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/|·|). 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.

In this paper we firstly prove that the CD_p curvature condition always satisfies for p\ge 2 on any connected locally finite graph. We show this property does not hold for . We also derive a lower bound for the first nonzero eigenvalue of the p-Laplace operator on a connected finite graph with the CD_p(m, K) condition for the case that and K > 0.