Geometric Aspects of the Moduli Space of Riemann Surfaces

摘要

This is a survey of our recent results on the geometry of moduli spaces andTeichmuller spaces of Riemann surfaces appeared in math.DG/0403068 andmath.DG/0409220. We introduce new metrics on the moduli and the Teichmullerspaces of Riemann surfaces with very good properties, study their curvaturesand boundary behaviors in great detail. Based on the careful analysis of thesenew metrics, we have a good understanding of the Kahler-Einstein metric fromwhich we prove that the logarithmic cotangent bundle of the moduli space isstable. Another corolary is a proof of the equivalences of all of the knownclassical complete metrics to the new metrics, in particular Yau's conjecturesin the early 80s on the equivalences of the Kahler-Einstein metric to theTeichmuller and the Bergman metric.