In this talk, we present a two dimensional analogue of the geodesic flow, namely the space of surfaces of constant curvature inside a closed negatively curved 3-manifold. The dynamical properties of this space are described in terms of actions of PSL(2,R) and the equidistribution properties allow to study the geometric rigidity associated to the counting of such surfaces according to their area. This is joint work with Graham Smith (PUC Rio de Janeiro) and Ben Lowe (University of Chicago).