A Discovery Tour in Random Riemannian Geometry

TL;DR

This paper investigates how random fractional Gaussian perturbations affect fundamental geometric and analytic properties of Riemannian manifolds, aiming to quantify these effects and define the fields on general manifolds.

Contribution

It introduces and analyzes Fractional Gaussian Fields on Riemannian manifolds, exploring their impact on geometry and spectral properties, and provides a framework for understanding noise influence.

Findings

Quantifies changes in diameter, volume, heat kernel, and spectral gap under Gaussian noise.

Establishes a detailed construction of Fractional Gaussian Fields on general manifolds.

Provides insights into the borderline case of Liouville geometry in two dimensions.

Abstract

We study random perturbations of Riemannian manifolds $(\mathsf{M},\mathsf{g})$ by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields $h^\bullet: \omega\mapsto h^\omega$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^\omega\colon\!\!= e^{2h^\omega}\,\mathsf{g}$. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will 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.