: There are many other natural metrics on Teichmüller space, e.g., the Teichmüller metric, the Bergman metric, the Arakelov metric, the McMullen metric, the Caratheodory metric, the Kobayashi metric, the Velling-Kirillov metric, the Takhtajan-Zograf metric -and the list probably goes on. A recent two paper series discussing various knowledge about some of these metrics on Teichmüller space is [44] and [45], which also contain many references to previous work on this subject. ...Article
- Nov 2009

In this paper we study the deformation problem of pairs consisting of a Riemann surface and a holomorphic line bundle over that surface, and also sections thereof. We emphasize a constructive approach throughout and work and use covering space techniques. In particular, we also describe the limits of such degenerations as the boundary of Teichm\"uller space is approached, and review the construction of augmented Teichm\"uller space in great detail. Comment: 59 pages, 4 figures - ... Yin and Zhang proved the four classical invariant metrics-the Carathéodoary metric, the Kobayashi metric, the Bergman metric and the Kähler-Einstein metric are all equivalent when the domains are convex [21]. Inspired by Liu-Sun-Yau's work [11], Yin proposed the following open problem: whether Cartan-Hartogs domains are homogeneous regular [22]? In this subsection, we give an affirmative answer to this question. ...ArticleFull-text available
- Feb 2012

In this article we continue the study of properties of squeezing functions and geometry of bounded domains. The limit of squeezing functions of a sequence of bounded domains is studied. We give comparisons of intrinsic positive forms and metrics on bounded domains in terms of squeezing functions. To study the boundary behavior of squeezing functions, we introduce the notions of (intrinsic) ball pinching radius, and give boundary estimate of squeezing functions in terms of these datum. Finally, we use these results to study geometric and analytic properties of some interesting domains, including planar domains, Cartan-Hartogs domains, and a strongly pseudoconvex Reinhardt domain which is not convex. As a corollary, all Cartan-Hartogs domains are homogenous regular, i.e., their squeezing functions admit positive lower bounds. - ... This appendix refers mainly to work of C. Earle and A. Marden in [13]. One might also see the following works: [9], [37], [46], [48], [32], [33], and [54]. In [13] , C. Earle and A. Marden have an alternative approach to the construction of Q Γ which is quite different from ours: it is based on Kleinian groups and quasiconformal techniques. ...In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.
- ... In the past decade, comparisons of various intrinsic metrics on Teichmüller spaces were extensively studied (see e.g. [4][13][14][23]). The equivalence of certain intrinsic measures on Teichmüller spaces was proved in [18]. ...The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points. We also study the stability of squeezing functions on a sequence of bounded domains, and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new and simple proofs of several well known results about geometry of strongly pseudoconvex domains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems that ask for further study are proposed.
- ... There have been a lots of studies on the geometry of Weil–Petersson (WP) metric on moduli spaces of Riemann surfaces, especially regarding its curvature properties [1] [10] [16] [14] [11] [6] [9] [17] [18] [15] [7] [8] [3] [4]. In a pioneering work, Ahlfors [1] showed that the Ricci, holomorphic sectional and scalar curvatures of the WP metric are all negative. ...In [4], Z. Huang showed that in the thick part of the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$, the sectional curvature of the Weil--Petersson metric is bounded below by a constant depending on injectivity radius, but independent of the genus $g$. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichm\"uller space equipped with Hilbert structure induced by Weil--Petersson metric, we prove that its sectional curvature is bounded below by a universal constant. Comment: 12 pages
- ... The Weil-Petersson metric is not fully compatible with this compactification in the sense that the local asymptotic behaviour of g WP near these divisors is somewhat complicated: normal to each divisor it has cusp-like behavior, but at intersections of the divisors, these normal cusps do not interact. Our goal in this paper is to sharpen the work of Masur [13], Yamada [24] and Wolpert [21, 22], and in a slightly different direction, Liu-Sun-Yau [11, 12], each of whom provided successively finer estimates. This work also refines Wolpert's very recent paper [23], which proves a certain uniformity of derivatives for this metric. ...We consider the Riemann moduli space $\mathcal M_{\gamma}$ of conformal structures on a compact surface of genus $\gamma>1$ together with its Weil-Petersson metric $g_{\mathrm{WP}}$. Our main result is that $g_{\mathrm{WP}}$ admits a complete polyhomogeneous expansion in powers of the lengths of the short geodesics up to the singular divisors of the Deligne-Mumford compactification of $\mathcal M_{\gamma}$.
- ... By definition, a bounded domain is called homogenous regular if its squeezing function has positive lower bound. The notion of homogenous regular domains was introduced and studied in [22,23], and was called uniform squeezing domains and systematically studied in [28]. ...Preprint
- Oct 2018

We show that if a bounded domain $\Omega$ is exhausted by a bounded strictly pseudoconvex domain $D$ with $C^2$ boundary, then $\Omega$ is holomorphically equivalent to $D$ or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman's invariant has certain growth condition near the boundary. - ... If the domain Ω does not admit any bounded injective holomorphic functions, then the squeezing function is called degenerate. The concept of squeezing function goes back to work by Liu-Sun-Yau, see [12] (2004), [13] (2005) and S.-K-Yeung [17] (2009). More recently, Deng-Guan-Zhang, see [2] (2012) initiated a basic study of the squeezing function. ...We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.
- Article
- Nov 2014
- INT J MATH

As is well-known, the Weil-Petersson metric ωWP on the moduli space ℳg has negative Ricci curvature. Hence, its negative first Chern form defines the so-called Ricci metric ωτ. Their combination , C > 0, introduced by Liu-Sun-Yau, is called the perturbed Ricci metric. It is a complete Kähler metric with finite volume. Furthermore, it has bounded geometry. In this paper, we investigate the finiteness of this new metric from another point of view. More precisely, we will prove in the thick part of ℳg, the holomorphic bisectional curvature of is bounded by a constant depending only on the thick constant and C0 when C ≥ (3g - 3)C0, but not on the genus g. - Article
- May 2007
- J REINE ANGEW MATH

We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and K\"uhn, in their extension of the arithmetic intersection theory of Gillet and Soul\'e, aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of Bost-Gillet-Soul\'e, as well as the properties established by Faltings for heights of points attached to hermitian line bundles whose metrics have logarithmic singularities. We also discuss various geometric constructions where such pre-log-log hermitian ample line bundles naturally arise. - Article
- May 2007
- MATH ANN

The family hyperbolic metric for the plumbing variety $\{zw=t\}$ and the non holomorphic Eisenstein series $E(\zeta;2)$ are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil-Petersson metric and a local form of symplectic reduction. - We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the curve X, in terms of the holomorphic sectional curvature of A_g and the second Gaussian map. Finally we extend the Kaehler form of the Siegel metric as a closed current on the Deligne-Mumford compatification of M_g and we determine its cohomology class as a multiple of the first Chern class of the Hodge bundle.
- Article
- Apr 2004
- J DIFFER GEOM

We prove the equivalences of several classical complete metrics on the Teichm\"uller and the moduli spaces of Riemann surfaces. We use as bridge two new K\"ahler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete K\"ahler metric with bounded negative holomorphic sectional curvature and bounded bisectional and Ricci curvature. As consequences we prove that these two new metrics are equivalent to several famous classical metrics, which inlcude the Teichm\"uller metric, therefore the Kabayashi metric, the K\"ahler-Einstein metric and the McMullen metric. This also solves a conjecture of Yau in the early 80s. - Article
- Jan 2019
- J GEOM ANAL

We extend the concept of a finite dimensional holomorphic homogeneous regular (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing function of all HHR bounded symmetric domains, including the two exceptional domains. We also show that uniformly elliptic domains in Hilbert spaces are HHR. - We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmuller space. Comment: 26 pages, 6 figures; v2: added references
- Article
- Aug 2010
- J DIFFER GEOM

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm\"{u}ller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichm\"{u}ller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with the earlier negative sectional curvature results to establish a classification of asymptotic flats. Furthermore, tangent subspaces of more than half the dimension of Teichm\"{u}ller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds. - Article
- Mar 2011
- ASIAN J MATH

In this note, we answer a question of Mirzakhani on asymptotic behavior of the one-point volume polynomial of moduli spaces of curves. We also present some applications of Mirzakhani's asymptotic formulae of Weil-Petersson volumes. - The main purpose of the present paper is to introduce the notion of squeezing functions of bounded domains and study some properties of them. The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved. Holomorphic homogeneous regular domains are exactly domains whose squeezing functions have positive lower bounds. Completeness of certain intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular domains are proved by alternative method. In dimension one case, we get a neat description of boundary behavior of squeezing functions of finitely connected planar domains. This leads to a necessary and sufficient conditions for a finitely connected planar domain to be a holomorphic homogeneous regular domain. Consequently, we can recover some important results in complex analysis. For annuli, we obtain some interesting properties of their squeezing functions. We finally exhibit some examples of bounded domains whose squeezing functions can be given explicitly.
- Article
- Mar 2012
- T AM MATH SOC

Let $X = M \times E$ where $M$ is an $m$-dimensional K\"ahler manifold with negative first Chern class and $E$ is an $n$-dimensional complex torus. We obtain $C^\infty$ convergence of the normalized K\"ahler-Ricci flow on $X$ to a K\"ahler-Einstein metric on $M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture. - Article
- Mar 2010
- ADV MATH

We prove topological transitivity for the Weil–Petersson geodesic flow for real two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that combines the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil–Petersson geodesic flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite. - Article
- Apr 2012
- ADV MATH

An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. For suitable stable curve families, the direct image sheaf on the base is locally free and the sequence of direct images is exact. Recent work of Hubbard-Koch and a formal argument provide that the direct image sheaf is naturally identified with the moduli cotangent sheaf. The result generalizes the role of holomorphic quadratic differentials as cotangents for smooth curve families. Formulas are developed for the pairing of an infinitesimal opening of a node and a section of the direct image sheaf. Applications include an analytic description of the conormal sheaf for the locus of noded stable curves and a formula comparing infinitesimal openings of a node. The moduli action of the automorphism group of a stable curve is described. An example of plumbing an Abelian differential and the corresponding period variation is presented. - Article
- Nov 2016
- J GEOM ANAL

J. E. Fornaess has posed the question whether the boundary point of smoothly bounded pseudoconvex domain is strictly pseudoconvex, if the asymptotic limit of the squeezing function is 1. The purpose of this paper is to give an affirmative answer when the domain is in C^2 with smooth boundary of finite type in the sense of D'Angelo. - Article
- Feb 2007
- SCI CHINA SER A

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau’s Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kähler metric. That means that the Yau’s conjecture is true on Cartan-Hartogs domains. - Article
- Apr 2008
- SCI CHINA SER A

Complex manifolds are topological spaces that are covered by coordinate charts where the coordinate changes are given by holomorphic transformations. For example, Riemann surfaces are one dimensional complex manifolds. In order to understand complex manifolds, it is useful to introduce metrics that are compatible with the complex structure. In general, we should have a pair (M,dsM2) where dsM2 is the metric. The metric is said to be canonical if any biholomorphisms of the complex manifolds are automatically isometries. Such metrics can naturally be used to describe invariants of the complex structures of the manifold. The first important examples of such metrics were constructed by Poincaré for Riemann surfaces with genus greater than one. Note that the flat metric on the torus is not quite anonical unless we require the biholomorphisms to preserve the area. (In higher dimensions, this is the same as preserving the Kähler class.) Poincaré?s metrics are metrics with constant negative curvature. It requires a proof of the existence theorem for conformal deformation of metrics. It is a nonlinear differential equation and hence a difficult higher dimensional problem. Generalizations of Poincaré?s work took some time. Besides Riemannian metrics that are compatible with the complex structure, there are other types of metrics which we shall discuss in the following. - Article
- Apr 2008
- SCI CHINA SER A

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the L 2-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces. - Article
- Mar 2010

In this note we discuss various canonical metrics on complex manifolds. A generalization of the Bergman metric is proposed and the relations of metrics on moduli spaces are commented. At last, we review some existence theorems of solutions to the Strominger system. Keywordsintrinsic metrics-generalized Bergman metrics-Weil-Petersson metric-Strominger system MSC(2000)53C55 - Article
- Jan 2011
- J MATH ANAL APPL

We make some comparisons concerning the induced infinitesimal Kobayashi metric, the induced Siegel metric, the L2 Bergman metric, the Teichmüller metric and the Weil–Petersson metric on the Teichmüller space of a compact Riemann surface of genus g⩾2. As a consequence, among others, we show that the moduli space has finite volume with respect to the L2 Bergman metric. This answers a question raised by Nag in 1989. - Article
- Aug 2016
- JPN J MATH

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and Hirzebruch–Riemann–Roch, and some modern development of uniformization theorems, Kähler–Einstein metric and the theory of Donaldson–Uhlenbeck–Yau on Hermitian Yang–Mills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions. - Article
- Jun 2016
- MATH ANN

We construct a class of bounded domains, on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point. - Article
- Jun 2019
- J GEOM ANAL

In this survey paper, we give a review of some recent developments on holomorphic invariants of bounded domains, which include squeezing functions, Fridman’s invariants, p-Bergman kernels, and spaces of Lp integrable holomorphic functions. - Article
- Apr 2008
- PAC J MATH

Here we show that the Carathéodory, Eisenman-Kobayashi, and Kähler-Einstein volume forms are equivalent on Teichmüller space. - We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmüller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argument Wolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist.
- Article
- Dec 2013
- GEOMETRIAE DEDICATA

We prove a Gauss-Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the Gauss-Bonnet formula of Allendoerfer and Weil for Riemannian polyhedra. - Article
- Oct 2008
- ASTERISQUE

In this paper we describe some recent results on the geometry of the moduli space of Riemann surfaces. We surveyed new and classical metrics on the moduli spaces of hyperbolic Riemann surfaces and their geometric properties. We then discussed the Mumford goodness and generalized goodness of various metrics on the moduli spaces and their deformation invariance. By combining with the dual Nakano negativity of the Weil-Petersson metric we derive various consequences such that the infinitesimal rigidity, the Gauss-Bonnet theorem and the log Chern number computations. - Article
- Nov 2013

We study the equivalent problem of the classical metric on a class of nonhomogeneous Hartogs domain. Firstly, it is proved that the equivalence between the Bergman and the Einstein-Kähler metrics on these domains; Secondly, it is shown that the Bergman metric, the Carath'eodary metric, the Kobayashi metric and the Einstein-Kähler metric are equivalent, when parameters of the domain satisfy mσ + nτ < 1. - Article
- Nov 2014

We give estimates for the squeezing function on strictly pseudoconvex domains, and derive some sharp estimates for the Caratheodory, Sibony and Azukawa metric near their boundaries. - Article
- Aug 2019
- ACTA MATH SIN

We show that if a bounded domain Ω is exhausted by a bounded strictly pseudoconvex domain D with C² boundary, then Ω is holomorphically equivalent to D or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman’s invariant has certain growth condition near the boundary.

- Riemann surfaces, volume 71 of Graduate Texts in Mathematics
- Jan 1992

- H M Farkas
- I Kra

H. M. Farkas and I. Kra. Riemann surfaces, volume 71 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1992. - Book
- Jan 1998

In the three decades since the introduction of the Kobayashi distance, the subject of hyperbolic complex spaces and holomorphic mappings has grown to be a big industry. This book gives a comprehensive and systematic account on the Caratheodory and Kobayashi distances, hyperbolic complex spaces and holomorphic mappings with geometric methods. A very complete list of references should be useful for prospective researchers in this area. - Without Abstract
- Article
- Feb 1985
- T AM MATH SOC

It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., [BLM, H]). In this paper, the known list of such properties is extended to the "Tits-Alternative", a property of linear groups established by J. Tits [T]. In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group. In addition, in order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston's projective lamination spaces [Th1]. This theory generalizes results known for pseudo-Anosov mapping classes [FLP]. - Article
- Jul 1980
- B AM MATH SOC

Closed, oriented surfaces of genus g > 2 admit many hyperbolic (constant Gaussian curvature -1) metrics in contrast to Mostow's rigidity theorems in higher dimensions. Only special hyperbolic surfaces have non-trivial groups of isometries, but many different, non-isomorphic groups arise for different symmetric metrics. The group of isometries of a closed hyperbolic surface M2 is always finite and the only isometry isotopic to the identity is the identity itself. As a result, hyperbolic surfaces with non-trivial groups of isometries have been a primary source for the construction of finite subgroups of the group of isotopy classes of diffeomorphisms of M2, ?TDiff(M2). An old question, usually referred to as the Nielsen Realization Problem, is whether every such finite subgroup arises as a group of isometries of some hyperbolic surface. In this paper we answer the question in the affirmative. - Article
- Oct 1998
- J DIFFER GEOM

this paper proved the following result in his Ph.D. thesis [43] in 1990: - Article
- Nov 2005
- COMMUN MATH PHYS

In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers. - Article
- Apr 2004
- J DIFFER GEOM

We prove the equivalences of several classical complete metrics on the Teichm\"uller and the moduli spaces of Riemann surfaces. We use as bridge two new K\"ahler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete K\"ahler metric with bounded negative holomorphic sectional curvature and bounded bisectional and Ricci curvature. As consequences we prove that these two new metrics are equivalent to several famous classical metrics, which inlcude the Teichm\"uller metric, therefore the Kabayashi metric, the K\"ahler-Einstein metric and the McMullen metric. This also solves a conjecture of Yau in the early 80s.