Skip to main content
SpringerLink
Log in

Explicit Formulas for GJMS-Operators and Q-Curvatures

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré–Einstein metrics and renormalized volume coefficients. As special cases, we derive explicit formulas for conformally covariant third and fourth powers of the Laplacian. Moreover, we prove related formulas for all Branson’s Q-curvatures. The results settle and refine conjectural statements in earlier works. The proofs rest on the theory of residue families introduced in Juhl (Progress in Mathematics, vol. 275. Birkhäuser Verlag, Basel, 2009).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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

    Google Scholar 

  2. H. Baum and A. Juhl, Conformal Differential Geometry: Q-Curvature and Conformal Holonomy. Oberwolfach Seminars, 40 (2010).

  3. Branson T.P: Differential operators canonically associated to a conformal structure. Mathematica Scandinavica, 57, 293–345 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Branson T.P.: Sharp inequalities, the functional determinant, and the complementary series. Transactions of the American Mathematical Society, 347, 3671–3742 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. T.P. Branson. Q-curvature and spectral invariants. Rendiconti del Circolo Matematico di Palermo (2) Supplementary, 75 (2005), 11–55.

  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.

  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).

  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

  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

  12. A.R. Gover. Laplacian operators and Q-curvature on conformally Einstein manifolds. Mathematische Annalen. (2)336 (2006), 311–334. arXiv:math/0506037v3

  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

  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

  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

  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.

    Google Scholar 

  17. C.R. Graham. Conformally invariant powers of the Laplacian. II. Nonexistence. Journal of the London Mathematical Society, (2)46 (1992), 566–576.

    Google Scholar 

  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

  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

  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

  22. C.R. Graham and A. Juhl. Holographic formula for Q-curvature. Advances in Mathematics, (2)216 (2007), 841–853. arXiv:0704.1673

  23. C.R. Graham and T. Willse. Parallel tractor extension and ambient metrics of holonomy split G 2. arXiv:1109.3504

  24. C.R. Graham and M. Zworski. Scattering matrix in conformal geometry. Inventiones Mathematicae, (1)152 (2003), 89–118. arXiv:math/0109089

  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).

  26. A. Juhl. Families of Conformally Covariant Differential Operators, Q-Curvature and Holography. Progress in Mathematics, vol. 275. Birkhäuser Verlag, Basel (2009).

  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

  29. A. Juhl. Holographic formula for Q-curvature. II. Advances in Mathematics. (4)226 (2011), 3409–3425. arXiv:1003.3989

  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

  32. C. Krattenthaler. Private communication (23.11.2010).

  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

  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

  36. V. Wünsch. On conformally invariant differential operators. Mathematische Nachrichten, 29 (1986), 269–281.

    Google Scholar 

  37. V. Wünsch. Some new conformal covariants. Journal for Analysis and its Applications, (2)19 (2000), 339–357.

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität, Unter den Linden, 10099, Berlin, Germany

    Andreas Juhl

  2. Department of Mathematics, University Uppsala, P.O. Box 480, 75106, Uppsala, Sweden

    Andreas Juhl

Authors
  1. Andreas Juhl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Juhl.

Additional information

The work was supported by grant 621–2003–5240 of the Swedish Research Council VR and SFB 647 “Space-Time-Matter“ at Humboldt-University Berlin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Juhl, A. Explicit Formulas for GJMS-Operators and Q-Curvatures. Geom. Funct. Anal. 23, 1278–1370 (2013). https://doi.org/10.1007/s00039-013-0232-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-013-0232-9

Mathematics Subject Classification (2010)

Search

Navigation