Abstract
We review how to construct the Paneitz operator in dimension four and the corresponding operator in dimension six, by constructing symmetric bilinear differential functionals that are conformally invariant.
References
Branson, T.P. (2001) “Automated symbolic computations in spin geometry”, in: F. Brackx, J.S.R. Chisholm & V. Souček (Eds.) Clifford Analysis and its Applications, NATO Science Series II, Vol. 25, Kluwer Academic Publishers, Dordrecht: 27–38.
Branson, T.P. (1985) “Differential operators canonically associated to a conformal structure”, Math. Scand. 57: 293–345.
Connes, A. (1995) “Quantized calculus and applications”, in: Proceedings of the XIth International Congress of Mathematical Physics, International Press, Cambridge, MA: 15–36.
Fefferman, C.; Graham, C.R. (1985) Conformal Invariants. In Élie Cartan et les mathématiques d’aujourd’hui , Astérisque, hors série, Société Mathé- matique de France: 95–116.
Graham, R.; Jenne, R.; Mason, L.; Sparling, G. (1992) “Conformally invariant powers of the Laplacian, I: Existence”, J. London Math. Soc. (2) 46: 557–565.
Gover, A. R.; Peterson, L.J. (2003) “Conformally invariant powers of the Laplacian, Q-curvature and tractor calculus”, Comm. Math. Phys. 235(2): 339–378.
Lee, J.M. (s.f.) “A Mathematica package for doing tensor calculations in differential geometry”, available at http://www.math.washington.edu/∼lee/Ricci/.
Paneitz, S. (1983) “A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds”, preprint, Massachusetts Institute of Technology.
Phillips, N.G.; Hu, B.L. (2003) “Noise kernel and stress energy bi-tensor of quantum fields in conformally-optical metrics: Schwarzschild black holes”, Phys. Rev. D (3) 67(10) 104002, 26 pp.
Phillips, N.G.; Hu, B.L. (2001) “Noise kernel and stress energy bi-tensor of quantum fields in hot flat space and Gaussian approximation in the optical Schwarzschild metric”, Phys. Rev. D (3) 63(10) 104001, 16 pp.
Tsantalis, E.; Puntigam, R.A.; Hehl, F.W. (1996) “A quadratic curvature Lagrangian of Pawlowski and Raczka: a finger exercise with math tensor”, Relativity and Scientific Computing, Springer, Berlin: 231–240.
Ugalde, W.J. (2006) “A construction of critical GJMS operators using Wodzicki’s residue” Comm. Math. Phys. 261(3): 771–788.