Geometric Aspects of the Moduli Space of Riemann Surfaces

Details Geometric Aspects of the Moduli Space of Riemann Surfaces Geometric Aspects of the Moduli Space of Riemann Surfaces

2004-11-11

arxiv.org/abs/math/0411247v2

Mathematics

Differential Geometry

Survey article of our recent results on the subject. Typoes corrrected

Scientific paper

This is a survey of our recent results on the geometry of moduli spaces and Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces with very good properties, study their curvatures and boundary behaviors in great detail. Based on the careful analysis of these new metrics, we have a good understanding of the Kahler-Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable. Another corolary is a proof of the equivalences of all of the known classical complete metrics to the new metrics, in particular Yau's conjectures in the early 80s on the equivalences of the Kahler-Einstein metric to the Teichmuller and the Bergman metric.

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