2018
DOI: 10.1112/topo.12058
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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 are… Show more

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Cited by 18 publications
(28 citation statements)
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“…Namely, we construct this H-surface S H as a limit of H-surfaces (S H ) n , with asymptotic boundary Γ n , with the property that Γ n is the graph of a quasi-symmetric homeomorphism conjugating two cocompact Fuchsian groups and Γ n converges to Γ in the Hausdorff topology. The existence of this approximating sequence (S H ) n is a consequence of some results in [BBZ07] and [BS16].…”
Section: Introductionmentioning
confidence: 73%
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“…Namely, we construct this H-surface S H as a limit of H-surfaces (S H ) n , with asymptotic boundary Γ n , with the property that Γ n is the graph of a quasi-symmetric homeomorphism conjugating two cocompact Fuchsian groups and Γ n converges to Γ in the Hausdorff topology. The existence of this approximating sequence (S H ) n is a consequence of some results in [BBZ07] and [BS16].…”
Section: Introductionmentioning
confidence: 73%
“…For the other values of H we choose κ < −1 such that √ −1 − κ > |H|. We claim that the past-convex space-like surface S + κ with constant curvature κ, whose existence is proved in [BS16], must be in the future of Σ. If not, the surfaces Σ and S + κ would intersect transversely, but, since constant curvature surfaces provide a foliation of D(Λ) \ C(Λ), there would exist a κ ′ < κ such that the surface S κ ′ with constant Gauss curvature κ ′ is tangent to Σ at a point x.…”
Section: Existence Of a Cmc Foliationmentioning
confidence: 98%
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