Geodesic
Conformally invariant powers of the Laplacian — A complete nonexistence theorem
Gover, A.1 ; Hirachi, Kengo2
1 Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand
2 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Megro, Tokyo 153-8914, Japan
Journal of the American Mathematical Society, Tome 17 (2004) no. 2, pp. 389-405

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-04-00450-3

Gover, A. 1 ; Hirachi, Kengo 2

1 Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand
2 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Megro, Tokyo 153-8914, Japan 
@article{10_1090_S0894_0347_04_00450_3,
     author = {Gover, A. and Hirachi, Kengo},
     title = {Conformally invariant powers of the {Laplacian} \^a€” {A} complete nonexistence theorem},
     journal = {Journal of the American Mathematical Society},
     pages = {389--405},
     publisher = {mathdoc},
     volume = {17},
     number = {2},
     year = {2004},
     doi = {10.1090/S0894-0347-04-00450-3},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-04-00450-3/}
}
TY  - JOUR
AU  - Gover, A.
AU  - Hirachi, Kengo
TI  - Conformally invariant powers of the Laplacian — A complete nonexistence theorem
JO  - Journal of the American Mathematical Society
PY  - 2004
SP  - 389
EP  - 405
VL  - 17
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-04-00450-3/
DO  - 10.1090/S0894-0347-04-00450-3
ID  - 10_1090_S0894_0347_04_00450_3
ER  - 
%0 Journal Article
%A Gover, A.
%A Hirachi, Kengo
%T Conformally invariant powers of the Laplacian — A complete nonexistence theorem
%J Journal of the American Mathematical Society
%D 2004
%P 389-405
%V 17
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-04-00450-3/
%R 10.1090/S0894-0347-04-00450-3
%F 10_1090_S0894_0347_04_00450_3
Gover, A.; Hirachi, Kengo. Conformally invariant powers of the Laplacian — A complete nonexistence theorem. Journal of the American Mathematical Society, Tome 17 (2004) no. 2, pp. 389-405. doi: 10.1090/S0894-0347-04-00450-3

[1] Bailey, T. N., Eastwood, M. G., Gover, A. R. Thomas’s structure bundle for conformal, projective and related structures Rocky Mountain J. Math. 1994 1191 1217

[2] Branson, Thomas P. Sharp inequalities, the functional determinant, and the complementary series Trans. Amer. Math. Soc. 1995 3671 3742

[3] ČAp, Andreas, Gover, A. Rod Tractor bundles for irreducible parabolic geometries 2000 129 154

[4] ČAp, Andreas, Gover, A. Rod Tractor calculi for parabolic geometries Trans. Amer. Math. Soc. 2002 1511 1548

[5] Chang, Sun-Yung A., Yang, Paul C. Partial differential equations related to the Gauss-Bonnet-Chern integrand on 4-manifolds 2002 1 30

[6] Eastwood, Michael, Slovã¡K, Jan Semiholonomic Verma modules J. Algebra 1997 424 448

[7] Fefferman, Charles, Graham, C. Robin Conformal invariants Astérisque 1985 95 116

[8] Fulton, William, Harris, Joe Representation theory 1991

[9] Gover, A. Rod Aspects of parabolic invariant theory Rend. Circ. Mat. Palermo (2) Suppl. 1999 25 47

[10] Gover, A. Rod Invariant theory and calculus for conformal geometries Adv. Math. 2001 206 257

[11] Graham, C. Robin Conformally invariant powers of the Laplacian. II. Nonexistence J. London Math. Soc. (2) 1992 566 576

[12] Graham, C. Robin, Jenne, Ralph, Mason, Lionel J., Sparling, George A. J. Conformally invariant powers of the Laplacian. I. Existence J. London Math. Soc. (2) 1992 557 565

[13] Jakobsen, Hans Plesner, Vergne, Michele Wave and Dirac operators, and representations of the conformal group J. Functional Analysis 1977 52 106

[14] Penrose, Roger, Rindler, Wolfgang Spinors and space-time. Vol. 1 1984

[15] Wã¼Nsch, V. Some new conformal covariants Z. Anal. Anwendungen 2000 339 357

Cité par Sources :