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Transactions of the American Mathematical Society

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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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  • 1. L. Bérard Bergery, J.P. Bourguignon. Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math., 26 (1982), 181-200. MR 0650387 (84m:58153)
  • 2. M. Berger, P. Gauduchon, E. Mazet. Le spectre d'une variété riemannienne. Lecture Notes in Math., n. 194, Springer-Verlag, 1971.MR 0282313 (43:8025)
  • 3. R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds of $ \mathbb{S}^3$. Int. J. Math., 12 (2001), 867-876.MR 1863283 (2002k:53123)
  • 4. R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds in spheres. Israel J. Math., 130 (2002), 109-123.MR 1919374 (2003c:53090)
  • 5. R. Caddeo, S. Montaldo, P. Piu. Biharmonic curves on a surface. Rend. Mat. Appl., (7) 21 (2001), no. 1-4, 143-157.MR 1884940 (2002k:58031)
  • 6. R. Caddeo, C. Oniciuc, P. Piu. Explicit formulas for non-geodesic biharmonic curves of the Heisenberg group. Rend. Sem. Mat. Univ. Politec. Torino 62 (2004), 265-277. MR 2129448
  • 7. J. Eells, J.H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86 (1964), 109-160. MR 0164306 (29:1603)
  • 8. H. Federer. Geometric measure theory. Springer-Verlag, 1969.MR 0257325 (41:1976)
  • 9. W. Fulton, J. Harris. Representation Theory. Springer-Verlag, 1991.MR 1153249 (93a:20069)
  • 10. J. Inoguchi. Submanifolds with harmonic mean curvature in contact $ 3-$manifolds. Colloq. Math., 100 (2004), 163-179.MR 2107514 (2005h:53105)
  • 11. G.Y. Jiang. $ 2-$harmonic maps and their first and second variational formulas. Chinese Ann. Math. Ser. A, 7 (1986), 389-402.MR 0886529 (88i:58039)
  • 12. E. Loubeau, Y.-L. Ou. The characterization of biharmonic morphisms. Differential Geometry and its Applications (Opava, 2001), Math. Publ., 3 (2001), 31-41.MR 1978760 (2004b:53111)
  • 13. E. Loubeau, C. Oniciuc. The index of biharmonic maps in spheres. Compositio Math. 141 (2005), 729-745. MR 2135286
  • 14. E. Mazet. La formule de la variation seconde de l'énergie au voisinage d'une application harmonique. J. Differential Geom., 8 (1973), 279-296.MR 0336767 (49:1540)
  • 15. C. Oniciuc. On the second variation formula for biharmonic maps to a sphere. Publ. Math. Debrecen, 61 (2002), no.3-4, 613-622.MR 1943720 (2003i:58031)
  • 16. C. Oniciuc. New examples of biharmonic maps in spheres. Colloq. Math., 97 (2003), 131-139. MR 2010548 (2004i:53091)
  • 17. T. Sasahara. Quasi-minimal Lagrangian surfaces whose mean curvature vectors are eigenvectors. Demonstratio Math. 38 (2005), 185-196.MR 2123733 (2005m:53110)
  • 18. T. Sasahara. Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors. Publ. Math. Debrecen 67 (2005), 285-303. MR 2162123
  • 19. T. Sasahara. Instability of biharmonic Legendre surfaces in Sasakian space forms. preprint.
  • 20. R.T. Smith. The second variation formula for harmonic mappings. Proc. Amer. Math. Soc., 47 (1975), 229-236. MR 0375386 (51:11580)
  • 21. H. Urakawa. Stability of harmonic maps and eigenvalues of the Laplacian. Trans. American Math. Soc., 301 (1987), 557-589.MR 0882704 (88g:58046)

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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

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

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

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