Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space

Abstract

We prove that any weakly acausal curve $\Gamma$ 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 $K\in (-\infty ,-1)$. The curve $\Gamma$ 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 $\theta \in (0,\pi )$, every quasisymmetric homeomorphism of the circle admits a unique extension which is a $\theta$-landslide of the hyperbolic plane. These extensions are quasiconformal.

47 pages, 18 figures. More details added to Remark 4.14, Remark 6.2 and Theorem 7.8 Step 2. Several references added and typos corrected. Final version. To appear in Journal of Topology

References in corpus (1)

• The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry

Cited by in corpus (4)

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• Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces
• On the maximal dilatation of quasiconformal minimal Lagrangian extensions
• The Anti-de Sitter proof of Thurston's earthquake theorem