Minimal Lagrangian diffeomorphisms between hyperbolic cone surfaces and Anti-de Sitter geometry

We study minimal diffeomorphisms between hyperbolic cone-surfaces (that is diffeomor-
phisms whose graph are minimal submanifolds). We prove that, given two hyperbolic
metrics with the same number of conical singularities of angles less than π, there always
exists a minimal diffeomorphism isotopic to the identity.
When the cone-angles of one metric are strictly smaller than the ones of the other, we
prove that this diffeomorphism is unique.
When the angles are the same, we prove that this diffeomorphism is unique and area-
preserving (so is minimal Lagrangian). The last result is equivalent to the existence of a
unique maximal space-like surface in some Globally Hyperbolic Maximal (GHM) anti-de
Sitter (AdS) 3-manifold with particles.