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A Discovery Tour in Random Riemannian Geometry

Lorenzo Dello Schiavo, Eva Kopfer, Karl-Theodor Sturm

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 $h^\bullet: ω\mapsto h^ω$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^ω\colon\!\!= e^{2h^ω}\,\mathsf{g}$. 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.

A Discovery Tour in Random Riemannian Geometry

Abstract

We study random perturbations of Riemannian manifolds by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields will act on the manifolds via conformal transformation . Our focus will be on the regular case with Hurst parameter , 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.

Paper Structure

This paper contains 39 sections, 64 theorems, 201 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Random Riemannian Geometry
  3. Fractional Gaussian Field (FGF)
  4. Higher Order Green Kernel
  5. The Riemannian Manifold
  6. Higher Order Green Operators
  7. The case of manifolds of bounded geometry
  8. Bessel potential spaces
  9. Standard Sobolev spaces
  10. Manifolds of bounded geometry
  11. Test functions
  12. Heat-kernel estimates
  13. The case of closed manifolds
  14. Grounding
  15. Eigenfunction expansion
  16. ...and 24 more sections

Key Result

Theorem 1.1

For every $\omega$, a regular, strongly local Dirichlet form is given by

Figures (9)

  • Figure 1: Gaussian random field over a toroid.
  • Figure 2: A realization of $h^\bullet_\ell$ in \ref{['eq:PartialSum']} on the unit sphere $\mbbS^2$ with, $m=s=1$ (critical case), and $\ell\in\left\{1,\dotsc, 20\right\}$.
  • Figure 3: A realization of the random metric ${\sf g}_\ell^\bullet=e^{2h^\bullet_\ell}{\sf g}$ on $\mbbS^2$, $\ell=30$.
  • Figure 4: The Green kernels $G^1_{s,1}$ for $2s=1,\dotsc,5$ (in reverse order w.r.t. the value at $0$). Note that $\lim_{r\rightarrow 0} G^1_{1/2,1}(r)=+\infty$.
  • Figure 5: The Green kernel $G^{\mbbT}_{1,1}(\tfrac{1}{2},y)$ with $y\in[0,1)$.
  • ...and 4 more figures

Theorems & Definitions (127)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 117 more