Fixing three dimensional geometries from entanglement entropies of CFT$_2$

In this paper, we propose a method of fixing the leading behaviors of three dimensional geometries from the dual CFT$_2$ entanglement entropies. We employ only the holographic principle and do not use any assumption about the AdS/CFT correspondence and bulk geometry. Our strategy involves using both UV and IR-like CFT$_2$ entanglement entropies to fix the bulk geodesics. With a simple trick, the metric can be extracted from the geodesics. As examples, we fix the leading behaviors of the pure AdS$_3$ metric from the entanglement entropies of free CFT$_2$ and, more importantly, the BTZ black hole from the entanglement entropies of finite temperature CFT$_2$. Consequently, CFT$_2$ with finite size or topological defects can be determined through simple transformations. Following the same steps, in principle, the leading behaviors of all three dimensional (topologically distinct) holographic classical geometries from the dual CFT$_2$ entanglement entropies can be fixed.