Random Riemannian Geometry
We will study random perturbations of Riemannian manifolds (M,g) by means of the so-called “Fractional Gaussian Fields” 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 H>0, the celebrated Liouville geometry in even dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, spectral bound, spectral gap, hitting probabilities of the Brownian motions, or heat kernel estimates will change under the influence of the noise. Another goal is to define and analyze in detail the Fractional Gaussian Field on a general Riemannian manifold, a fascinating object of independent interest.