On Weil-petersson Symmetry of Moduli Spaces of Riemann Surfaces

Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces

Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models

We prove that Mirzakhani’s recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix recursion relations, and therefore we confirm again that Kontsevich’s integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(Mg,0).

The Weil-petersson Geometry on the Thick Part of the Moduli Space of Riemann Surfaces

In the thick part of the moduli space of Riemann surface, we show the sectional curvature of the Weil-Petersson metric is bounded independent of the genus of the surface.

Moduli spaces of hyperbolic surfaces and their Weil–Petersson volumes

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.

On the Weil-petersson Curvature of the Moduli Space of Riemann Surfaces of Large Genus

Let Sg be a closed surface of genus g and Mg be the moduli space of Sg endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of Mg for large genus g. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in Mg as g → ∞. Then we prove two curvature properties on the whole spac...