## Conformally invariant powers of the Laplacian — A complete nonexistence theorem

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- by A. Rod Gover and Kengo Hirachi
- J. Amer. Math. Soc.
**17**(2004), 389-405 - DOI: https://doi.org/10.1090/S0894-0347-04-00450-3
- Published electronically: January 9, 2004
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## Abstract:

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.## References

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## Bibliographic Information

**A. Rod Gover**- Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand
- MR Author ID: 335695
- Email: gover@math.auckland.ac.nz
**Kengo Hirachi**- Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Megro, Tokyo 153-8914, Japan
- Email: hirachi@ms.u-tokyo.ac.jp
- Received by editor(s): April 10, 2003
- Published electronically: January 9, 2004
- Additional Notes: The first author gratefully acknowledges support from the Royal Society of New Zealand via a Marsden Grant (grant no. 02-UOA-108). The second author gratefully acknowledges support from the Japan Society for the Promotion of Science.
- © Copyright 2004 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**17**(2004), 389-405 - MSC (2000): Primary 53A30; Secondary 53A55, 35Q99
- DOI: https://doi.org/10.1090/S0894-0347-04-00450-3
- MathSciNet review: 2051616