Constant Gauss curvature foliations of AdS spacetimes with particles

Chen, Qiyu; Schlenker, Jean-Marc

doi:10.1090/tran/8018

Cited by 2 publications

(8 citation statements)

References 45 publications

(107 reference statements)

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“…The map φ K relates deeply to the minimal Lagrangian maps between two hyperbolic surfaces with cone singularities in T Σ,θ , which provides the embedding data to construct a hyperbolic end with particles. With the result in [11,Lemma 3.19], we have the following proposition.…”

Section: Proof Of Statement (2)mentioning

confidence: 72%

“…We prove this property by applying the Maximum Principle outside the singular locus and a specialized analysis near cone singularities. The idea is similar to that given in [11,Section 3.2] for the case of AdS manifold with particles. Indeed, this argument is applicable to two concave surfaces which behave "umbilically" (i.e.…”

Section: 2mentioning

confidence: 94%

“…Note that the hyperbolic metrics near the cone singularities throughout this paper are assumed to satisfy a regularity condition. This ensures the existence of harmonic maps from Riemann surfaces with marked points to hyperbolic surfaces with cone singularities at the marked points (see [15,Theorem 2]), so that we can relate minimal Lagrangian maps (see Definition 2.12) to harmonic maps, and apply the result in [11,Lemma 3.19] to show the continuity of the parametrization map φ K of HE θ (see Section 4.3). This regularity condition is defined by using the weighted Hölder spaces (see [15,Section 2.2] and [34, Definition 2.1]).…”

Section: Background Materialsmentioning

confidence: 99%

“…It can be checked as Lemma 3.3 in [11] that for any (τ, τ ′ ) ∈ T Σ,θ × T Σ,θ , if (h, h ′ ) and (h 1 , h ′ 1 ) are two normalized representatives of (τ, τ ′ ), then the hyperbolic end with particles associated to (h, h ′ ) and (h 1 , h ′ 1 ), as described in Lemma 5.1, are isotopic. Now we are ready to give the definition of the parametrization map φ K .…”

Section: Parametrization Of He θ By T σθ × T σθ In Terms Of Constanmentioning

confidence: 99%

“…The following theorem is an alternative version of the Maximum Principle Theorem (see e.g. [3, Lemma 2.3], [11,Theorem 3.10]) for the case of hyperbolic ends with particles.…”

Section: 2mentioning

confidence: 99%

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Hyperbolic ends with particles and grafting on singular surfaces

Chen

Schlenker

2019

Ann. Inst. H. Poincaré 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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Section: Proof Of Statement (2)mentioning

confidence: 72%

Section: 2mentioning

confidence: 94%

Section: Background Materialsmentioning

confidence: 99%

Section: Parametrization Of He θ By T σθ × T σθ In Terms Of Constanmentioning

confidence: 99%

“…The following theorem is an alternative version of the Maximum Principle Theorem (see e.g. [3, Lemma 2.3], [11,Theorem 3.10]) for the case of hyperbolic ends with particles.…”

Section: 2mentioning

confidence: 99%

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Hyperbolic ends with particles and grafting on singular surfaces

Chen

Schlenker

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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Anti-de Sitter Geometry and Teichmüller Theory

Bonsante

Seppi

2020

In the Tradition of Thurston

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The aim of these notes is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.

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Constant Gauss curvature foliations of AdS spacetimes with particles

Cited by 2 publications

References 45 publications

Hyperbolic ends with particles and grafting on singular surfaces

Hyperbolic ends with particles and grafting on singular surfaces

Anti-de Sitter Geometry and Teichmüller Theory

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