Maximal surfaces in anti-de Sitter 3-manifolds with particles

Toulisse, Jérémy

doi:10.5802/aif.3040

Cited by 14 publications

(20 citation statements)

References 23 publications

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“…More precisely, for n = 1, maximal representations are exactly the holonomies of globally hyperbolic Cauchy-compact anti-de Sitter 3manifolds (see [Mes07]). In this particular case, our theorem is due to Barbot, Béguin and Zeghib [BBZ03] (see also [Tou16] for the case with cone singularities).…”

Section: Introductionmentioning

confidence: 86%

The geometry of maximal representations of surface groups into SO0(2,n)

Collier¹,

Tholozan²,

Toulisse³

2019

Duke Math. J.

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In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest.We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

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Section: Introductionmentioning

confidence: 86%

The geometry of maximal representations of surface groups into SO0(2,n)

Collier¹,

Tholozan²,

Toulisse³

2019

Duke Math. J.

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“…By equations (44) and (41) it follows that the ratio cos(t + ξ(t)/2) t is bounded. Thus using equation (42) we deduce that lim t→0 σ 11 (t) = 0.…”

Section: The Transformation δ(T) : Z → E ϕ(T) Z Sends P(t) To Q(t) Anmentioning

confidence: 94%

“…In [1,14,39] the case in which S is a maximal surface, and correspondingly Φ is a minimal Lagrangian diffeomorphism, has been extensively studied. A similar construction has also been applied to surfaces with certain singularities in [27,41]. More recently, progresses have been made on the problem of characterizing the area-preserving maps Φ obtained by means of this construction, satisfying certain equivariance properties, see [5,17,38].…”

Section: A Representation Formula For Convex Surfacesmentioning

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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“…Remark The same picture holds for GHMC anti‐de Sitter manifolds with particles: their deformation space is parameterised by a pair of hyperbolic metrics on

normalΣ

with cone singularities of angle

θ

less than

π

at the punctures, or equivalently by the vector bundle over

{scriptT}_{θ} false(normalΣ false)

of meromorphic quadratic differentials on

normalΣ

with at most simple poles at the punctures. See also .…”

Section: Background Materialsmentioning

confidence: 99%

Regular globally hyperbolic maximal anti‐de Sitter structures

Tamburelli

2020

Journal of Topology

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Let normalΣ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti‐de Sitter structures on Σ×R and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of normalΣ and as the bundle over the Teichmüller space of normalΣ whose fibre consists of meromorphic quadratic differentials with poles of order at most 2 at the punctures.

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Maximal surfaces in anti-de Sitter 3-manifolds with particles

Cited by 14 publications

References 23 publications

The geometry of maximal representations of surface groups into SO0(2,n)

The geometry of maximal representations of surface groups into SO0(2,n)

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

Regular globally hyperbolic maximal anti‐de Sitter structures

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