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

• Published in: Journal of topology 2018-06, Vol.11 (2), p.420-468
• Main Authors: Bonsante, Francesco, Seppi, Andrea
• Format: Article
• Language: English
• Subjects: 30F60; 32G15; 53C42 (secondary); 53C50 (primary); Mathematics
• ISSN: 1753-8416; 1753-8424; 1753-8424; 1753-8416
• DOI: 10.1112/topo.12058

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.