On the Weil-petersson Curvature of the Moduli Space of Riemann Surfaces of Large Genus
نویسنده
چکیده
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 space Mg as g → ∞ in a probabilistic way.
منابع مشابه
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.
متن کاملThe Geometry of the Moduli Space of Riemann Surfaces
We wish to describe how the hyperbolic geometry of a Riemann surface of genus g y g > 2, leads to a symplectic geometry on Tg, the genus g Teichmüller space, and ~Mg, the moduli space of genus g stable curves. The symplectic structure has three elements: the Weil-Petersson Kahler form, the FenchelNielsen vector fields t+, and the geodesic length functions I*. Weil introduced a Kahler metric for...
متن کاملOn Weil-petersson Symmetry of Moduli Spaces of Riemann Surfaces
In this article, we give a perspective on several results, old and new, concerning geometric structures of moduli spaces of Riemann surfaces with respect to the L2 metric (Weil-Petersson metric) on deformations of hyperbolic metrics. In doing so, we aim to demonstrate that the Weil-Petersson metric is suited to account for the geometry of moduli spaces while the topological type, genus in parti...
متن کاملSpectral Theory for the Weil-petersson Laplacian on the Riemann Moduli Space
We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric gWP on Mγ , the Riemann moduli space of surfaces of genus γ > 1. This space has a singular compactification with respect to gWP, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar...
متن کاملGrowth of Weil-petersson Volumes and Random Hyperbolic Surfaces of Large Genus
In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume Vg,n of the moduli spaces of hyperbolic surfaces of genus g with n...
متن کامل