Some open questions on anti-de Sitter geometry

We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.Comment: Not a research article in the usual sense but rather a list of open questions. 19 page

Maximal surfaces in anti-de Sitter 3-manifolds with particles

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 $\pi$.Comment: Accepted for publication at "Annales de l'institut Fourier