Liouville quantum gravity and the Brownian map III: the conformal structure is determined
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440_2021_Article_1026.pdf
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Author(s)
Miller, Jason
Sheffield, Scott
Date Issued
March 25, 2021
Publisher
Springer Berlin Heidelberg
Version
Final published version
Abstract
Abstract
Previous works in this series have shown that an instance of a
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-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 Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the
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-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
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-LQG surfaces with other topologies.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00440-021-01026-8
