Scalar curvature and $Q$-curvature of random metrics

TL;DR

This paper investigates the properties of scalar curvature and Q-curvature in random Riemannian metrics on compact surfaces and higher-dimensional manifolds, motivated by comparison geometry.

Contribution

It extends the study of curvature properties to random metrics, including scalar and Q-curvature, across different dimensions and conformal classes.

Findings

Analysis of Gaussian curvature for random metrics on surfaces

Extension to scalar curvature in higher dimensions

Investigation of Q-curvature in the context of random metrics

Abstract

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.