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

Understand

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).