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Biharmonic submanifolds in spheres. (English) Zbl 1038.58011

Harmonic maps \(\phi\) are critical points of the energy functional \(E(\phi ) = \int | d\phi | ^2\), and \(\phi\) is harmonic if and only if \(\tau ( \phi) = 0\), where \(\tau (\phi )\) is the tension field of \(\phi\). Biharmonic maps are critical ones of the bienergy functional \(\int | \tau( \phi )| ^2\).
The authors study biharmonic maps into a manifold \(N\) of constant curvature, in particular an \(n\)-dimensional standard sphere. This paper consists of two parts: (1) non-existence results of non-harmonic biharmonic maps. (2) examples of non-harmonic biharmonic maps.

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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References:

[1] R. Caddeo, S. Montaldo and C. Oniciuc, biharmonic submanifolds ofS^3, International Journal of Mathematics, to appear. · Zbl 1111.53302
[2] Chen, B. Y., Some open problems and conjectures on submanifolds of finite type, Soochow Journal of Mathematics, 17, 169-188 (1991) · Zbl 0749.53037
[3] Chen, B. Y.; Ishikawa, S., Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu Journal of Mathematics, 52, 167-185 (1998) · Zbl 0892.53012
[4] Chen, B. Y.; Yano, K., Minimal submanifolds of a higher dimensional sphere, Tensor (N.S.), 22, 369-373 (1971) · Zbl 0218.53073
[5] Dimitric, I., Submanifolds of E^m with harmonic mean curvature vector, Bulletin of the Institute of Mathematics, Academic Sinica, 20, 53-65 (1992) · Zbl 0778.53046
[6] Eells, J.; Sampson, J. H., Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, 86, 109-160 (1964) · Zbl 0122.40102
[7] Gluck, H., Geodesics in the unit tangent bundle of a round sphere, L’Enseignement Mathématique, 34, 233-246 (1988) · Zbl 0675.53041
[8] Hasanis, T.; Vlachos, T., Hypersurfaces inE · Zbl 0839.53007
[9] Jiang, G. Y., 2-harmonic isometric immersions between Riemannian manifolds, Chinese Annals of Mathematics, 7, 130-144 (1986) · Zbl 0596.53046
[10] Jiang, G. Y., 2-harmonic maps and their first and second variational formulas, Chinese Annals of mathematics, 7, 389-402 (1986) · Zbl 0628.58008
[11] Lawson, H. B., Complete minimal surfaces inS · Zbl 0205.52001
[12] C. Oniciuc,Biharmonic maps between Riemannian manifolds, Analele Stiintifice ale University Al. I. Cuza Iasi. Mat. (N.S.), to appear. · Zbl 1061.58015
[13] Simons, J., Minimal varieties in Riemannian manifolds, Annals of Mathematics, 88, 62-105 (1968) · Zbl 0181.49702
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