Fixing the AdS${}_3$ metric from the pure state entanglement entropies of CFT${}_2$

In this paper, based on RT formula, by identifying the pure state UV and IR entanglement entropies of a perturbed CFT${}_2$ with geodesic lengths in the bulk, we demonstrate that the dual geometry is uniquely determined to be asymptotically AdS${}_3$. The pure AdS${}_3$ geometry is recovered by taking the massless limit of the system. Our derivations hold in both static and covariant scenarios.