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...

For an even integer k, let r2k(n) be the number of representations of n as a sum of 2k squares. The quantity r2k(n) is approximated by the classical singular series ρ2k(n) nk−1. Deligne’s bound on the Fourier coefficients of Hecke eigenforms gives that r2k(n) = ρ2k(n) + O(d(n)n k−1 2 ). We determine the optimal implied constant in this estimate provided that either k/2 or n is odd. The proof re...

The Mumford class κ1 on Mg,0 was shown to be proportional to the cohomology class [ωWP ] of the Weil-Petersson form by Wolpert in [WO]. Furthermore he showed that the restriction of this class to any component of the compactyfying divisor coincides with the corresponding Weil-Petersson class. Arbarello and Cornalba introduced classes κ1 on Mg,n, proved a similar restriction property for these a...

We study the large-scale geometry of Weil-Petersson space, that is, Teichmüller space equipped with the Weil-Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space, we recover a result of Eskin Masur and Rafi which gives the coarse rank of the space. We g...