A Discovery Tour in Random Riemannian Geometry

We study random perturbations of a Riemannian manifold \((\textsf{M},\textsf{g})\) $(\mathsf{M},\mathsf{g})$ by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields \(h^\bullet : \omega \mapsto h^\omega \) $h^{\bullet }:\omega \mapsto h^{\omega }$ will act on the manifold via the conformal transformation \(\textsf{g}\mapsto \textsf{g}^\omega := e^{2h^\omega }\,\textsf{g}\) $\mathsf{g}\mapsto {\mathsf{g}}^{\omega }:=e^{2h^{\omega }}\mathsf{g}$ . Our focus will be on the regular case with Hurst parameter \(H>0\) $H>0$ , the critical case  \(H=0\) $H=0$ being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.