Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient ...
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This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric in n+1 dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.Less
This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric in n+1 dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0001
- Subject:
- Mathematics, Geometry / Topology
This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A ...
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This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.Less
This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter presents the full infinite-order formal theory for ambient metric forms, including the freedom at order n/2 in all dimensions and the precise description of the log terms when n ≤ 4 is ...
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This chapter presents the full infinite-order formal theory for ambient metric forms, including the freedom at order n/2 in all dimensions and the precise description of the log terms when n ≤ 4 is even. The description of the solutions with freedom at order n/2 and log terms extends and sharpens results of Kichenassamy [K]. Convergence of the formal series determined by singular nonlinear initial value problems of this type has been considered by several authors; these results imply that the formal series converge if the data are real-analytic.Less
This chapter presents the full infinite-order formal theory for ambient metric forms, including the freedom at order n/2 in all dimensions and the precise description of the log terms when n ≤ 4 is even. The description of the solutions with freedom at order n/2 and log terms extends and sharpens results of Kichenassamy [K]. Convergence of the formal series determined by singular nonlinear initial value problems of this type has been considered by several authors; these results imply that the formal series converge if the data are real-analytic.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0004
- Subject:
- Mathematics, Geometry / Topology
This chapter considers the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). It shows that even Poincaré metrics are in one-to-one correspondence with straight ambient ...
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This chapter considers the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). It shows that even Poincaré metrics are in one-to-one correspondence with straight ambient metrics, if both are in normal form. Thus, the formal theory for Poincaré metrics is a consequence of the results of Chapter 3. The derivation of a Poincaré metric from an ambient metric was described in [FG], and the inverse construction of an ambient metric as the cone metric over a Poincaré metric was given in § 5 of [GrL].Less
This chapter considers the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). It shows that even Poincaré metrics are in one-to-one correspondence with straight ambient metrics, if both are in normal form. Thus, the formal theory for Poincaré metrics is a consequence of the results of Chapter 3. The derivation of a Poincaré metric from an ambient metric was described in [FG], and the inverse construction of an ambient metric as the cone metric over a Poincaré metric was given in § 5 of [GrL].
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0007
- Subject:
- Mathematics, Geometry / Topology
This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even ...
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This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even dimensional conformal structures of these types. It shows that for these special conformal classes, there is a way to uniquely specify the formally undetermined term at order n/2 in an invariant way and thereby obtain a unique ambient metric up to terms vanishing to infinite order and up to diffeomorphism, just like in odd dimensions. It derives a formula of Skenderis and Solodukhin [SS] for the ambient or Poincaré metric in the locally conformally flat case which is in normal form relative to an arbitrary metric in the conformal class, and proves an elated unique continuation result for hyperbolic metrics in terms of data at conformal infinity. The case n = 2 is special for all of these considerations. The chapter also derives the form of the GJMS operators for an Einstein metric.Less
This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even dimensional conformal structures of these types. It shows that for these special conformal classes, there is a way to uniquely specify the formally undetermined term at order n/2 in an invariant way and thereby obtain a unique ambient metric up to terms vanishing to infinite order and up to diffeomorphism, just like in odd dimensions. It derives a formula of Skenderis and Solodukhin [SS] for the ambient or Poincaré metric in the locally conformally flat case which is in normal form relative to an arbitrary metric in the conformal class, and proves an elated unique continuation result for hyperbolic metrics in terms of data at conformal infinity. The case n = 2 is special for all of these considerations. The chapter also derives the form of the GJMS operators for an Einstein metric.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0003
- Subject:
- Mathematics, Geometry / Topology
This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics ...
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This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ > 0 or ρ < 0 involving positive powers of ¦ ρ r ρ and log ¦ ρ r ρ, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].Less
This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ > 0 or ρ < 0 involving positive powers of ¦ ρ r ρ and log ¦ ρ r ρ, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0006
- Subject:
- Mathematics, Geometry / Topology
This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form ...
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This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.Less
This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0003
- Subject:
- Mathematics, Geometry / Topology
This chapter proves (1.17) when the worst terms in P(g) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need ...
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This chapter proves (1.17) when the worst terms in P(g) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need both a local conformal invariant W(g) and a divergence divᵢTⁱ(g) to prove (1.17). One obvious difficulty is how, upon inspection of P(g)subscript worst-piece, to separate the piece that must be cancelled out by a local conformal invariant from the piece that is cancelled out by a divergence. In a first step, we prove that we can first explicitly construct a local conformal invariant and a divergence and subtract them from P(g)subscript worst-piece, to be left with a new worst piece, which has some additional algebraic properties. In a second step, we show that this new worst piece can be cancelled out by subtracting a divergence.Less
This chapter proves (1.17) when the worst terms in P(g) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need both a local conformal invariant W(g) and a divergence divᵢTⁱ(g) to prove (1.17). One obvious difficulty is how, upon inspection of P(g)subscript worst-piece, to separate the piece that must be cancelled out by a local conformal invariant from the piece that is cancelled out by a divergence. In a first step, we prove that we can first explicitly construct a local conformal invariant and a divergence and subtract them from P(g)subscript worst-piece, to be left with a new worst piece, which has some additional algebraic properties. In a second step, we show that this new worst piece can be cancelled out by subtracting a divergence.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0005
- Subject:
- Mathematics, Geometry / Topology
As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic ...
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As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic self-dual Einstein metric in dimension 4 defined near the boundary with prescribed real-analytic conformal infinity.Less
As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic self-dual Einstein metric in dimension 4 defined near the boundary with prescribed real-analytic conformal infinity.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0008
- Subject:
- Mathematics, Geometry / Topology
A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered ...
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A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors. This chapter proves an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a “conformal normal form” for metrics about a point.Less
A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors. This chapter proves an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a “conformal normal form” for metrics about a point.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0009
- Subject:
- Mathematics, Geometry / Topology
This chapter shows how to derive a characterization of scalar invariants of conformal structures by reduction to the relevant results of [BEGr]. In [FG], the authors conjectured that when n is odd, ...
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This chapter shows how to derive a characterization of scalar invariants of conformal structures by reduction to the relevant results of [BEGr]. In [FG], the authors conjectured that when n is odd, all scalar conformal invariants arise as Weyl invariants constructed from the ambient metric. The second main goal of this book is to prove this together with an analogous result when n is even. These results are contained in Theorems 9.2, 9.3, and 9.4. The parabolic invariant theory needed to prove these results was developed in [BEGr], including the observation of the existence of exceptional invariants. But substantial work is required to reduce the theorems in the chapter to the results of [BEGr]. To understand this, it briefly reviews how Weyl's characterization of scalar Riemannian invariants is proved.Less
This chapter shows how to derive a characterization of scalar invariants of conformal structures by reduction to the relevant results of [BEGr]. In [FG], the authors conjectured that when n is odd, all scalar conformal invariants arise as Weyl invariants constructed from the ambient metric. The second main goal of this book is to prove this together with an analogous result when n is even. These results are contained in Theorems 9.2, 9.3, and 9.4. The parabolic invariant theory needed to prove these results was developed in [BEGr], including the observation of the existence of exceptional invariants. But substantial work is required to reduce the theorems in the chapter to the results of [BEGr]. To understand this, it briefly reviews how Weyl's characterization of scalar Riemannian invariants is proved.
