March 2024 The Minkowski content measure for the Liouville quantum gravity metric
Ewain Gwynne, Jinwoo Sung
Author Affiliations +
Ann. Probab. 52(2): 658-712 (March 2024). DOI: 10.1214/23-AOP1667


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 γ(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.

Funding Statement

E.G. was partially supported by a Clay research fellowship. J.S. was partially supported by a scholarship from Kwanjeong Educational Foundation.


We thank the anonymous referee for useful remarks on an earlier version of this work and Greg Lawler for helpful discussions.


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Ewain Gwynne. Jinwoo Sung. "The Minkowski content measure for the Liouville quantum gravity metric." Ann. Probab. 52 (2) 658 - 712, March 2024.


Received: 1 November 2022; Revised: 1 September 2023; Published: March 2024
First available in Project Euclid: 4 March 2024

MathSciNet: MR4718403
Digital Object Identifier: 10.1214/23-AOP1667

Primary: 60D05 , 60J67
Secondary: 60G57

Keywords: Liouville quantum gravity , LQG metric , mating of trees , Minkowski content , Schramm–Loewner evolution

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • March 2024
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