The Ambient Metric

By Charles Fefferman, C. Graham. 2007

542 citations recorded by the source corpus.

Chapter 1. Introduction 1 Chapter 2. Ambient Metrics 9 Chapter 3. Formal Theory 17 Chapter 4. Poincar'e Metrics 42 Chapter 5. Self-dual Poincar'e Metrics 50 Chapter 6. Conformal Curvature Tensors 56 Chapter 7. Conformally Flat and Conformally Einstein Spaces 67 Chapter 8. Jet Isomorphism 82 Chapter 9. Scalar Invariants 97 Bibliography 107 Index 113

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/855d1a688957240b12a821d46dc68aa042206561

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Citations in the corpus

Relative entropy and holography (2013, methodology): “As commented above, our general analysis will be formulated in the context of the Fefferman-Graham expansion of the asymptotic bulk solutions [35, 41].”
Comments on holographic complexity (2016, methodology): “According to the Fefferman-Graham (FG) construction [29, 30], any asymptotically AdS geometry can be described with the following metric:”
Fractional Laplacian in conformal geometry (2010, background): “ows to deﬁne Pk. Indeed, Ress=n/2+kS(s) = ckPk, ck = (−1) k[22kk!(k−1)!]−1. These are the conformally invariant powers of the Laplacian constructed by GrahamJenne-Mason-Sparling [15], Feﬀerman-Graham [11], that are local operators, and satisfy Pk = (−∆)k + lower order terms. In particular, when k= 1”
Quantum extremal islands made easy. Part I. Entanglement on the brane (2020, methodology): “The gravitational action can be determined as follows: First, one considers a FeffermanGraham (FG) expansion near the boundary of an asymptotic AdS geometry [51, 52].”
Holographic c-theorems in arbitrary dimensions (2010, background): “To properly understand these issues, it seems that one should analyse the asymptotic geometry of a general solution of our ‘unitary’ higher curvature theories using a Fefferman-Graham-like expansion [51], e.”
Juhl's formulae for GJMS operators and Q-curvatures (2012, background): “t2 +2tdtdρ+t2g ρ on R + × M× (−ǫ,ǫ), where t∈ R +, ρ∈ (−ǫ,ǫ), and gρ = hr with ρ= −r2/2. The asymptotic vanishing of Ric(g +)+ng + at r= 0 translates into asymptotic vanishing of Ric(eg) at ρ= 0. See [FG] for details. If f∈ C∞(M) and fe∈ C∞(M× (−ǫ,ǫ)) satisﬁes fe(x,0) = f, then the GJMS deﬁnition is”
Complexity of formation in holography (2016, methodology): “As described in appendix A, we do so by describing both geometries with the corresponding metric in the canonical form given by the Fefferman-Graham expansion [38, 39] and then we set the radial cutoff surface at z = δ in both cases.”
Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes (2013, methodology): “Using Fefferman-Graham coordinates, the resulting asymptotically AdS metric may 6As 5D Einstein gravity is a consistent truncation of 10D IIB supergravity on AdS5 × S5, all the 5D pure Einstein gravity solutions we discuss may be viewed as supergravity solutions which describe the dynamics of N = 4…”
On Fractional GJMS Operators (2014, methodology): “One can also define weighted analogues of the GJMS operators [19], which, analogous to the GJMS operators for Einstein manifolds [11, 15], admit factorizations for the smooth metric measure spaces .”
Holographic calculations of Rényi entropy (2011, background): “At this point, it is most systematic to work in the Fefferman-Graham (FG) gauge [44] — see also [45] — where the asymptotically AdS3 metric reads”
Some calculable contributions to holographic entanglement entropy (2011, methodology): “This asymptotic geometry can then be described with the Fefferman-Graham expansion, as follows [14] — see also [15]:”
Divergences in holographic complexity (2016, background): “es in the CV and CA calculations. We consider an asymptotically AdS d+1 solution of the vacuum Einstein equations. The metric in the asymptotic region can then be written in the Feerman-Graham gauge [19,20] ds2 = ‘2 z2 (dz2 + g (x;z)dxdx); (19) where the metric along the boundary directions has a po”
Comments on holographic entanglement entropy and RG flows (2012, methodology): “To further simplify this expression, we use the Fefferman-Graham expansion near the boundary [30].”
On holographic entanglement entropy and higher curvature gravity (2011, background): “This normalization ensures that integrated over a d-dimensional sphere: H Sdd d x p g Ed = 2. A general construction of the In can be found in [66, 67 ].”
Flat holography and Carrollian fluids (2018, methodology): “The mathematical foundations of holography are based on the existence of the Fefferman-Graham expansion for asymptotically anti-de Sitter Einstein spaces [1, 2].”
Extended obstruction tensors and renormalized volume coefficients (2008, methodology): “The definition and basic properties of the extended obstruction tensors are part of the theory of conformal curvature tensors developed in §6 of [ FG2 ].”
Fractional conformal Laplacians and fractional Yamabe problems (2010, background): “R[ ˆh] of the metric ˆ which undergoes the change Pˆh 1 w= n−2 4(n−1) R[hˆ w]w n+2 n−2 when taking conformal change of metrics, provided that (Xn+1,g+) is a Poincar´e-Einstein as established in [27], [19]. The conformal fractional Laplacians and fractional scalar curvatures should also be compared t”