Gaussian measures on the of space of Riemannian metrics

Lior Silberman, Brian Clarke, Dmitry Jakobson, Niky Kamran, Jonathan Taylor, Yaiza Canzani

Abstract: We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension ≥3. For this random model we compute the characteristic function for the L2 (Ebin) distance to the reference metric. In the Appendix, we study Lipschitz-type distance between Riemannian metrics and give applications to the diameter, eigenvalue and volume entropy functionals.