Random k-Surfaces
  The main result is the construction of ergodic transversal measures of full support on the space of all k-surfaces of a compact hyperbolic 3-manifold. This space is a laminated space, each of its leaf being identified with a "complete" k-surface, i.e. a surface of constant (extrinsic) curvature k, where k belongs to ]0,1[. Elsewhere, this space has been shown to have chaotic properties mimicking those of the geodesic flow. This construction is achieved through a coding of the space. 
Francois Labourie