Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces

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...

Compact polyhedral surfaces of an arbitrary genus and determinants of Laplacians

Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as a functional on the moduli space of these surfaces. An explicit formula for this determinant is obtained.

On Some Properties of Analytic Spaces Connected with Bergman Metric Ball

Spectral 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...

1 9 O ct 1 99 8 Ω - Admissible Theory II : New metrics on determinant of cohomology And Their applications to moduli spaces of punctured Riemann surfaces

For singular metrics, Ray and Singer’s analytic torsion formalism cannot be applied. Hence we do not have the so-called Quillen metric on determinant of cohomology with respect to a singular metric. In this paper, we introduce a new metric on determinant of cohomology by adapting a totally different approach. More precisely, by strengthening results in the first paper of this series, we develop...