Approximation of length metrics by conformally flat Riemannian metrics

TL;DR

This paper proves that any length metric on Euclidean space can be uniformly approximated by conformally flat Riemannian metrics, facilitating applications in quantum gravity research.

Contribution

It provides a rigorous proof of approximation of length metrics by conformally flat Riemannian metrics, a result with implications for quantum gravity models.

Findings

Any length metric on ℝ^d can be approximated uniformly by conformally flat Riemannian metrics.

The approximation technique is used to explore Liouville quantum gravity.

The result bridges metric geometry and Riemannian geometry in a novel way.

Abstract

We present a proof of the folklore result that any length metric on $\mathbb R^d$ can be approximated by conformally flat Riemannian distance functions in the uniform distance. This result is used to study Liouville quantum gravity in another paper by the same authors.