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Laplacian operators and \(Q\)-curvature on conformally Einstein manifolds. (English) Zbl 1125.53032
The 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.

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
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