A cyclic extension of the earthquake flow I

Abstract. We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d + 3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP 2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmüller space.

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“…In [BMS12] it is shown that this landslide flow is Hamiltonian for the product of Weil-Petersson symplectic structures.…”

Section: Complements and Conjecturesmentioning

confidence: 99%

Polynomial cubic differentials and convex polygons in the projective plane

2015

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“…From the proof Proposition 5.5 we have actually shown that, under the assumption Φ(x) = x and that dΦ x = −b, if dσ Φ,b is non-singular at x, then under the identification T id Isom(H 2 ) ∼ = so(2, 1) ∼ = R 2,1 , the future unit normal vector at σ Φ,b (x) = id is N (x) = 2Λ(x). In fact, N (x) = 2Λ(x) is orthogonal to the surface at σ Φ,b (x) by the above observation, and is unit, since the AdS metric at id is identified to 1/4 the Minkowski product, see equation (13).…”

Section: A Representation Formula For Convex Surfacesmentioning

confidence: 91%

“…Let us check the factor in equation (13). By (12) for any x, y ∈ R 2,1 we have ad(Λ(x))(Λ(y)) = Λ(Λ(x)y), that is,…”

Section: Two Useful Modelsmentioning

confidence: 99%

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

Bonsante

Seppi

2018

Journal of Topology

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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...

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“…Smooth grafting on hyperbolic surfaces with cone singularities. Constant Gauss curvature surfaces in hyperbolic ends are related to the "smooth grafting" map SGr : (0, 1)×T ×T → CP, see [7,Section 1.2]. The properties of K-surfaces in hyperbolic ends with particles as described here show that this "smooth grafting" map is still well-defined on hyperbolic surfaces with cone singularities of angles less than π, as a map SGr θ from (0, 1) × T Σ,θ × T Σ,θ to CP θ .…”

Section: Introductionmentioning

confidence: 99%

“…This implies that for all r ∈ (0, 1), the map SGr θ (r, ·, ·) is a homeomorphism from T Σ,θ × T Σ,θ to CP θ . We do not elaborate on this point here, since it follows from the same arguments as in the non-singular case, see [7]. The relations among all the spaces we consider throughout this paper are presented in Figure 1, which is a combination of Figure 2 1.7.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic ends with particles and grafting on singular surfaces

Chen

Schlenker

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by "smooth grafting". Theorem 1.5. Let M ∈ HE θ be a non-degenerate hyperbolic end with particles, and let M d ∈ DS θ be the dual future-complete convex GHM de Sitter spacetime with particles. Given a closed, strictly concave surface S ⊂ M , there is a unique strictly future-convex spacelike surface S d and a unique diffeomorphism u : S → S d such that u * I d = III and u * III d = I, where I, III are the induced metric and third fundamental form on S, and I d and III d are the induced metric and third fundamental form on S d .Conversely, given any space-like, strictly future-convex S d surface in M d , there is a unique strictly concave surface S in M such that S d is the dual of S in the sense of Theorem 1.5. Proposition 1.6. Let S be a strictly concave surface in M , and let S d be the dual surface in M d . Then S has constant curvature K ∈ (−1, 0) if and only if S d has constant curvature K d = K/(K + 1) ∈ (−∞, 0).

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A cyclic extension of the earthquake flow I

Cited by 30 publications

References 38 publications

Polynomial cubic differentials and convex polygons in the projective plane

Polynomial cubic differentials and convex polygons in the projective plane

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

Hyperbolic ends with particles and grafting on singular surfaces

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