The Anti-de Sitter space is the Lorentzian analogue of the hyperbolic space, namely it is the model for negatively curved manifolds in signature $(n,1)$. As its Riemannian counterpart, it comes with a conformal asymptotic boundary.
The asymptotic Plateau problem in the the Anti-de Sitter space consists in finding hypersurfaces with constant mean curvature (CMC) and with a prescribed boundary at infinity.
In this talk, we show that there exists a unique hypersurface having constant mean curvature $H\in\mathbb{R}$, for any suitable prescribed boundary data.
Furthermore, all the hypersurfaces mentioned above are complete : this result extends the Cheng-Yau theorem from the flat case (Minkowski space) to the negative constant sectional curvature case.
In the second part of the talk, we focus on the $(2+1)-$dimensional case, which is deeply linked to Teichmuller theory. In this case, CMC-surfaces induce a special class of quasi-conformal maps on the hyperbolic plane, called $\theta-$landslides.
We will present some partial result of a on-going project, whose goal is to estimate the maximal dilation of a $\theta-$landslide with respect to the cross-ratio norm of its extension to the asymptotic boundary of $\mathbb{H}^2$. These estimates are obtained by studying the eigenvalues of the second fundamental form of the corresponding CMC-hypersurface