Conformal boundary conditions from cutoff AdS3

We construct a particular flow in the space of 2D Euclidean QFTs on torus, which we argue is dual to class solutions 3D gravity with conformal boundary conditions. This new comes from Legendre transform kernel implements $T\bar{T}$ deformation, and motivated by need for conditions be elliptic, i.e. that they have well-defined propagators metric fluctuations. demonstrate equivalence between our equation variants Wheeler de-Witt torus universe so-called Constant Mean Curvature (CMC) slicing. derive flow, compute corresponding ground state energy low-temperature limit. Once deformation parameters are fixed, existence independent initial data, provided seed theory CFT. The high-temperature density states has Cardy-like behavior, rather than Hagedorn growth characteristic $T\bar{T}$-deformed theories.