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