Gaussian measures on the of space of Riemannian metrics

TL;DR

This paper introduces Gaussian measures on the space of Riemannian metrics with fixed volume, computes their characteristic functions, and explores applications to geometric functionals like diameter and eigenvalues.

Contribution

It presents a novel construction of Gaussian measures on the space of metrics and analyzes their properties and applications in Riemannian geometry.

Findings

Computed characteristic functions for the $L^2$ distance to the reference metric.

Studied Lipschitz-type distances and their implications for geometric functionals.

Provided applications to diameter, eigenvalues, and volume entropy.

Abstract

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.