Conformally invariant powers of the Laplacian -- A complete non-existence theorem

Details Conformally invariant powers of the Laplacian -- A complete non-existence theorem Conformally invariant powers of the Laplacian -- A complete non-existence theorem

2003-04-07

arxiv.org/abs/math/0304082v2

J.Am.Math.Soc. 17 (2004) 389-405

Mathematics

Differential Geometry

18 pages; Minor changes and typographical corrections throughout article; To appear in Jour. Amer. Math. Soc

Scientific paper

We show that on conformal manifolds of even dimension $n\geq 4$ there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian $\Delta^{k}$ for $k>n/2$. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of $\Delta^k$ for $1\le k\le n/2$, is sharp.

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