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Conformally invariant powers of the Laplacian–A complete nonexistence theorem. (English) Zbl 1066.53037
In 1992, C. R. Graham, R. Jenne, L. J. Mason and G. A. J. Sparling [Conformally invariant powers of the Laplacian. I: Existence, J. Lond. Math. Soc., II. Ser. 46, No. 3, 557–565 (1992; Zbl 0726.53010)] constructed natural conformally invariant operators between densities of weights $$k-n/2$$ and $$-k-n/2$$ for all $$n, k$$ except for the case $$n\geqslant4$$ even and $$k>n/2$$. They conjectured that this result is sharp.
In the article of A. R. Gover and K. Hirachi this conjecture is completely proved. The authors use so-called Fefferman-Graham obstruction tensor. For a class of conformal metrics for which obstruction tensor vanishes they build a curvature expression and by direct calculation using the tractor calculus show that it is nonzero. However it must be zero under the conjecture conditions, that is a sequence of some classical invariant theory. The authors point out that their idea can be used in wide range of settings where similar nonexistence statements can appear, such as irreducible bundles on conformally flat manifolds and CR structures.

MSC:
 53A30 Conformal differential geometry (MSC2010) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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References:
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