The Minkowski content measure for the Liouville quantum gravity metric

TL;DR

This paper demonstrates that the LQG measure can be obtained as the Minkowski measure from the LQG metric, establishing that the metric structure uniquely determines the conformal structure of the surface.

Contribution

It proves the Minkowski content measure equals the LQG measure and shows the metric determines the conformal structure for all  in (0,2), answering a key open question.

Findings

LQG measure is the Minkowski measure of the LQG metric

The metric structure determines the conformal structure of the LQG surface

Established H"older continuity of space-filling SLE with respect to the LQG metric

Abstract

A Liouville quantum gravity (LQG) surface is a natural random two-dimensional surface, initially formulated as a random measure space and later as a random metric space. We show that the LQG measure can be recovered as the Minkowski measure with respect to the LQG metric, answering a question of Gwynne and Miller (arXiv:1905.00383). As a consequence, we prove that the metric structure of a $\gamma$-LQG surface determines its conformal structure for every $\gamma \in (0,2)$. Our primary tool is the continuum mating-of-trees theory for space-filling SLE. In the course of our proof, we also establish a H\"older continuity result for space-filling SLE with respect to the LQG metric.