Liouville quantum gravity and the Brownian map III: the conformal structure is determined

TL;DR

This paper proves that the conformal structure of a Liouville quantum gravity sphere can be uniquely recovered from its metric measure space, establishing the equivalence between the Brownian map and LQG in encoding the same geometric and conformal data.

Contribution

It demonstrates that the conformal structure of a $ frac{ ext{8}}{3}$-LQG sphere is almost surely determined by its metric measure space, confirming the canonical conformal parameterization of the Brownian map.

Findings

The conformal structure is almost surely recoverable from the mm-space.

The Brownian map and $ frac{ ext{8}}{3}$-LQG sphere are equivalent in law.

The results extend to other topologies of Brownian and LQG surfaces.

Abstract

Previous works in this series have shown that an instance of a $\sqrt{8/3}$-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 M\"obius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $\sqrt{8/3}$-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 $\sqrt{8/3}$-LQG surfaces with other topologies.