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

Abstract Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has well-defined distance function, and the resulting metric measure space (mm-space) agrees law with Brownian map (TBM). In work, we show given just mm-space structure, one can a.s. recover LQG sphere. This implies there is canonical way to parameterize TBM by Euclidean (up Möbius transformation). other words, conformal structure. The conclusion -LQG are equivalent. They ultimately encode same structure (a topological measure, metric, structure) law. From point view, fact determines vice-versa be understood as property unified results work also imply analogous facts hold for surfaces topologies.