  We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter
(AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles
less than $\pi$ along time-like curves), the complement of the convex core in
$N$ admits a unique foliation by constant Gauss curvature surfaces. This
extends, and provides a new proof of, a result of \cite{BBZ2}. We also describe
a parametrization of the space of convex GHM AdS metrics on a given manifold,
with particles of given angles, by the product of two copies of the
Teichm\"uller space of hyperbolic metrics with cone singularities of fixed
angles. Finally, we use the results on $K$-surfaces to extend to hyperbolic
surfaces with cone singularities of angles less than $\pi$ a number of results
concerning landslides, which are smoother analogs of earthquakes sharing some
of their key properties.
