Area-preserving diffeomorphisms of the hyperbolic plane and K-surfaces in anti-de Sitter space

We prove that any weakly acausal curve (Formula presented.) in the boundary of anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike (Formula presented.) -surfaces, one of which is past-convex and the other future-convex, for every (Formula presented.). The curve (Formula presented.) is the graph of a quasisymmetric homeomorphism of the circle if and only if the (Formula presented.) -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 (Formula presented.), every quasisymmetric homeomorphism of the circle admits a unique extension which is a (Formula presented.) -landslide of the hyperbolic plane. These extensions are quasiconformal.