THE MINKOWSKI CONTENT MEASURE FOR THE LIOUVILLE QUANTUM GRAVITY METRIC

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Sung, Jinwoo
• 署名单位: University of Chicago
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1667
• 发表日期: 2024
• 页码: 658-712
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS; brownian-motion; sle; geodesics; geometry

摘要

A Liouville quantum gravity (LQG) surface is a natural random twodimensional 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 gamma -LQG surface determines its conformal structure for every gamma is an element of (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 Holder continuity result for space -filling SLE with respect to the LQG metric.