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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)].
Reviewer: Andreas Cap (Wien)

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
Zbl 1276.53042
Full Text: DOI arXiv
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