Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space

We prove that any weakly acausal curve $\mathrm{\Gamma }$ 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 $K\in (-\mathrm{\infty },-1)$. The curve $\mathrm{\Gamma }$ 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.