The critical Liouville quantum gravity metric induces the Euclidean topology

Understand

We show that every possible metric associated with critical ($\gamma=2$) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric.

• More precisely, we show that the optimal modulus of continuity of the critical LQG metric with respect to the Euclidean metric is a power of $1/\log(1/|\cdot|)$.
• Our result applies to every possible subsequential limit of critical Liouville first passage percolation, a natural approximation scheme for the LQG metric which was recently shown to be tight.