  We find the most general solution to Chern-Simons AdS$_3$ gravity in
Fefferman-Graham gauge. The connections are equivalent to geometries that have
a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a
spin connection. We define a variational principle for Dirichlet boundary
conditions and find the boundary stress tensor in the Chern-Simons formalism.
Using this variational principle as the departure point, we show how to treat
other choices of boundary conditions in this formalism, such as, including the
mixed boundary conditions corresponding to a $T \bar T$-deformation.
