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Approximation of length metrics by conformally flat Riemannian metrics

Andres A. Contreras Hip, Ewain Gwynne

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.

Approximation of length metrics by conformally flat Riemannian metrics

Abstract

We present a proof of the folklore result that any length metric on 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.

Paper Structure

This paper contains 3 sections, 2 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['app']}
  3. Construction of an approximating graph

Key Result

Theorem 1.2

Let $\bar{D}$ be a boundedly compact length metric on $\mathbb{R}^d$ which induces the same topology as the Euclidean metric $D_0.$ Let $\varepsilon>0$ and $R>0.$ Then there exists a bounded continuous function $f:\mathbb{R}^d\to \mathbb{R}$ such that

Figures (4)

  • Figure 1: old geodesics (red and blue paths)
  • Figure 2: New geodesics, cut from the red and blue geodesics
  • Figure 3: Approximation by line segments
  • Figure 4: Left: Values of $F_{\eta,\bar{\eta}}^{\mathrm{ext}}$. Right: Values of $f_{e,\eta}$

Theorems & Definitions (4)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof