Geometric aspects of the moduli space of Riemann surfaces

摘要

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kaehler metrics were introduced on the moduli space and Teichmueller 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 Kaehler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kaehler-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.