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# Canonical Metrics on the Moduli Space of Riemann Surfaces I

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*in*Journal of differential geometry 10(2) · April 2004

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DOI: 10.4310/PAMQ.2014.v10.n2.a2 · Source: arXiv

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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.

- ... We first recall the deformation theory of Riemann surfaces as well as canonical metrics on the moduli spaces. Most of the results can be found in [2], [3], [7] and [1]. Let M g,k be the moduli space of Riemann surfaces of genus g with k punctures such that 2g − 2 + k > 0. We know there is a unique Kähler-Einstein metric on such a Riemann surface. ...... There are two kinds of local holomorphic coordinate on a collar or near a node. We first recall the rs-coordinate defined by Wolpert in [7]. In the node case, given a nodal surface X with a node p ∈ X, we let a, b be two punctures which are glued together to form p. In the collar case, given a closed surface X, we assume there is a closed geodesic γ ⊂ X such that its length l = l(γ) < c * where c * is the collar constant. ...... (Now we describe the pinching coordinate chart of M g near the divisor D [7]. Let X 0 be a nodal surface corresponding to a codimension m boundary point and let p 1 , . . . ...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. - ... These include the Teichmüller (Finsler) metric, the Bergman metric, the Kobayashi metric (which was shown by Royden [71, 72] to be equal to the Teichmüller metric), the Carathéodory metric, the Kähler-Einstein metric, the McMullen metric, the Ricci metric, and the Liu-Sun-Yau metric (a perturbed Ricci metric). It was shown in [58] that all of these are, in fact, quasi-isometric. 3 Looijenga [59] and Saper-Stern [77] proved that the L 2 -cohomology of Γ\D is canonically isomorphic to the (topological) intersection cohomology of Γ\D BB for what is called the middle perversity. ...... We wish to remind the reader that the mapping class groups, the topology of M g,n , and its Weil-Petersson metric have been studied extensively by many people, too numerous to list here. On the other hand, the complete Riemannian metrics on M g,n mentioned above have only recently been studied systematically in [58], where the aforementioned quasi-isometries were proved. Some related results had also been proved in [90]. ...... For any two points in T g,n (representing two marked Riemann surfaces of genus g with n punctures, S and S , there is a unique quasi-conformal mapping S → S that has least " distortion " K ≥ 1 and is compatible with the markings. Then the Teichmüller distance d(S, S ) is defined to be log K. Other well-known examples of complete metrics include: the Bergman metric , Kähler-Einstein metric, McMullen metric [64], Ricci metric (the negative of the Ricci curvature of the Weil-Petersson), the perturbed Ricci metric of Liu, Sun and Yau [58], and any Poincaré metric adapted to the boundary of M DM g ...ArticleFull-text available
- Nov 2010

Let M g,n be the moduli space of algebraic curves of genus g with n punctures, which is a noncompact orbifold. Let M DM g,n denote its Deligne-Mumford compactification. Then M g,n admits a class of canon-ical Riemannian and Finsler metrics. We probe the analogy between M g,n (resp., Teichmüller spaces) with these metrics and certain non-compact locally symmetric spaces (resp. symmetric spaces of noncom-pact type) with their natural metrics. In this chapter, we observe that for all 1 < p < ∞, the L p -cohomology of M g,n with respect to these Riemannian metrics that are complete can be identified with the (or-dinary) cohomology of M DM g,n , and hence the L p -cohomology is the same for different values of p. This suggests a "rank-one nature" of the moduli space M g,n from the point of view of L p -cohomology. On the other hand, the L p -cohomology of M g,n with respect to the in-complete Weil-Petersson metric is either the cohomology of M DM g,n or that of M g,n itself, depending on whether p ≤ 4 3 or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics. - ... [25] [6] [20] [19] [16] [1] [22]). Motivated by the works in [13], Yin proposed a problem that whether all Cartan-Hartogs domains are homogenous regular ([26]). ...... , Yin computed the automorphism groups and Bergman kernels of Cartan- Hartogs domains explicitly. Motivated by Liu-Sun-Yau's work [13], Yin proposed the following open problem: whether Cartan-Hartogs domains are homogeneous regular [26]? In this section, we give an affirmative answer to this question. ...... We also prove a weaker result for a sequence of decreasing domains. A homogenous regular domain (introduced in [13]) is a bounded domain whose squeezing function is bounded below by a positive constant. By the famous Bers embedding ([3]), Teichmüller spaces of compact Riemann surfaces are homogenous regular. ...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.
- ... In fact we will present proofs of the goodness of the metrics induced by the Weil-Petersson metric, as well as the Ricci and perturbed Ricci metrics on the logarithmic cotangent bundle over the compactified moduli space of Riemann surfaces (the DM moduli spaces). These works depend on our very accurate estimates of the asymptotic of the curvature and connection forms of these metrics in [6] and [7] together with the estimates of derivatives of the hyperbolic metric on Riemann surfaces. The computations and proofs are quite involved and very subtle. ...... Now we briefly describe the organization of this note. In Section 2 we recall some known works on the geometry of moduli spaces which include the degeneration of Riemann surfaces and hyperbolic metrics, the Ricci, perturbed Ricci and Kähler-Einstein metrics as well as their curvature properties, the asymptotic of Weil-Petersson metric, the Ricci and perturbed Ricci metrics as established in [6] and [7]. We also review Mumford's definition of good metrics. ...... In this section we review the necessary backgrounds and setup our notations . Most of the results can be found in [6], [7], [20] and [11]. Let M g,k be the moduli space of Riemann surfaces of genus g with k punctures such that 2g − 2 + k > 0. We know there is a unique hyperbolic metric on such a Riemann surface. ...
- ... Keywords: squeezing function, extremal map, holomorphic homogeneous regular domain. 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]. ...... 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: Definition 1.1. ...... 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: Definition 1.1. Let D be a bounded domain in n . ...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.
- ... Tromba [30] and Wolpert [34] found a formula for the Weil-Petersson curvature tensor, which has been applied to study a variety of curvature properties of Teich(S g ) over the past several decades. (See also [13, 16, 28, 31] for alternative proofs of the curvature formula.) In their papers, they deduced from their formula that the holomorphic sectional curvatures are bounded above by a negative number which only depends on the genus of the surface, confirming a conjecture of Royden. ...... The following property is well-known to experts – see for example Lemma 4.3 in [16]. For completeness, we include the proof. ...... 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 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.
- ... [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. ...... 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), 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. - ... Note that 0 < s Ω (z) ≤ 1 for any z ∈ Ω. This concept first appeared in [13, 14] in the context concerning the holomorphic homogeneous regular manifolds. But the name squeezing function comes from [17]; this concept is closely related to the concept of " bounded geometry " by Cheng and Yau [2]. ...... Note that 0 < s Ω (z) ≤ 1 for any z ∈ Ω. This concept first appeared in[13,14]in the context concerning the holomorphic homogeneous regular manifolds. But the name squeezing function comes from[5]; this concept is closely related to the concept of bounded geometry by Cheng and Yau[2], as one sees from[17]. ...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. - ... There are many other related works in a similar vein including, just to name a few, Albin- Rochon [4], Brüning-Seeley [6], Gil-Krainer-Mendoza [11], Lesch [22], Schultze [34], and Grieser's notes on parametrix constructions for heat kernels [14]. For analysis of moduli space, to give just a sample recent work, we refer the reader to the papers of Liu-Sun-Yao, for example [24, 23]. As described above we work on a smooth manifold with boundary M whose boundary is the total space of a fiber bundle with base Y and typical fiber Z. (Again, we assume that Y, Z are both compact and without boundary.) ...ArticleFull-text available
- Sep 2015

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of $L^2$-harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of $L^2$-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near $t = 0$. - ... Explicitly, one may take −β −ε , where β is the function in (4.6) and 0 < ε < 1. The existence of a bounded plurisubharmonic function has important implications for the equivalence of invariant metrics on Teichmüller space (see [24, 118, 210]). ...ArticleFull-text available
- May 2006

This is a survey article to appear in the "Handbook on Teichmueller Theory". - ... In [15] and [16], Liu, Sun and Yau showed that the Teichmüller metric, the Carathéodory metric, the Bergman metric, the McMullen metric [18], the Kähler–Einstein metric, the Ricci metric and the perturbed Ricci metric are quasi-isometric to each other on T (X). Similar results are obtained by Yeung [27]. ...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. - ... 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.
- ... moduli space which are quasi-isometric to the Teichmüller metric (see e.g. [26]). It is conceivable that a Gauss- Bonnet formula also holds for certain of these other metrics (some of which seem to be more canonical from the point of view of possible applications). ...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. - ... 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 $\mathcal M_{\gamma}$.
- ... Bounded domains where s D ≥ c > 0 are called holomorphic homoge- neous regular domains[12,13]. On those, the Kobayashi, Carathéodory and several other invariant metrics are equivalent. Those domains in- clude several well-known classes: Teichmüller spaces, bounded domains covering compact Kähler manifolds, and strictly convex domains with C 2 -boundary[18].S. Kobayashi defines in[10, (5.4)] a projective analogue of the Koba- yashi-Royden metric. ...PreprintFull-text available
- Sep 2019

In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong-Rosay theorem. - ... Remark 9. Theorem 1 can be proved by an alternative approach, which combines Theorem 1 in [14] and Theorem 7.2 in [15] or Theorem 2 in [22]. However, our approach has the further consequence which holds to any closed complex submanifold S in E = E(m, n, p): Since we can restrict W. Yin's complete invariant Kähler metric Y to S, which still has the negative pinched holomorphic sectional curvature range on S. Then by Theorem 2 and Theorem 3 in [20], we have Corollary 2. ...Preprint
- Dec 2018

We provide a class of geometric convex domains on which the Carathéodory-Reiffen metric, the Bergman metric, the complete Kähler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Poincaré-disk. - ... 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 . ...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 recently, some new Kähler metrics with more desirable properties such as Kähler hyperbolicity have been found where the Kähler hyperbolicity means that the Kähler metric is complete with bounded curvatures and it has a bounded Kähler primitive. Such Kähler hyperbolic metrics are studied by McMullen [14] and Liu-Sun-Yau [13]. ...Preprint
- Feb 2019

Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\geq 2$. We construct a new mapping class group invariant K\"ahler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichm\"uller space $\mathcal T(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the tautological directions. - ... The Ricci curvature of the Weil-Petersson metric itself defines a Kähler metric on the moduli space; the quasi-isometry class was found by Trapani [19] and the leading asymptotics at a divisor by Liu, Sun and Yau [9,8]. Near the intersection of k divisors, written out in terms of the singular coordinate basis ...ArticleFull-text available
- Jun 2016

The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown here to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knudsen-Deligne-Mumford compactification. This is accomplished by finding a full expansion for the hyperbolic metrics on the fibers of the universal curve as they approach the complete metrics on the nodal curves above the exceptional divisors and using a push-forward theorem for conormal functions. This refines a two-term expansion due to Obitsu-Wolpert for the conformal factor relative to the model plumbing metric which in turn refined the bound obtained by Masur. - ... Even more, bounded pseudoconvex domains with α-Hörder bound- ary for all α < 1 are hyperconvex [1]. Recall that a bounded domain Ω ⊂ C n is called homogeneous regular, a concept proposed by Liu-Sun-Yau [27], if there is a constant c ∈ (0, 1) such that for any z ∈ Ω, there is a holomorphic injective map f : Ω → B n with f (z) = 0 and B n (c) ⊂ f (Ω), where B n (c) = {z ∈ C n ; z < c}. It is shown in [38] that all homogeneous regular domains are hyperconvex. ...PreprintFull-text available
- Jan 2019

For a bounded domain D and a real number p > 0, we denote by A p (D) the space of L p integrable holomorphic functions on D, equipped with the L p-pseudonorm. We prove that two bounded hyperconvex domains D 1 ⊂ C n and D 2 ⊂ C m are biholomorphic (in particular n = m) if there is a linear isometry between A p (D 1) and A p (D 2) for some 0 < p < 2. The same result holds for p > 2, p = 2, 4, · · · , provided that the p-Bergman kernels on D 1 and D 2 are exhaustive. With this as a motivation, we show that, for all p > 0, the p-Bergman kernel on a strongly pseudoconvex domain with C 2 boundary or a simply connected homogeneous regular domain is exhaustive. These results shows that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important invariants of complex manifolds. The second part of the present work devotes to studying variations of these invariants. We show that the direct image sheaf of the twisted relative m-pluricanonical bundle associated to a holomorphic family of Stein manifolds or compact Kähler manifolds is positively curved, with respect to the canonical singular Finsler metric. - ... For further results about the squeezing function the reader may also consult the references [1], [2],[3],[4],[5],[6],[7], [8], [9]. In the last section we will post some open problems. ...Article
- Nov 2016
- PAC J MATH

In recent work by Zimmer it was proved that if $\Omega\subset\mathbb C^n$ is a bounded convex domain with $C^\infty$-smooth boundary, then $\Omega$ is strictly pseudoconvex provided that the squeezing function approaches one as one approaches the boundary. We show that this result fails if $\Omega$ is only assumed to be $C^2$-smooth. - ... In [12, 13] , the authors introduced the notion of holomorphic homogeneous regular . Then in [14], the equivalent notion of uniformly squeezing was introduced. ...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. - ... Even more, bounded pseudoconvex domains with α-Hörder bound- ary for all α < 1 are hyperconvex [1]. Recall that a bounded domain Ω ⊂ C n is called homogeneous regular, a concept proposed by Liu-Sun-Yau [27], if there is a constant c ∈ (0, 1) such that for any z ∈ Ω, there is a holomorphic injective map f : Ω → B n with f (z) = 0 and B n (c) ⊂ f (Ω), where B n (c) = {z ∈ C n ; z < c}. It is shown in [38] that all homogeneous regular domains are hyperconvex. ...PreprintFull-text available
- Jan 2019

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$ for some $0<p<2$. The same result holds for $p>2, p\neq 2,4,\cdots$, provided that the $p$-Bergman kernels on $D_1$ and $D_2$ are exhaustive. With this as a motivation, we show that, for all $p>0$, the $p$-Bergman kernel on a strongly pseudoconvex domain with $\mathcal C^2$ boundary or a simply connected homogeneous regular domain is exhaustive. These results shows that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important invariants of complex manifolds. The second part of the present work devotes to studying variations of these invariants. We show that the direct image sheaf of the twisted relative $m$-pluricanonical bundle associated to a holomorphic family of Stein manifolds or compact K\"ahler manifolds is positively curved, with respect to the canonical singular Finsler metric. - Article
- Apr 2016
- INT J MATH

We study bounded domains with certain smoothness conditions and the properties of their squeezing functions in order to prove that the domains are biholomorphic to the ball. - ArticleThere are surprisingly rich properties of these holomorphic functions. The possibility of holomorphic continuation of holomorphic functions forces us to consider multi-valued holomorphic functions. The concept of Riemann Surfaces was introduced to understand such phenomena. The ideas of branch cuts and branch points immediately relate topology of these surfaces to complex variables. The possibility of two Riemann surfaces can be homeomorphic to each other with- out being equal was realized in nineteenth century where remarkable uniformization theorems were proved by Riemann for simply connected surfaces. Although it took Hilbert many years later to make Riemann's work on variational principle to be rigor- ous, the Dirichlet principle of constructing harmonic functions and hence holomorphic functions has tremendous in∞uence up to modern days.
- Article
- Apr 2008
- PAC J MATH

Here we show that the Carathéodory, Eisenman-Kobayashi, and Kähler-Einstein volume forms are equivalent on Teichmüller space. - ArticleFull-text available
- Feb 2013

We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We will also show that the moduli space $\mathbb{M}(S_{g,n})$ of surface $S_{g,n}$ of $g$ genus with $n$ punctures admits no complete visible CAT(0) Riemannian metric if $3g+n\geq 5$, which answers the Brock-Farb-McMullen question in the visible case. - Article
- Sep 2012
- COLLECT MATH

We define a natural singular hermitian metric ${\|\cdot\|_{s}}$ (s > 0) on the boundary divisor ${{\delta=\mathcal{O}(\partial\mathcal{M}_{1,1})}}$ of the moduli stack of 1-pointed stable curves of genus 1, ${{\overline{\mathcal{M}}_{1,1}}}$ . For s > 3/2 we prove that ${\|\cdot\|_{s}}$ is a log-singular hermitian metric in the sense of Burgos–Kramer–Kühn, with singularities along ${{\partial\mathcal{M}_{1,1}}}$ . We compute the arithmetic intersection number of ${(\delta,\|\cdot\|_{s})}$ with the first tautological hermitian line bundle ${\overline{\kappa}_{1,1}}$ on ${{\overline{\mathcal{M}}_{1,1}.}}$ The result involves the special values ${{\zeta^{\prime}(-1), \zeta^{\prime}(-2)}}$ and ${{\zeta(2, s)}}$ , where ${\zeta(s)}$ is Riemann’s zeta function and ${\zeta(\sigma,s)}$ is Hurwitz’ zeta function. - 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
- 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 - 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
- Jan 2012
- INT J MATH

We give a simple proof of a theorem of McMullen on Kähler hyperbolicity of moduli space of Riemann surfaces by using the Bergman metric on Teichmüller space. - Article
- Mar 2017
- MATH ANN

In this paper we consider the following question: For bounded domains with smooth boundary, can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain? Our approach to answering this question is based on understanding the dynamical behavior of real geodesics in the Kobayashi metric and allows us to prove a number of results for domains with low regularity. For instance, we show that for convex domains with $C^{2,\epsilon}$ boundary strong pseudoconvexity can be characterized in terms of the behavior of the squeezing function near the boundary, the behavior of the holomorphic sectional curvature of the Bergman metric near the boundary, or any other reasonable measure of the complex geometry near the boundary. The first characterization gives a partial answer to a question of Forn{\ae}ss and Wold. As an application of these characterizations, we show that a convex domain with $C^{2,\epsilon}$ boundary which is biholomorphic to a strongly pseudoconvex domain is also strongly pseudoconvex. - In this paper, we investigate the curvature relations between K\"ahler metrics and background Riemannian metrics. We prove that Siu's strongly non-positivity (resp. non-negativity) is equivalent to the non-positivity (resp. non-negativity) of the complex sectional curvature of the background Riemannain metric; the semi-dual-Nakano-positivity (resp. the semi-dual-Nakano-negativity) implies the non-negativity (resp. non-positivity) of the Riemannian curvature operator. As applications, we show that the moduli space $(\sM_g, \omega_{WP})$ of curves with genus $g>1$ has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. We also prove that any submanifold in $\sM_g$ which is totally geodesic in $\sA_g$ with finite volume must be a ball quotient.
- We construct a strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic.
- Article
- Dec 2016

We study the K\"ahler geometry of the classical Hurwitz space $\mathcal{H}^{n,b}$ of simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. A generalized Weil-Petersson metric on the Hurwitz space was recently introduced. Deformations of simple branched coverings fit into the more general framework of Horikawa's deformation theory of holomorphic maps, which we equip with distinguished representatives in the presence of hermitian metrics. In the article we will investigate the curvature of the generalized Weil-Petersson K\"ahler metric on the Hurwitz space. - Article
- Aug 2010
- INT J MATH

A characteristic for a complex Randers metric to be a complex Berwald metric is obtained. The formula of the holomorphic curvature for complex Randers metrics is given. It is shown that a complex Berwald Randers metric with isotropic holomorphic curvature must be either usually Kählerian or locally Minkowskian. The Deicke and Brickell theorems in complex Finsler geometry are also proved. - Article
- Nov 2016

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images. This generalizes the result of Schumacher, where he computed the curvature of $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(K_{\mathcal{X}/S}^{\otimes m})$ for a family of canonically polarized manifolds. For $p=n$, it coincides with a formula of Berndtsson. Thus, when $L$ is globally ample, we reprove his result on the Nakano positivity of $f_*(K_{\mathcal{X}/F}\otimes L)$. - 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
- Sep 2016
- T AM MATH SOC

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some $\epsilon > 0$ so that if the squeezing function on a smoothly bounded convex domain is greater than $1-\epsilon$ outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension $d$ there exists some $\epsilon > 0$ so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within $\epsilon$ of $-4/(d+1)$ outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to $-4/(d+1)$ on the boundary). - Let $S_g$ be a closed surface of genus $g$ and $\mathbb{M}_g$ be the moduli space of $S_g$ endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of $\mathbb{M}_g$ for large genus $g$. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in $\mathbb{M}_g$ as $g \to \infty$. Then we prove two curvature properties on the whole space $\mathbb{M}_g$ as $g\to \infty$ in a probabilistic way.
- In this note we show that the moduli space M(Sg,n) of surface Sg,n of genus g with n punctures, satisfying 3g + n ≥ 5, admits no complete Riemannian metric of nonpositive sectional curvature such that the Teichm¨uller space T(Sg,n) is a mapping class group Mod(Sg,n)-invariant visibility manifold.
- 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. - ChapterFull-text available
- Aug 2015

We consider the Bergman curvatures estimate for bounded domains in terms of the squeezing function. As applications, we give the asymptotic boundary behaviors of the curvatures near strictly pseudoconvex boundary points, using a recent result given by Fornaess and Wold. - Article
- May 2016
- CHINESE ANN MATH B

In this paper, the author considers a class of bounded pseudoconvex domains, i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Kähler metric g Ω(μ, m) of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Kähler-Einstein metric, the Carathéodary metric, and the Koboyashi metric are equivalent for Ω(μ, m). - In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g\geq 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that $ds^2 \succ ds_{T}^2$ where $ds_{T}^2$ is the Teichm\"uller metric. Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_{g}$ of a closed Riemann surface $S_{g}$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichm\"uller metric, which implies a conjecture of Farb-Weinberger.

- Article
- Apr 1996
- P AM MATH SOC

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved. - Completeness of the K¨ ahler-Einstein metric on bounded domains and the character-ization of domains of holomorphy by curvature conditions. In The mathematical heritage of Henri Poincar´ e
- 41-59

- N Mok
- S.-T Yau

N. Mok and S.-T. Yau. Completeness of the K¨ ahler-Einstein metric on bounded domains and the character-ization of domains of holomorphy by curvature conditions. In The mathematical heritage of Henri Poincar´ e, Part 1 (Bloomington, Ind., 1980), volume 39 of Proc. Sympos. Pure Math., pages 41–59. Amer. Math. Soc., Providence, RI, 1983. - Article
- Jan 1986

For compact Riemann surfaces of genus at least two, using Petersson’s Hermitian pairing for automorphic forms, Weil introduced a Hermitian metric for the Teiclmüller space, now known as the Weil-Petersson metric. Ahlfors [1,2] showed that the Weil-Petersson metric is Kähler and that its Ricci and holomorphic section curvatures are negative. By using a different method of curvature computation, Royden [8] later showed that the holomorphic sectional curvature of the Weil-Petersson metric is bounded away from zero and conjectured the best bound to be \(- \frac{1}{{2\pi \left( {g - 1} \right)}}\) , where g is the genus. Recently Wolpert [12] and also Royden proved Royden’s conjecture on the bound of the holomorphic sectional curvature and obtained in addition the negativity of the Riemannian sectional curvature. Wolpert’s method used some SL(2,R) invariant first-order differential operators obtained by Maass [7]. Royden’s computation is based on the fact that the Pbincaré metric on a compact Riemann surface of genus at least two is Einstein. - Article
- Jan 1993

The Petersson-Weil metric is a main tool for investigating the geometry of moduli spaces. When A. Weil considered the classical Teichmüller space from the viewpoint of deformation theory, he suggested, in 1958, investigating the Petersson inner product on the space of holomorphic quadratic differentials. He conjectured that it induced a Kähler metric on the Teichmüller space. After proving this property, Ahlfors showed, in 1961, that the holomorphic sectional and Ricci curvatures were negative. Royden’s conjecture of a precise upper bound for the holomorphic sectional curvature was proven by Wolpert and Tromba in 1986 along with the negativity of the sectional curvature. - 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. - Mathematics Version of Record
- Mathematics
- Article
- Jun 1987
- COMMUN MATH PHYS

Let Z( s, R) be the Selberg zeta function of a compact Riemann surface R. We study the behavior of Z( s, R) as R tends to infinity in the moduli space of stable curves. The main result is an estimate for Z( s, R) valid for s in a neighborhood, depending only on the genus, of s=1. Our analysis gives an alternate proof of the Belavin-Knizhnik double pole result, [5]. - Article
- Apr 2000
- ANN MATH

Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant curvature -1 in the Teichm\"uller (=Kobayashi) metric. When $r=\dim_{\cx} \cM_{g,n}$ is greater than one, however, $\cM_{g,n}$ carries no complete metric of bounded negative curvature. Instead, Dehn twists give chains of subgroups $\zed^r \subset \pi_1(\cM_{g,n})$ reminiscent of flats in symmetric spaces of rank $r>1$. In this paper we introduce a new K\"ahler metric on moduli space that exhibits its hyperbolic tendencies in a form compatible with higher rank. - 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
- Oct 2004
- J DIFFER GEOM

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.