The Minkowski content measure for the Liouville quantum gravity 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 (Invent. Math. 223 (2021) 213–333). As a consequence, we prove that the metric structure of a γ-LQG surface determines its conformal structure for every $\mathit{\gamma }\in (0,2)$. Our primary tool is the continuum mating-of-trees theory for space-filling SLE. In the course of our proof, we also establish a Hölder continuity result for space-filling SLE with respect to the LQG metric.