Liouville Action for Hyperbolic Ideal Tetrahedra

For a given ideal tetrahedron in the hyperbolic space \(\mathbb {H}^3\) , we show that the boundary of the ideal tetrahedron is the same as the image of the Epstein map for a certain conformal metric called Thurston metric on the complement of vertices in the conformal boundary of \(\mathbb {H}^3\) . We also introduce the Liouville action for the Thurston metric and prove the holography principle, that is, the equality between the Liouville action for the Thurston metric and the renormalized Einstein-Hilbert action given by the Epstein map for the Thurston metric.