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**Explicit formulas for GJMS-operators and \(Q\)-curvatures.**
*(English)*
Zbl 1293.53049

The GJMS-operators are conformally covariant powers of the Laplacian, which generalize the Yamabe operator. Together with a closely related family of curvature invariants known as Branson’s \(Q\)-curvature, they have received a lot of attention during the last years, both in conformal differential geometry and in analysis. In spite of a relatively simple construction via the Fefferman-Graham ambient metric, it turned out that it is notoriously difficult to find explicit formulae for the higher-order GJMS-operators.

This article is part of a big project of the author in which he uses his theory of residue families to obtain recursive formulae for the GJMS-operators. Moreover, based on so-called Poincaré-Einstein metrics (which are closely related to the ambient metric) and the associated renormalized volume, the author introduces a family of second-order operators depending naturally on a Riemannian metric, from which the GJMS-operators can be obtained as linear combinations of iterated compositions. Determining the coefficients for these expressions requires difficult combinatorics. The family of these operators is in a precise way related to the family of all GJMS-operators which then leads to recursive expressions for the latter. The \(Q\)-curvatures can be treated in a similar way. The technical difficulties which have to be overcome to carry out this program are quite formidable.

It should be mentioned that a way to deduce the formulae obtained in this article directly from the original construction of the GJMS-operators from the ambient metric has been recently found by C. Fefferman and C. R. Graham [J. Am. Math. Soc. 26, No. 4, 1191–1207 (2013; Zbl 1276.53042)].

This article is part of a big project of the author in which he uses his theory of residue families to obtain recursive formulae for the GJMS-operators. Moreover, based on so-called Poincaré-Einstein metrics (which are closely related to the ambient metric) and the associated renormalized volume, the author introduces a family of second-order operators depending naturally on a Riemannian metric, from which the GJMS-operators can be obtained as linear combinations of iterated compositions. Determining the coefficients for these expressions requires difficult combinatorics. The family of these operators is in a precise way related to the family of all GJMS-operators which then leads to recursive expressions for the latter. The \(Q\)-curvatures can be treated in a similar way. The technical difficulties which have to be overcome to carry out this program are quite formidable.

It should be mentioned that a way to deduce the formulae obtained in this article directly from the original construction of the GJMS-operators from the ambient metric has been recently found by C. Fefferman and C. R. Graham [J. Am. Math. Soc. 26, No. 4, 1191–1207 (2013; Zbl 1276.53042)].

Reviewer: Andreas Cap (Wien)

### MSC:

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

53A30 | Conformal differential geometry (MSC2010) |

05A15 | Exact enumeration problems, generating functions |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

35Q76 | Einstein equations |

53A55 | Differential invariants (local theory), geometric objects |

53B20 | Local Riemannian geometry |

### Citations:

Zbl 1276.53042
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\textit{A. Juhl}, Geom. Funct. Anal. 23, No. 4, 1278--1370 (2013; Zbl 1293.53049)

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