The ambient metric.

*(English)*Zbl 1243.53004
Annals of Mathematics Studies 178. Princeton, NJ: Princeton University Press (ISBN 978-0-691-15314-8/pbk; 978-0-691-15313-1/hbk; 978-1-400-84058-8/ebook). v, 113 p. (2012).

Conformal geometry studies pseudo-Riemannian metrics up to rescaling by positive smooth functions. While this is a classical topic in differential geometry and has been intensively studied since the beginning of the 20th century, there was surprisingly little progress on some fundamental questions of conformal geometry for a long time. For example, apart from the Weyl curvature, conformal invariants were only known in special dimensions until the 1980s. This changed with the introduction of the ambient metric associated to a conformal structure by the two authors of this book in 1985. A conformal class on a smooth manifold \(M\) defines a ray sub-bundle of \(S^2T^*M\) which carries a tautological degenerate metric. Forming the product with a small interval, one views the ray bundle as a hypersurface in an “ambient space” and tries to extend the degenerate metric to a Ricci flat pseudo-Riemannian metric with suitable homogeneity in fiber directions off the initial surface. It turns out that in odd dimensions there is an essentially unique infinite order power series solution to this problem, while in even dimensions there is a formal obstruction to solving the problem at order half the dimension, but up to this order, the solution is again essentially unique. The obstruction can be expressed by the so-called obstruction tensor, which is naturally associated to an even-dimensional conformal structure. Forming Riemannian invariants of the ambient metric (which is easy) one can then obtain conformal invariants.

In the original 1985 article [C. Fefferman and C. R. Graham, Astérisque, No. Hors Sér. 1985, 95–116 (1985; Zbl 0602.53007)] by the authors of the book, many results on the ambient metric and its equivalent description in terms of the so-called Poincaré metric were stated without complete proofs. One of the aims of the book under review is to provide these complete proofs. An important application of the ambient metric has been the construction of conformally invariant powers of the Laplacian, usually referred to as GJMS-operators, in [C. R. Graham, R. Jenne, L. J. Mason and G. A. J. Sparling, J. Lond. Math. Soc., II. Ser. 46, No. 3, 557–565 (1992; Zbl 0726.53010)] which led to the introduction of \(Q\)-curvature. These objects are not only intensively studied by geometers but have also found a lot of interest in analysis due to their relation to higher order elliptic PDE and scattering theory. For all these applications, the book under review provides a fundamental reference work.

Let us describe the contents of the book in a bit more detail: After an introduction in Chapter 1, Chapter 2 explains the basic setup, defines ambient metrics and introduces some initial normalizations. Chapter 3 discusses the formal solution of the Ricci-flatness equation and the main existence and uniqueness theorems for ambient metrics. Chapter 4 discusses the equivalent reformulation of an ambient metric as a Poincaré metric, which is obtained by factoring the fiber direction using homogeneity of the ambient metric. Then, the initial conformal structure is realized (again formally and up to a certain order in even dimensions) as the conformal infinity of a negative Einstein metric. Chapter 5 specializes the discussion of the Poincaré metric to dimension four, in which the additional concept of self-duality is available, to prove a formal power series version of a result of LeBrun on existence of self-dual Einstein metrics with given conformal infinity. The next chapter discusses the special cases of conformal classes containing a flat metric or an Einstein metric. It is shown that the obstruction tensor vanishes and distinguished infinite order solutions are obtained in both cases. The last two chapters study the question whether all conformal invariants can be obtained via the ambient metric. Chapter 8 proves the jet isomorphism theorem relating the jets of conformal structures modulo diffeomorphisms to jets of the ambient curvature. Together with parabolic invariant theory as developed in [T. N. Bailey, M. G. Eastwood and C. R. Graham, Ann. Math. (2) 139, No. 3, 491–552 (1994; Zbl 0814.53017)], this is then applied in Chapter 9 to prove that in odd dimensions all conformal invariants can be obtained from the ambient metric, while in even dimensions this holds up to a critical weight.

In the original 1985 article [C. Fefferman and C. R. Graham, Astérisque, No. Hors Sér. 1985, 95–116 (1985; Zbl 0602.53007)] by the authors of the book, many results on the ambient metric and its equivalent description in terms of the so-called Poincaré metric were stated without complete proofs. One of the aims of the book under review is to provide these complete proofs. An important application of the ambient metric has been the construction of conformally invariant powers of the Laplacian, usually referred to as GJMS-operators, in [C. R. Graham, R. Jenne, L. J. Mason and G. A. J. Sparling, J. Lond. Math. Soc., II. Ser. 46, No. 3, 557–565 (1992; Zbl 0726.53010)] which led to the introduction of \(Q\)-curvature. These objects are not only intensively studied by geometers but have also found a lot of interest in analysis due to their relation to higher order elliptic PDE and scattering theory. For all these applications, the book under review provides a fundamental reference work.

Let us describe the contents of the book in a bit more detail: After an introduction in Chapter 1, Chapter 2 explains the basic setup, defines ambient metrics and introduces some initial normalizations. Chapter 3 discusses the formal solution of the Ricci-flatness equation and the main existence and uniqueness theorems for ambient metrics. Chapter 4 discusses the equivalent reformulation of an ambient metric as a Poincaré metric, which is obtained by factoring the fiber direction using homogeneity of the ambient metric. Then, the initial conformal structure is realized (again formally and up to a certain order in even dimensions) as the conformal infinity of a negative Einstein metric. Chapter 5 specializes the discussion of the Poincaré metric to dimension four, in which the additional concept of self-duality is available, to prove a formal power series version of a result of LeBrun on existence of self-dual Einstein metrics with given conformal infinity. The next chapter discusses the special cases of conformal classes containing a flat metric or an Einstein metric. It is shown that the obstruction tensor vanishes and distinguished infinite order solutions are obtained in both cases. The last two chapters study the question whether all conformal invariants can be obtained via the ambient metric. Chapter 8 proves the jet isomorphism theorem relating the jets of conformal structures modulo diffeomorphisms to jets of the ambient curvature. Together with parabolic invariant theory as developed in [T. N. Bailey, M. G. Eastwood and C. R. Graham, Ann. Math. (2) 139, No. 3, 491–552 (1994; Zbl 0814.53017)], this is then applied in Chapter 9 to prove that in odd dimensions all conformal invariants can be obtained from the ambient metric, while in even dimensions this holds up to a critical weight.

Reviewer: Andreas Cap (Wien)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53A30 | Conformal differential geometry (MSC2010) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

35Q76 | Einstein equations |