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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.

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
Full Text: DOI
[1] R. Caddeo, S. Montaldo and C. Oniciuc, biharmonic submanifolds ofS 3, International Journal of Mathematics, to appear. · Zbl 1111.53302
[2] B. Y. Chen,Some open problems and conjectures on submanifolds of finite type, Soochow Journal of Mathematics17 (1991), 169–188. · Zbl 0749.53037
[3] B. Y. Chen and S. Ishikawa,Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu Journal of Mathematics52 (1998), 167–185. · Zbl 0892.53012 · doi:10.2206/kyushujm.52.167
[4] B. Y. Chen and K. Yano,Minimal submanifolds of a higher dimensional sphere, Tensor (N.S.)22 (1971), 369–373. · Zbl 0218.53073
[5] I. Dimitric,Submanifolds of E m with harmonic mean curvature vector, Bulletin of the Institute of Mathematics. Academic Sinica20 (1992), 53–65. · Zbl 0778.53046
[6] J. Eells and J.H. Sampson,Harmonic mappings of Riemannian manifolds, American Journal of Mathematics86 (1964), 109–160. · Zbl 0122.40102 · doi:10.2307/2373037
[7] H. Gluck,Geodesics in the unit tangent bundle of a round sphere, L’Enseignement Mathématique34 (1988), 233–246. · Zbl 0675.53041
[8] T. Hasanis and T. Vlachos, Hypersurfaces inS 4 with harmonic mean curvature vector field, Mathematische Nachrichten172 (1995), 145–169. · Zbl 0839.53007 · doi:10.1002/mana.19951720112
[9] G. Y. Jiang,2-harmonic isometric immersions between Riemannian manifolds, Chinese Annals of Mathematics. Series A7 (1986), 130–144. · Zbl 0596.53046
[10] G. Y. Jiang,2-harmonic maps and their first and second variational formulas, Chinese Annals of mathematics. Series A7 (1986), 389–402. · Zbl 0628.58008
[11] H. B. Lawson, Complete minimal surfaces inS 3, Annals of Mathematics (2)92 (1970), 335–374. · Zbl 0205.52001 · doi:10.2307/1970625
[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] J. Simons,Minimal varieties in Riemannian manifolds, Annals of Mathematics88 (1968), 62–105. · Zbl 0181.49702 · doi:10.2307/1970556
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