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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the biharmonic and harmonic indices of the Hopf map


Authors: E. Loubeau and C. Oniciuc
Journal: Trans. Amer. Math. Soc. 359 (2007), 5239-5256
MSC (2000): Primary 58E20, 31B30
Published electronically: June 4, 2007
MathSciNet review: 2327029
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Abstract: Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map $ \psi:\mathbb{S}^3\to \mathbb{S}^2$ and modify it into a nonharmonic biharmonic map $ \phi:\mathbb{S}^3\to \mathbb{S}^3$. We show $ \phi$ to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.


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Additional Information

E. Loubeau
Affiliation: Département de Mathématiques, Laboratoire C.N.R.S. U.M.R. 6205, Université de Bretagne Occidentale, 6, Avenue Victor Le Gorgeu, CS 93837, 29238 Brest Cedex 3, France
Email: loubeau@univ-brest.fr

C. Oniciuc
Affiliation: Faculty of Mathematics, “Al.I. Cuza" University of Iasi, Bd. Carol I, no. 11, 700506 Iasi, Romania
Email: oniciucc@uaic.ro

DOI: http://dx.doi.org/10.1090/S0002-9947-07-03934-7
PII: S 0002-9947(07)03934-7
Keywords: Harmonic and biharmonic maps, Riemannian submersions, stability
Received by editor(s): October 9, 2004
Received by editor(s) in revised form: July 1, 2005
Published electronically: June 4, 2007
Additional Notes: The authors are grateful to T. Levasseur for his help with representation theory.
The second author thanks the C.N.R.S. for a grant which made possible a three-month stay at the Université de Bretagne Occidentale in Brest.
Dedicated: In memoriam James Eells
Article copyright: © Copyright 2007 American Mathematical Society