  We show that any element of the universal Teichm\"uller space is realized by
a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself.
The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We
show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is
the boundary at infinity of a space-like hypersurface bounds a maximal
space-like hypersurface. In $AdS^3$, if $E$ is the graph of a quasi-symmetric
homeomorphism, then this maximal surface is unique, and it has negative
sectional curvature. As a by-product, we find a simple characterization of
quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional
projective geometry.
