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Conformally invariant powers of the Laplacian -- A complete nonexistence theorem


Authors: A. Rod Gover and Kengo Hirachi
Translated by:
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
Published electronically: January 9, 2004
MathSciNet review: 2051616
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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.


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Additional Information

A. Rod Gover
Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand
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

DOI: https://doi.org/10.1090/S0894-0347-04-00450-3
Keywords: Conformal geometry, invariant differential operators
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.
Article copyright: © Copyright 2004 American Mathematical Society

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