Explicit formulas for GJMS-operators and \(Q\)-curvatures.

*(English)*Zbl 1293.53049The 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
PDF
BibTeX
XML
Cite

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

**OpenURL**

##### References:

[1] | E. Aubry and C. Guillarmou. Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity. Journal of the European Mathematical Society, 13 (2011), 911-957. arXiv:0808.0552 · Zbl 1225.53033 |

[2] | H. Baum and A. Juhl, Conformal Differential Geometry: Q-Curvature and Conformal Holonomy. Oberwolfach Seminars, 40 (2010). · Zbl 1189.53045 |

[3] | Branson, T.P, Differential operators canonically associated to a conformal structure, Mathematica Scandinavica,, 57, 293-345, (1985) · Zbl 0596.53009 |

[4] | Branson, T.P., Sharp inequalities, the functional determinant, and the complementary series, Transactions of the American Mathematical Society,, 347, 3671-3742, (1995) · Zbl 0848.58047 |

[5] | T.P. Branson. \(Q\)-curvature and spectral invariants. Rendiconti del Circolo Matematico di Palermo (2) Supplementary, 75 (2005), 11-55. · Zbl 1101.53016 |

[6] | T.P. Branson and A.R. Gover. Origins, applications and generalisations of the \(Q\)-curvature. Acta Applicandae Mathematicae, (2-3)102 (2008), 131-146. · Zbl 1145.53022 |

[7] | A. Čap, A.R. Gover and V. Souček. Conformally invariant operators via curved Casimirs: examples. Pure and Applied Mathematics Quarterly, (3)6 (2010), 693-714. arXiv:0808.1978 |

[8] | Z. Djadli, C. Guillarmou and M. Herzlich. Opérateurs géométriques, invariants conformes et variétés asymptotiquement hyperboliques. Panoramas et Synthèses, vol. 26. Société Mathématique de France (2008). · Zbl 1196.53002 |

[9] | C. Falk and A. Juhl. Universal recursive formulae for \(Q\)-curvature. Journal für die reine und angewandte Mathematik, 652 (2011), 113-163. arXiv:0804.2745v2 · Zbl 1218.53039 |

[10] | C. Feffermann and C.R. Graham. Conformal invariants. The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque, Numero Hors Serie, (1985), 95-116. |

[11] | C. Fefferman and C.R. Graham. The ambient metric. Annals of Mathematics Studies, 178 (2012). arXiv:0710.0919v2 · Zbl 1243.53004 |

[12] | A.R. Gover. Laplacian operators and \(Q\)-curvature on conformally Einstein manifolds. Mathematische Annalen. (2)336 (2006), 311-334. arXiv:math/0506037v3 · Zbl 1125.53032 |

[13] | A.R. Gover. Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature. Journal of Geometry and Physics, (2)60 (2010), 182-204. arXiv:0803.3510 · Zbl 1194.53038 |

[14] | A.R. Gover and K. Hirachi. Conformally invariant powers of the Laplacian-a complete nonexistence theorem. Journal of American Mathematical Society, (2)17 (2004), 389-405. arXiv:math/0304082v2 · Zbl 1066.53037 |

[15] | A.R. Gover and L. Peterson. Conformally invariant powers of the Laplacian, \(Q\)-curvature, and tractor calculus. Communications in Mathematical Physics, (2)235 (2003), 339-378. arXiv:math-ph/0201030v3 · Zbl 1022.58014 |

[16] | C.R. Graham, R. Jenne, L.J. Mason and G.A.J. Sparling. Conformally invariant powers of the Laplacian. I. Existence. Journal of the London Mathematical Society, (2)46 (1992), 557-565. · Zbl 0726.53010 |

[17] | C.R. Graham. Conformally invariant powers of the Laplacian. II. Nonexistence. Journal of the London Mathematical Society, (2)46 (1992), 566-576. · Zbl 0726.53011 |

[18] | C.R. Graham. Volume and area renormalizations for conformally compact Einstein metrics. Rendiconti del Circolo Matematico di Palermo (2) Supplementary, 63 (2000), 31-42. arXiv:math/9909042 · Zbl 0984.53020 |

[19] | C.R. Graham. Conformal powers of the Laplacian via stereographic projection. SIGMA Symmetry Integrability Geom. Methods Applications, 3 (2007), Paper 121, 4p. arXiv:0711.4798v2 |

[20] | C.R. Graham. Extended obstruction tensors and renormalized volume coefficients. Advances in Mathematics, (6)220 (2009), 1956-1985. arXiv:0810.4203 · Zbl 1161.53062 |

[21] | C.R. Graham and K. Hirachi. The ambient obstruction tensor and \(Q\)-curvature. AdS/CFT correspondence: Einstein metrics and their conformal boundaries. IRMA Lectures in Mathematics and Theoretical Physics, 8, (2005), 59-71. arXiv:math/0405068 · Zbl 1074.53027 |

[22] | C.R. Graham and A. Juhl. Holographic formula for \(Q\)-curvature. Advances in Mathematics, (2)216 (2007), 841-853. arXiv:0704.1673 · Zbl 1147.53030 |

[23] | C.R. Graham and T. Willse. Parallel tractor extension and ambient metrics of holonomy split \(G\)_{2}. arXiv:1109.3504 · Zbl 1268.53075 |

[24] | C.R. Graham and M. Zworski. Scattering matrix in conformal geometry. Inventiones Mathematicae, (1)152 (2003), 89-118. arXiv:math/0109089 · Zbl 1030.58022 |

[25] | R.L. Graham, D.E. Knuth and O. Patashnik. Concrete Mathematics. A Foundation for Computer Science. Addison-Wesley Publishing Company Advanced Book Program (1989). · Zbl 0668.00003 |

[26] | A. Juhl. Families of Conformally Covariant Differential Operators, \(Q\)-Curvature and Holography. Progress in Mathematics, vol. 275. Birkhäuser Verlag, Basel (2009). · Zbl 1177.53001 |

[27] | A. Juhl. On conformally covariant powers of the Laplacian. arXiv:0905.3992v3 |

[28] | A. Juhl. On Branson’s \(Q\)-curvature of order eight. Conformal Geometry and Dynamics. 15 (2011), 20-43. arXiv:0912.2217 · Zbl 1229.53018 |

[29] | A. Juhl. Holographic formula for \(Q\)-curvature. II. Advances in Mathematics. (4)226 (2011), 3409-3425. arXiv:1003.3989 · Zbl 1244.53018 |

[30] | A. Juhl. On the recursive structure of Branson’s \(Q\)-curvature. arXiv:1004.1784.v2 |

[31] | A. Juhl and C. Krattenthaler. Summation formulas for GJMS-operators and \(Q\)-curvatures on the Möbius sphere. arXiv:0910.4840 · Zbl 1317.53014 |

[32] | C. Krattenthaler. Private communication (23.11.2010). · Zbl 0596.53009 |

[33] | T. Leistner and P. Nurowski. Ambient metrics for \(n\)-dimensional pp-waves. Communications in Mathematical Physics, (3)296 (2010), 881-898. arXiv:0810.2903 · Zbl 1207.53028 |

[34] | B. Michel. Masse des opérateurs GJMS. arXiv:1012.4414 |

[35] | S. Paneitz. A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA Symmetry Integrability Geometry Methods and Applications 4 (2008), paper 036, 3 p. arXiv:0803.4331 · Zbl 1145.53053 |

[36] | V. Wünsch. On conformally invariant differential operators. Mathematische Nachrichten, 29 (1986), 269-281. · Zbl 0619.53008 |

[37] | V. Wünsch. Some new conformal covariants. Journal for Analysis and its Applications, (2)19 (2000), 339-357. · Zbl 0978.53034 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.