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 . 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.
Acknowledgments
We thank the anonymous referee for useful remarks on an earlier version of this work and Greg Lawler for helpful discussions.
Citation
Ewain Gwynne. Jinwoo Sung. "The Minkowski content measure for the Liouville quantum gravity metric." Ann. Probab. 52 (2) 658 - 712, March 2024. https://doi.org/10.1214/23-AOP1667
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