Liouville quantum gravity and the Brownian map III: the conformal structure is determined
成果类型：
Article
署名作者：
Miller, Jason; Sheffield, Scott
署名单位：
University of Cambridge; Massachusetts Institute of Technology (MIT)
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-021-01026-8
发表日期：
2021
页码：
1183-1211
关键词：
摘要：
Previous works in this series have shown that an instance of a root 8/3-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Mobius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the root 8/3-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and root 8/3-LQG surfaces with other topologies.
来源URL：