Conformal covariance of the Liouville quantum gravity metric for ??(0,2)

For ??(0,2), U?C, and an instance h of the Gaussian free field (GFF) on U, ?-Liouville quantum gravity (LQG) surface associated with (U,h) is formally described by Riemannian metric tensor e?h(dx2+dy2) U. Previous work authors showed that one can define a canonical (distance function) Dh U ?-LQG surface. We show this conformally covariant in sense it respects coordinate change formula for surfaces. That is, if U˜ are domains, ?:U?U˜ conformal transformation, Q=2/?+?/2, h˜=h???1+Qlog|(??1)?|, then Dh(z,w)=Dh˜(?(z),?(w)) all z,w?U. This proves intrinsic to structure (U,h), i.e., does not depend particular choice parameterization.