  We prove the existence of a unique maximal surface in each anti-de Sitter
(AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that
is, with conical singularities along time-like lines) for cone angles less than
$\pi$. We interpret this result in terms of Teichm\"uller theory, and prove the
existence of a unique minimal Lagrangian diffeomorphism isotopic to the
identity between two hyperbolic surfaces with cone singularities when the cone
angles are the same for both surfaces and are less than $\pi$.
