Area-preserving diffeomorphisms of the hyperbolic plane and K-surfaces in Anti-de Sitter space.
Résumé
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 qua-sisymmetric homeomorphism of the circle admits a unique extension which is a θ-landslide of the hyperbolic plane. These extensions are quasiconformal.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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