Laplacian operators and \(Q\)-curvature on conformally Einstein manifolds.

*(English)*Zbl 1125.53032The Laplacian \(\Delta\) is conformally invariant in two dimensions and, more generally on an \(n\)-dimensional Riemannian manifold with scalar curvature \(R\), the Yamabe operator \(\Delta-\frac{n-2}{4(n-1)}R\) is conformally invariant. In odd dimensions any power of the Laplacian admits lower order curvature correction terms so that the resulting operator is conformally invariant. In even dimensions, however, there are conformally invariant operators \(P_k=\Delta^k+\cdots\) defined on any Riemannian manifold only for \(1 \leq k\leq n/2\). In particular, the existence of the operators \(P_{n/2}\) [C. R. Graham, R. Jenne, L. J. Mason, and G. A. J. Sparling, J. Lond. Math. Soc. 46, 557–565 (1992; Zbl 0726.53010)] is extremely subtle as is the non-existence of higher powers in general [A. R. Gover and K. Hirachi, J. Am. Math. Soc. 17, 389–405 (2004; Zbl 1066.53037)].

In 2003, at a workshop of the American Institute of Mathematics, C. R. Graham noticed that his ambient metric construction established with C. Fefferman [Conformal invariants. The mathematical heritage of Élie Cartan, Sémin. Lyon 1984, Astérisque, No. Hors Sér. 1985, 95–116 (1985; Zbl 0602.53007)] could be extended to all orders in the case of a conformal Einstein metric. As a consequence, it follows that the operators \(P_k \) can be defined for all \(k\) including the previously unavailable \(k>n/2\). Graham also sketched an argument at this meeting whereby these new operators admit a convenient factorisation as products of terms of the form \(\Delta+cR \) for suitable constants \(c\) just as they do on the round sphere.

In this article, the author presents an alternative approach to these results. Instead of the ambient construction, he uses the tractor calculus and various associated tools that he has developed in earlier work. As well as the operators \(P_k\) of above, he also finds an explicit formula for the Branson’s \(Q\)-curvature, which is constant in an Einstein scale and again mimics the formula on the round sphere. A subtle point here is that an Einstein metric can be conformal to another Einstein metric by means of a non-constant conformal rescaling. In other words a Riemannian metric can have two Einstein representatives in a non-trivial way.

Perhaps the most interesting aspect of this article is that for \(k>n/2 \) the operators \(P_k\) in the conformally Einstein case might only be described with respect to an Einstein metric in the conformal class. Specifically, he shows for the operator \(P_6\) in four dimensions that there is no formula for the operator in terms of the metric connection of an arbitrary metric in the conformal class: one is obliged to choose an Einstein representative. This is a bizarre state of affairs and (by means of another observation due to C. R. Graham) does not arise for conformally flat metrics. This shows that one has to be very careful with this particular meaning of invariant.

In 2003, at a workshop of the American Institute of Mathematics, C. R. Graham noticed that his ambient metric construction established with C. Fefferman [Conformal invariants. The mathematical heritage of Élie Cartan, Sémin. Lyon 1984, Astérisque, No. Hors Sér. 1985, 95–116 (1985; Zbl 0602.53007)] could be extended to all orders in the case of a conformal Einstein metric. As a consequence, it follows that the operators \(P_k \) can be defined for all \(k\) including the previously unavailable \(k>n/2\). Graham also sketched an argument at this meeting whereby these new operators admit a convenient factorisation as products of terms of the form \(\Delta+cR \) for suitable constants \(c\) just as they do on the round sphere.

In this article, the author presents an alternative approach to these results. Instead of the ambient construction, he uses the tractor calculus and various associated tools that he has developed in earlier work. As well as the operators \(P_k\) of above, he also finds an explicit formula for the Branson’s \(Q\)-curvature, which is constant in an Einstein scale and again mimics the formula on the round sphere. A subtle point here is that an Einstein metric can be conformal to another Einstein metric by means of a non-constant conformal rescaling. In other words a Riemannian metric can have two Einstein representatives in a non-trivial way.

Perhaps the most interesting aspect of this article is that for \(k>n/2 \) the operators \(P_k\) in the conformally Einstein case might only be described with respect to an Einstein metric in the conformal class. Specifically, he shows for the operator \(P_6\) in four dimensions that there is no formula for the operator in terms of the metric connection of an arbitrary metric in the conformal class: one is obliged to choose an Einstein representative. This is a bizarre state of affairs and (by means of another observation due to C. R. Graham) does not arise for conformally flat metrics. This shows that one has to be very careful with this particular meaning of invariant.

Reviewer: Michael G. Eastwood (Adelaide)

##### MSC:

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

53A30 | Conformal differential geometry (MSC2010) |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

**OpenURL**

##### References:

[1] | Bailey T.N., Eastwood M.G., Gover A.R. (1994) Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24: 1191–1217 · Zbl 0828.53012 |

[2] | Branson T. (1985) Differential operators canonically associated to a conformal structure. Math. Scand. 57: 293–345 · Zbl 0596.53009 |

[3] | Branson T. (1995) Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc. 347: 3671–3742 · Zbl 0848.58047 |

[4] | Branson T., Gover A.R. (2005) Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature. Comm. Partial Differential Equations 30: 1611–1669 · Zbl 1226.58011 |

[5] | Branson T., Ørsted B. (1991) Explicit functional determinants in four dimensions. Proc. Amer. Math. Soc. 113: 669–682 · Zbl 0762.47019 |

[6] | Brinkmann H.W. (1925) Einstein spaces which are mapped conformally on each other. Math. Ann. 94: 119–145 · JFM 51.0568.03 |

[7] | Cartan E. (1923) Les espaces à connexion conforme. Ann. Soc. Pol. Math. 2: 171–202 · JFM 50.0493.01 |

[8] | Čap A., Gover A.R. Tractor bundles for irreducible parabolic geometries. In: Global analysis and harmonic analysis (Marseille-Luminy, 1999), pp. 129–154. Sémin. Congr., vol. 4: Soc. Math. France, Paris (2000) |

[9] | Čap A., Gover A.R. (2002) Tractor calculi for parabolic geometries. Trans. Amer. Math. Soc. 354: 1511–1548 · Zbl 0997.53016 |

[10] | Čap A., Gover A.R. (2003) Standard tractors and the conformal ambient metric construction. Ann. Global Anal. Geom. 24(3): 231–259 · Zbl 1039.53021 |

[11] | Chang S.-Y.A., Gursky M., Yang P. (2002) An equation of Monge-Ampere type in conformal geometry, and four manifolds of positive Ricci curvature. Ann. Math. 155: 709–787 · Zbl 1031.53062 |

[12] | Chang S.-Y.A., Qing J., Yang P. (2000) On the Chern–Gauss–Bonnet integral for conformal metrics on R 4. Duke Math. J. 103: 523–544 · Zbl 0971.53028 |

[13] | Djadli Z., Malchiodi A. Existence of conformal metrics with constant Q-curvature. Preprint math.AP/0410141, http://www.arxiv.org · Zbl 1186.53050 |

[14] | Dirac P.A.M. (1936) Wave equations in conformal space. Ann. Math. 37: 429–442 · Zbl 0014.08004 |

[15] | Eastwood M.G. Notes on conformal differential geometry. In: The Proceedings of the 15th Winter School ”Geometry and Physics” (Srni, 1995). Rend. Circ. Mat. Palermo (2) Suppl. No. 43 pp. 57–76 (1996) · Zbl 0911.53020 |

[16] | Eastwood M.G., Rice J.W. Conformally invariant differential operators on Minkowski space and their curved analogues. Comm. Math. Phys. 109, 207–228 (1987). Erratum, Comm. Math. Phys. 144, 213 (1992). · Zbl 0659.53047 |

[17] | Eastwood M.G., Slovák J. (1997) Semiholonomic Verma modules. J. Algebra 197: 424–448 · Zbl 0907.17010 |

[18] | Fefferman C., Graham C.R. Conformal invariants. The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque 1985, Numero Hors Série, pp. 95–116 |

[19] | Fefferman C., Hirachi K. (2003) Ambient metric construction of Q-curvature in conformal and CR geometries. Math. Res. Lett. 10: 819–832 · Zbl 1166.53309 |

[20] | Gauduchon P. Connexion canonique et structures de Weyl en geometrie conforme. Report: CNRS UA766 (1990) |

[21] | Gover A.R., Hirachi K. (2004) Conformally invariant powers of the Laplacian – A complete theorem. J. Amer. Math. Soc. 17: 389–405 · Zbl 1066.53037 |

[22] | Gover A.R., Nurowski P. (2006) Obstructions to conformally Einstein metrics in n dimensions. J. Geom. Phys. 56: 450–484 · Zbl 1098.53014 |

[23] | Gover A.R., Peterson L.J. (2003) Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus. Commun. Math. Phys. 235: 339–378 · Zbl 1022.58014 |

[24] | Gover A.R., Peterson L.J. The ambient obstruction tensor and the conformal deformation complex. Pacific J. Math., (in Press). math.DG/0408229, http://arXiv.org · Zbl 1125.53010 |

[25] | Gover A.R. Aspects of parabolic invariant theory. In: The 18th Winter School ”Geometry and Physics” (Srní 1998), pp. 25–47. Rend. Circ. Mat. Palermo (2) Suppl. No. 59 (1999) · Zbl 0967.53033 |

[26] | Gover A.R. (2001) Invariant theory and calculus for conformal geometries. Adv. Math. 163: 206–257 · Zbl 1004.53010 |

[27] | Gover A.R., Šilhan J. In progress |

[28] | Graham C.R., Hirachi K. The ambient obstruction tensor and Q-curvature. In: AdS/CFT correspondence: Einstein metrics and their conformal boundaries, pp. 59–71, IRMA Lect. Math. Theor. Phys., vol. 8, Eur. Math. Soc., ZÃijrich, 2005 · Zbl 1074.53027 |

[29] | Graham C.R., Jenne R., Mason L.J., Sparling G.A. (1992) Conformally invariant powers of the Laplacian, I: Existence. J. London Math. Soc. 46: 557–565 · Zbl 0788.53011 |

[30] | Paneitz S. A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983) · Zbl 1145.53053 |

[31] | Riegert R. (1984) A nonlocal action for the trace anomaly. Phys. Lett. B 134: 56–60 · Zbl 0966.81550 |

[32] | Thomas T.Y. (1926) On conformal geometry. Proc. Natl. Acad. Sci. USA 12: 352–359 · JFM 52.0736.01 |

[33] | Wünsch V. (1986) On conformally invariant differential operators. Math. Nachr. 129: 269–281 · Zbl 0619.53008 |

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.