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 M , the Riemann moduli space of surfaces of genus > 1. This space has a singular compactication with respect to gWP, and this metric has crossing cusp-edge singularities along a nite collection of simple normal crossing divisors. We prove rst that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete. The second theorem is a Weyl asymptotic formula for the counting function for this spectrum.