Polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe

Tamburelli, Andrea

doi:10.1016/j.aim.2019.06.015

Cited by 11 publications

(21 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By definition, this means that the equivariant maximal embeddings associated to

(h, q)

and

(h^{'}, q^{'})

have the same holonomy representation and the same boundary at infinity. On the other hand, the same argument as in [, Lemma 4.2] shows that given a locally achronal curve

normalΓ

\partial_{\infty} {normalAdS}_{3}

the maximal surface bounding

normalΓ

is unique. This gives a contradiction because the pair

(h, q)

is uniquely determined by the embedding data of the maximal surface.

□

…”

Section: Parameterising Regular Anti‐de Sitter Structuresmentioning

confidence: 69%

“…If the induced metric is complete, the space‐like condition implies that, identifying

{\hat{A d S}}_{3}

with

D \times S^{1}

via

F

, the surface is the graph of a 2‐Lipschitz map [, Proposition 3.1] and its boundary at infinity

normalΓ

is a topological circle in

\partial_{\infty} {normalAdS}_{3}

[, Corollary 3.3]. We also have control on the causal geometry of the curve at infinity.…”

Section: Background Materialsmentioning

confidence: 99%

“…The frame field can be written explicitly in the special case when

q

is a constant holomorphic quadratic differential, and the associated maximal surface in

A d S_{3}

appears in the literature as the horospherical surface .…”

Section: Description Of the Domain Of Dependencementioning

confidence: 99%

“…Since the way of completing the limit set of the holonomy is determined by the decoration, this implies that the boundary at infinity of the maximal surfaces with embedding data

I_{n} = 2 e^{2 v_{n}} h_{n}

and

I I_{n} = 2 R e false(q_{n} false)

are converging in the Hausdorff topology. The techniques introduced in [, Section 4.1] show that the maximal surfaces bounding such curves are actually converging smoothly on compact sets. Hence their embedding data converge and

normalΨ

is proper.…”

Section: Parameterising Regular Anti‐de Sitter Structuresmentioning

confidence: 99%

“…Passing to the universal cover, this means that we need to determine the behaviour of the left and right Gauss maps along sequences that converge to a point on the boundary at infinity of the equivariant maximal surface

\overset{S}{true}

lying on a light‐like segment. From equation ,

\overset{S}{true}

is asymptotic to an isometric copy of the model horospherical surface in a neighbourhood of the puncture, thus

G_{l}

and

G_{r}

can be approximated by the Gauss map of the horospherical surface, which has been studied in [, Section 5]. In order to recall that result, let us first introduce some notation.…”

Section: Application To Minimal Lagrangian Mapsmentioning

confidence: 99%

See 4 more Smart Citations

Regular globally hyperbolic maximal anti‐de Sitter structures

Tamburelli

2020

Journal of Topology

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…By definition, this means that the equivariant maximal embeddings associated to

(h, q)

and

(h^{'}, q^{'})

have the same holonomy representation and the same boundary at infinity. On the other hand, the same argument as in [, Lemma 4.2] shows that given a locally achronal curve

normalΓ

\partial_{\infty} {normalAdS}_{3}

the maximal surface bounding

normalΓ

is unique. This gives a contradiction because the pair

(h, q)

is uniquely determined by the embedding data of the maximal surface.

□

…”

Section: Parameterising Regular Anti‐de Sitter Structuresmentioning

confidence: 69%

“…If the induced metric is complete, the space‐like condition implies that, identifying

{\hat{A d S}}_{3}

with

D \times S^{1}

via

F

, the surface is the graph of a 2‐Lipschitz map [, Proposition 3.1] and its boundary at infinity

normalΓ

is a topological circle in

\partial_{\infty} {normalAdS}_{3}

[, Corollary 3.3]. We also have control on the causal geometry of the curve at infinity.…”

Section: Background Materialsmentioning

confidence: 99%

“…The frame field can be written explicitly in the special case when

q

is a constant holomorphic quadratic differential, and the associated maximal surface in

A d S_{3}

appears in the literature as the horospherical surface .…”

Section: Description Of the Domain Of Dependencementioning

confidence: 99%

“…Since the way of completing the limit set of the holonomy is determined by the decoration, this implies that the boundary at infinity of the maximal surfaces with embedding data

I_{n} = 2 e^{2 v_{n}} h_{n}

and

I I_{n} = 2 R e false(q_{n} false)

normalΨ

is proper.…”

Section: Parameterising Regular Anti‐de Sitter Structuresmentioning

confidence: 99%

\overset{S}{true}

lying on a light‐like segment. From equation ,

\overset{S}{true}

is asymptotic to an isometric copy of the model horospherical surface in a neighbourhood of the puncture, thus

G_{l}

and

G_{r}

can be approximated by the Gauss map of the horospherical surface, which has been studied in [, Section 5]. In order to recall that result, let us first introduce some notation.…”

Section: Application To Minimal Lagrangian Mapsmentioning

confidence: 99%

See 3 more Smart Citations

Regular globally hyperbolic maximal anti‐de Sitter structures

Tamburelli

2020

Journal of Topology

Self Cite

View full text Add to dashboard Cite

show abstract

Anti-de Sitter Geometry and Teichmüller Theory

Bonsante

Seppi

2020

In the Tradition of Thurston

View full text Add to dashboard Cite

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.

show abstract

Fenchel–Nielsen coordinates on the augmented moduli space of anti-de Sitter structures

Tamburelli

2020

Math. Z.

View full text Add to dashboard Cite

Polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe

Cited by 11 publications

References 25 publications

Regular globally hyperbolic maximal anti‐de Sitter structures

Regular globally hyperbolic maximal anti‐de Sitter structures

Anti-de Sitter Geometry and Teichmüller Theory

Fenchel–Nielsen coordinates on the augmented moduli space of anti-de Sitter structures

Contact Info

Product

Resources

About