On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

Chu, Yawei; Zhai, Shujie

doi:10.1016/j.difgeo.2011.08.002

Cited by 6 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…in the induced metric of the hypersurface. Thus our ansatz (73) implies that the integral lines of the normalised vector-field perpendicular to the direction m a have to be geodesics which foliate the hypersurface. So far, we are explicitly talking about the geodesic equation with respect to the induced metric, but as (73) implies K ab l a l b = 0 these curves also have vanishing extrinsic curvature in the normal direction to the hypersurface.…”

Section: Solution Method Totally Geodesic Foliationsmentioning

confidence: 99%

“…To summarise, we have shown that in AdS 3 , the constant curvature surfaces that we are trying to find as solutions of (71) (which implies (73) in three bulk dimensions) are foliated by curves that are geodesics both with respect to the ambient space and the induced metric. This is just the AdS equivalent of the well known statement in R 3 that all developable surfaces (i.e.…”

Section: Solution Method Totally Geodesic Foliationsmentioning

confidence: 99%

“…Of course, the problem of constant curvature surfaces embedded into maximally symmetric ambient spaces is well studied in the mathematical literature, see e.g. [68][69][70][71][72][73][74][75] and references therein for interesting results. However due to differences in notation and nomenclature in the mathematical literature, in the following sections we will spell out the most relevant facts for our case in our own language and try to give them a physical interpretation from the perspective of holography.…”

Section: Equations Of Motionmentioning

confidence: 99%

“…This satisfies(71). For this solution the sign of K switches in between points on the surface, hence k in(73) can not be globally absorbed into the normalisation of m. Note we are working in the Euclidean case, so the norm of m has to be positive. Right: Example of (88) for t bd y [φ] =1 8 cos(4φ) −…”

mentioning

confidence: 99%

See 3 more Smart Citations

Cost of holographic path integrals

et al. 2023

View full text Add to dashboard Cite

We consider proposals for the cost of holographic path integrals. Gravitational path integrals within finite radial cutoff surfaces have a precise map to path integrals in T\overline{T}TT¯ deformed holographic CFTs. In Nielsen’s geometric formulation cost is the length of a not-necessarily-geodesic path in a metric space of operators. Our cost proposals differ from holographic state complexity proposals in that (1) the boundary dual is cost, a quantity that can be 'optimised' to state complexity, (2) the set of proposals is large: all functions on all bulk subregions of any co-dimension which satisfy the physical properties of cost, and (3) the proposals are by construction UV-finite. The optimal path integral that prepares a given state is that with minimal cost, and cost proposals which reduce to the CV and CV2.0 complexity conjectures when the path integral is optimised are found, while bounded cost proposals based on gravitational action are not found. Related to our analysis of gravitational action-based proposals, we study bulk hypersurfaces with a constant intrinsic curvature of a specific value and give a Lorentzian version of the Gauss-Bonnet theorem valid in the presence of conical singularities.

show abstract

Section: Solution Method Totally Geodesic Foliationsmentioning

confidence: 99%

Section: Solution Method Totally Geodesic Foliationsmentioning

confidence: 99%

Section: Equations Of Motionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Cost of holographic path integrals

et al. 2023

View full text Add to dashboard Cite

show abstract

Section: Equations Of Motionmentioning

confidence: 99%

Cost of holographic path integrals

Chandra¹,

Boer²,

Flory³

et al. 2022

Preprint

View full text Add to dashboard Cite

We consider proposals for the cost of holographic path integrals. Gravitational path integrals within finite radial cutoff surfaces have a precise map to path integrals in T T deformed holographic CFTs. In Nielsen's geometric formulation cost is the length of a not-necessarily-geodesic path in a metric space of operators. Our cost proposals differ from holographic state complexity proposals in that (1) the boundary dual is cost, a quantity that can be 'optimised' to state complexity, (2) the set of proposals is large: all functions on all bulk subregions of any co-dimension which satisfy the physical properties of cost, and (3) the proposals are by construction UV-finite. The optimal path integral that prepares a given state is that with minimal cost, and cost proposals which reduce to the CV and CV2.0 complexity conjectures when the path integral is optimised are found, while bounded cost proposals based on gravitational action are not found. Related to our analysis of gravitational action-based proposals, we study bulk hypersurfaces with a constant intrinsic curvature of a specific value and give a Lorentzian version of the Gauss-Bonnet theorem valid in the presence of conical singularities.

show abstract

Ricci curvature of spacelike hypersurfaces in de Sitter space

Zhu

2013

Differential Geometry and its Applications

View full text Add to dashboard Cite

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

Cited by 6 publications

References 19 publications

Cost of holographic path integrals

Cost of holographic path integrals

Cost of holographic path integrals

Ricci curvature of spacelike hypersurfaces in de Sitter space

Contact Info

Product

Resources

About