Constant mean curvature foliation of domains of dependence in 𝐴𝑑𝑆₃

Tamburelli, Andrea

doi:10.1090/tran/7295

Cited by 9 publications

(9 citation statements)

References 9 publications

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“…Let

M

be a GHMC anti‐de Sitter manifold with holonomy

ρ = (ρ_{l}, ρ_{r})

and let

S

be the unique maximal surface embedded in

M

. Lifting to the universal cover, the Gauss map

G : \tilde{S} \to {double-struckH}^{2} \times {double-struckH}^{2}

provides a pair of

(ρ_{r}, ρ_{l})

‐equivariant harmonic maps with Hopf differentials

\pm i q

, where

R e (q)

is the second fundamental form of

S

. Denoting with

π_{l}

and

π_{r}

the projections onto the left and right factor, the metrics

{(G \circ π_{l})}^{*} g_{H^{2}} and {(G \circ π_{r})}^{*} g_{H^{2}}

descend to hyperbolic metrics

h_{l}

and

h_{r}

S

with holonomy

ρ_{l}

and

ρ_{r}

, respectively.…”

Section: Background Materialsmentioning

confidence: 99%

See 1 more Smart Citation

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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“…Let

M

be a GHMC anti‐de Sitter manifold with holonomy

ρ = (ρ_{l}, ρ_{r})

and let

S

be the unique maximal surface embedded in

M

. Lifting to the universal cover, the Gauss map

G : \tilde{S} \to {double-struckH}^{2} \times {double-struckH}^{2}

provides a pair of

(ρ_{r}, ρ_{l})

‐equivariant harmonic maps with Hopf differentials

\pm i q

, where

R e (q)

is the second fundamental form of

S

. Denoting with

π_{l}

and

π_{r}

the projections onto the left and right factor, the metrics

{(G \circ π_{l})}^{*} g_{H^{2}} and {(G \circ π_{r})}^{*} g_{H^{2}}

descend to hyperbolic metrics

h_{l}

and

h_{r}

S

with holonomy

ρ_{l}

and

ρ_{r}

, respectively.…”

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%

Regular globally hyperbolic maximal anti‐de Sitter structures

Tamburelli

2020

Journal of Topology

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“…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

{AdS}_{3}

appears in the literature as the horospherical surface [7, 19]. See also [21] and [23].…”

Section: Description Of the Boundary At Infinitymentioning

confidence: 99%

Wild globally hyperbolic maximal anti‐de Sitter structures

Tamburelli

2020

J. London Math. Soc.

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Let normalΣ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti‐de Sitter structures on Σ×R and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichmüller space of crowned hyperbolic surfaces and as the bundle over the Teichmüller space of normalΣ of meromorphic quadratic differentials with poles of order at least 3 at the punctures.

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“…The material covered here is classical and can be found for instance in [BBS11] and [BS10]. See also [Tam19a] for generalisations to constant mean curvature surfaces and [CTT17] for higher signature.…”

Section: From Polynomial Quadratic Differentials To Light-like Polygonsmentioning

confidence: 99%

“…The horospherical surface. The solution to Equation (1) can be written explicitly in the special case when q is a constant polynomial quadratic differential, and the associated maximal surface in AdS 3 appears in the literature as the horospherical surface ([BS10], [Sep16], [Tam19a]). Let us describe in detail the related frame field F 0 : C → SL(4, C) in this case.…”

Section: The Vectorsmentioning

confidence: 99%