In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the Kahler–Einstein metric, we show that the logarithmic cotangent bundle over the Deligne–Mumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the Kahler–Einstein metric by using the Kahler–Ricci flow and a priori estimates of the complex Monge-Ampere equation.

/pdf/canonical-metrics-on-the-moduli-space-of-riemann-surfaces-ii-140y15sji9.pdf

Canonical Metrics on the Moduli Space of Riemann Surfaces II

1 Holomorphic Functions.- 1.1. Holomorphic Functions.- 1.2. Holomorphic Map.- 2 Complex Manifolds.- 2.1. Complex Manifolds.- 2.2. Compact Complex Manifolds.- 2.3. Complex Analytic Family.- 3 Differential Forms, Vector Bundles, Sheaves.- 3.1. Differential Forms.- 3.2. Vector Bundles.- 3.3. Sheaves and Cohomology.- 3.4. de Rham's Theorem and Dolbeault's Theorem.- 3.5. Harmonic Differential Forms.- 3.6. Complex Line Bundles.- 4 Infinitesimal Deformation.- 4.1. Differentiable Family.- 4.2. Infinitesimal Deformation.- 5 Theorem of Existence.- 5.1. Obstructions.- 5.2. Number of Moduli.- 5.3. Theorem of Existence.- 6 Theorem of Completeness.- 6.1. Theorem of Completeness.- 6.2. Number of Moduli.- 6.3. Later Developments.- 7 Theorem of Stability.- 7.1. Differentiable Family of Strongly Elliptic Differential Operators.- 7.2. Differentiable Family of Compact Complex Manifolds.- Appendix Elliptic Partial Differential Operators on a Manifold.- 1. Distributions on a Torus.- 2. Elliptic Partial Differential Operators on a Torus.- 3. Function Space of Sections of a Vector Bundle.- 4. Elliptic Linear Partial Differential Operators.- 5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation.- 6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations.

Complex manifolds and deformation of complex structures

This paper presents a number of problems about mapping class groups and moduli space. The paper will appear in the book "Problems on Mapping Class Groups and Related Topics", ed. by B. Farb, Proc. Symp. Pure Math. series, Amer. Math. Soc.

Some problems on mapping class groups and moduli space

Geometry of domains with the uniform squeezing property

*S. Oh, *E.K. Choi, *J.J. Kwak, *Y.S. Choi *Department of Internal Medicine, Seoul National University College of Medicine, Seoul, Korea

「Oral Communication」の実践と課題

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.

Geometric aspects of the moduli space of riemann surfaces

In this paper we continue our study on the canonical metrics on the Teichmuller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Caratheodory metric to the Kahler-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 Kahler-Einstein metric and prove that it has bounded geometry, and all of the covariant derivatives of its curvature are uniformly bounded on the Teichmuller 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.

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, \omega_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. As applications, we prove that any submanifold in $M_g$ which is totally geodesic in $A_g$ with finite volume must be a ball quotient.

Xiaofeng Sun

Papers

Canonical Metrics on the Moduli Space of Riemann Surfaces II

Geometric aspects of the moduli space of riemann surfaces

Canonical Metrics on the Moduli Space of Riemann Surfaces II

Curvatures of moduli space of curves and applications

Curvatures of moduli space of curves and applications