Gaussian measures on the of space of Riemannian metrics

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

Published 2013-09-05, updated 2015-09-06Version 3

We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (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.