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Analytic and Geometric Aspects of Moduli Space of Riemann Surfaces

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Speaker: Xiaofeng Sun, Harvard University
Location: 693 Kerr
Start time: Thu, Jan 6 2005, 11:00AM

We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces, and study their curvatures and boundary behaviors by using the singular perturbation techniques from partial differential equations. These new metrics have Poincare growth near the boundary of the moduli space and have bounded geometry. Based on the detailed analysis of these new metrics, we obtain good understanding of all of the known classical complete Kahler metrics, in particular the Kahler-Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable in the sense of Mumford. By studying the Monge-Ampere equation together with the Kahler-Ricci flow on complete non-compact manifolds, we derive C^k estimates directly without using the C0 estimate. Based on these analysis, we prove that the Kahler-Einstein metric has strongly bounded geometry. Another corollary is a proof of the equivalences of all of the known classical complete metrics to these new metrics, in particular Yau's conjectures on the equivalences of the Kahler-Einstein metric to the Teichmuller and the Bergman metric.