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# New results on the geometry of the moduli space of Riemann surfaces

**Article**

*in*Science in China Series A Mathematics 51(4):632-651 · April 2008

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Abstract

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.

- ... 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.) ...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 $t = 0$.
- ... The algebro-geometric corollaries such as the stability of the logarithmic cotangent bundles and the infinitesimal rigidity of the moduli spaces will also be briefly discussed. Similar to our previous survey articles [15, 14], we will briefly describe the basic ideas of our proofs, the details of the proofs will be published soon, see [16, 17]. After introducing the definition of Weil-Petersson metric in Section 2, we discuss the fundamental curvature formula of Wolpert for the Weil-Petersson metric. ...Article
- Dec 2009

We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds. - 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
- 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.

- L 2-cohomology of the Weil-Petersson metric. In: Mapping class groups and moduli spaces of Riemann surfaces
- L Saper

- 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. - Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class. In: Contributions to several complex variables
- Jan 1986
- 261-298

Y T. Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class. In: Contributions to several complex variables, Aspects Math, E9, pages 261–298. Vieweg, Braunschweig, 1986 - 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. - 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
- Sep 2005

We prove that the moduli stack of stable curves of genus g with n marked points is rigid, i.e., has no infinitesimal deformations. This confirms the first case of a principle proposed by Kapranov. It can also be viewed as a version of Mostow rigidity for the mapping class group. Comment: 11 pages. v2: Proof rewritten to avoid use of log structures. Example of nonrigid moduli space of surfaces added - 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.