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 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.Then σ Φ,b is an embedding of Ω ⊆ H 2 onto a convex surface in AdS 3 . If π l is the left projection of the image of σ Φ,b , then π l • σ Φ,b = id, and π r • σ Φ,b = Φ.In fact, given a tensor b satisfying equations (1)-(3), one can still define σ Φ,b by means of equations (5) and (6), and the image of the differential of σ Φ,b is orthogonal to the family of timelike lines {L x,Φ(x) }, but in general dσ Φ,b will not be injective. Actually, given b which satisfies equations (1)-(3), for every angle ρ, also the (1,1)-tensor R ρ • b satisfies equations (1) and (2), where R ρ denotes the counterclockwise rotation of angle ρ. Changing b by postcomposition with R ρ corresponds to changing the surface S to a parallel surface, moving along the timelike geodesics {L x,Φ(x) }. The condition tr b ∈ (−2, 2) (which in general is only satisfied for some choices of b) ensures that σ Φ,b is an embedding.We now restrict our attention to K-surfaces and θ-landslides. A θ-landslide is a diffeomorphism Φ for which there exists b satisfying the conditions of equations (1)-(3) and moreover its trace is constant. More precisely, tr b = 2 cos θ and tr Jb < 0for θ ∈ (0, π). It turns out that θ-landslides are precisely the maps associated to past-convex K-surfaces, for K = −1/cos 2 (θ/2). On the other hand, by means of the map defined in equations (5) and (6), one associates a K-surface to a θ-landslide. Changing b by −b in the defining equations (5) and (6) enables to pass from the K-surface to its dual surface, which is still a surface of constant curvature K * = −K/(K + 1). Hence a θ-landslide is also the map associated with a future-convex K * -surface. A special case of θ-landslides are minimal Lagrangian maps, for θ = π/2. In this case, we get two (−2)-surfaces, dual to one another. It is well known (see [14]) that a minimal Lagrangian map is associated to a maximal surface S 0 in AdS 3 , and that the two (−2)-surfaces are obtained as parallel surfaces at distance π/4 from S 0 . Since in this case tr b = 0, changing b by Jb, one has that Jb is self-adjoint for the hyperbolic metric, and the map σ Φ,J...

Constant mean curvature foliation of domains of dependence in 𝐴𝑑𝑆₃

“…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].…”

“…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.…”

“…In general, we will use the notation with a tilda to indicate the lift of an object to the universal cover. By Theorem 7.8 in [BS16], the sequence (Σ ± κ ) n converges to constant curvature surfacesΣ ± κ with boundary at infinityΓ. We denote with K ′ the domain…”

“…More recently, with the work of Bonsante and Schlenker [BS10] and of Bonsante and Seppi [BS16], Anti-de Sitter geometry turned out to be a useful tool to construct quasi-conformal extensions of quasi-symmetric homeomorphisms of the unit disc. Indeed, the graph of a quasi-symmetric map φ : S 1 → S 1 describes a curve (called quasi-circle) c φ on the boundary at infinity of the 3-dimensional Anti-de Sitter space AdS 3 .…”

“…Moreover, some remarkable properties of the quasi-conformal extension can be deduced from the geometry of the surface itself: for example, when the surface S is maximal (i.e. it has vanishing mean curvature) the quasi-conformal extension induced by S is minimal Lagrangian ( [BS10]); when S is a smooth convex κ-surface, the corresponding quasi-conformal extension is a landslide ( [BS16]); when the surface S is the past-convex boundary of the convex-hull of c φ , the quasi-symmetric homeomorphism φ is exactly the boundary map of the earthquake E 2λ : H 2 → H 2 , where λ is the pleating locus of S ( [Mess07]). …”