  We prove that any weakly acausal curve $\Gamma$ in the boundary of Anti-de
Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces,
one of which is past-convex and the other future-convex, for every
$K\in(-\infty,-1)$. The curve $\Gamma$ is the graph of a quasisymmetric
homeomorphism of the circle if and only if the $K$-surfaces have bounded
principal curvatures. Moreover in this case a uniqueness result holds.
  The proofs rely on a well-known correspondence between spacelike surfaces in
Anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic
plane. In fact, an important ingredient is a representation formula, which
reconstructs a spacelike surface from the associated area-preserving
diffeomorphism.
  Using this correspondence we then deduce that, for any fixed
$\theta\in(0,\pi)$, every quasisymmetric homeomorphism of the circle admits a
unique extension which is a $\theta$-landslide of the hyperbolic plane. These
extensions are quasiconformal.
