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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.

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