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/9780691153131.003.0005
- 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:
- 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/9780691153131.003.0007
- 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.
Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony
Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0020
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of ...
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This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.Less
This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.
Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony
Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0021
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated ...
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This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface is finite. The chapter then proves the finiteness of the entropy in the local setting near a singular point in any dimension, using a division of a neighborhood of a singular point into adapted cells. Next, the chapter estimates the modulus of continuity for the Poincaré metric along the leaves of the foliation, using notion of conformally (R,δ)-close maps. The estimate holds for foliations on manifolds of higher dimension.Less
This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface is finite. The chapter then proves the finiteness of the entropy in the local setting near a singular point in any dimension, using a division of a neighborhood of a singular point into adapted cells. Next, the chapter estimates the modulus of continuity for the Poincaré metric along the leaves of the foliation, using notion of conformally (R,δ)-close maps. The estimate holds for foliations on manifolds of higher dimension.
