Minimal Lagrangian diffeomorphism via Anti-de Sitter geometry

Minimal Lagrangian maps have played an important role in the study of hyperbolic structures on surfaces. As observed independently by Labourie and Schoen, given a closed hyperbolic surfaces and two metrics on it, there exists a unique minimal Lagrangian diffeomorphism isotopic to the identity. Alternative proofs have been provided later, and in my thesis I concentrate on the proof in the context of Anti-de Sitter three dimensional geometry.