On the biharmonic and harmonic indices of the Hopf map

Loubeau, E.; Oniciuc, C.

doi:10.1090/S0002-9947-07-03934-7

On the biharmonic and harmonic indices of the Hopf map
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by E. Loubeau and C. Oniciuc PDF

Trans. Amer. Math. Soc. 359 (2007), 5239-5256 Request permission

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.

References

Lionel Bérard-Bergery and Jean-Pierre Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math. 26 (1982), no. 2, 181–200. MR 650387
Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of $S^3$, Internat. J. Math. 12 (2001), no. 8, 867–876. MR 1863283, DOI 10.1142/S0129167X01001027
R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math. 130 (2002), 109–123. MR 1919374, DOI 10.1007/BF02764073
R. Caddeo, S. Montaldo, and P. Piu, Biharmonic curves on a surface, Rend. Mat. Appl. (7) 21 (2001), no. 1-4, 143–157 (English, with English and Italian summaries). MR 1884940
R. Caddeo, C. Oniciuc, and P. Piu, Explicit formulas for non-geodesic biharmonic curves of the Heisenberg group, Rend. Sem. Mat. Univ. Politec. Torino 62 (2004), no. 3, 265–277. MR 2129448
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Jun-Ichi Inoguchi, Submanifolds with harmonic mean curvature vector field in contact 3-manifolds, Colloq. Math. 100 (2004), no. 2, 163–179. MR 2107514, DOI 10.4064/cm100-2-2
Guo Ying Jiang, $2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389–402 (Chinese). An English summary appears in Chinese Ann. Math. Ser. B 7 (1986), no. 4, 523. MR 886529
E. Loubeau and Y.-L. Ou, The characterization of biharmonic morphisms, Differential geometry and its applications (Opava, 2001) Math. Publ., vol. 3, Silesian Univ. Opava, Opava, 2001, pp. 31–41. MR 1978760
E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (2005), no. 3, 729–745. MR 2135286, DOI 10.1112/S0010437X04001204
E. Mazet, La formule de la variation seconde de l’énergie au voisinage d’une application harmonique, J. Differential Geometry 8 (1973), 279–296 (French). MR 336767
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
C. Oniciuc, New examples of biharmonic maps in spheres, Colloq. Math. 97 (2003), no. 1, 131–139. MR 2010548, DOI 10.4064/cm97-1-12
Toru Sasahara, Quasi-minimal Lagrangian surfaces whose mean curvature vectors are eigenvectors, Demonstratio Math. 38 (2005), no. 1, 185–196. MR 2123733
Toru Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), no. 3-4, 285–303. MR 2162123
T. Sasahara. Instability of biharmonic Legendre surfaces in Sasakian space forms. preprint.
R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229–236. MR 375386, DOI 10.1090/S0002-9939-1975-0375386-2
Hajime Urakawa, Stability of harmonic maps and eigenvalues of the Laplacian, Trans. Amer. Math. Soc. 301 (1987), no. 2, 557–589. MR 882704, DOI 10.1090/S0002-9947-1987-0882704-8