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

Abstract

We prove that any weakly acausal curve Γ 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 ∈ ( − ∞ , − 1 ) . The curve Γ 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 θ ∈ ( 0 , π ) , every quasisymmetric homeomorphism of the circle admits a unique extension which is a θ-landslide of the hyperbolic plane. These extensions are quasiconformal.