Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes

Abstract

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.

37 pages - submitted to Handbook of Moduli (edited by G. Farkas and I. Morrison)

References in corpus (4)

• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
• A simple proof of Witten conjecture through localization
• Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus
• The asymptotic Weil-Petersson form and intersection theory on M_{g,n}