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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Weil-Petersson volumes and intersection theory on the moduli space of curves


Author: Maryam Mirzakhani
Journal: J. Amer. Math. Soc. 20 (2007), 1-23
MSC (2000): Primary 32G15, 14H15
Published electronically: March 8, 2006
MathSciNet review: 2257394
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Abstract: In this paper, we establish a relationship between the Weil- Petersson volume $ V_{g,n}(b)$ of the moduli space $ \mathcal{M}_{g,n}(b)$ of hyperbolic Riemann surfaces with geodesic boundary components of lengths $ b_{1},\ldots, b_{n}$, and the intersection numbers of tautological classes on the moduli space $ \overline{\mathcal{M}}_{g,n}$ of stable curves. As a result, by using the recursive formula for $ V_{g,n}(b)$ obtained in the author's Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, preprint, 2003, we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten-Kontsevich formula.


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Additional Information

Maryam Mirzakhani
Affiliation: Department of Mathematics, Princeton University, Princeton, NJ 08544

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00526-1
PII: S 0894-0347(06)00526-1
Received by editor(s): April 6, 2004
Published electronically: March 8, 2006
Additional Notes: The author is supported by a Clay fellowship.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.