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$ k$-harmonic maps into a Riemannian manifold with constant sectional curvature

Author: Shun Maeta
Journal: Proc. Amer. Math. Soc. 140 (2012), 1835-1847
MSC (2010): Primary 58E20; Secondary 53C43
Published electronically: September 26, 2011
MathSciNet review: 2869168
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Abstract: J. Eells and L. Lemaire introduced $ k$-harmonic maps, and Shaobo Wang showed the first variational formula. When $ k=2$, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and $ k$-harmonic maps, and we show the non-existence theorem of 3-harmonic maps. We also give the definition of $ k$-harmonic submanifolds of Euclidean spaces and study $ k$-harmonic curves in Euclidean spaces. Furthermore, we give a conjecture for $ k$-harmonic submanifolds of Euclidean spaces.

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  • 1. Bang-Yen Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169–188. MR 1143504
  • 2. Bang-Yen Chen and Susumu Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math. 52 (1998), no. 1, 167–185. MR 1609044,
  • 3. Ivko Dimitrić, Submanifolds of 𝐸^{𝑚} with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 53–65. MR 1166218
  • 4. James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510
  • 5. Toshiyuki Ichiyama, Jun-ichi Inoguchi, and Hajime Urakawa, Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009), no. [2008 on verso], suppl. 1, 233–275. MR 2640583
  • 6. Jiang Guoying, 2-harmonic maps and their first and second variational formulas, Note Mat. 28 (2009), no. [2008 on verso], suppl. 1, 209–232. Translated from the Chinese by Hajime Urakawa. MR 2640582
  • 7. Sh. Maeta, The second variational formula of the $ k$-energy and $ k$-harmonic curves, arXiv:1008.3700v1 [math.DG], 22 Aug 2010.
  • 8. Y.-L. Ou, Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces, arXiv:0912.1141v1, [math.DG] 6 Dec 2009.
  • 9. Hajime Urakawa, Calculus of variations and harmonic maps, Translations of Mathematical Monographs, vol. 132, American Mathematical Society, Providence, RI, 1993. Translated from the 1990 Japanese original by the author. MR 1252178
  • 10. S. B. Wang, The first variation formula for $ K$-harmonic mapping, Journal of Nanchang University 13, No. 1, 1989.

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Additional Information

Shun Maeta
Affiliation: Graduate School of Information Sciences, Tohoku University, Aoba 6-3-09 Aramaki Aoba-ku Sendai-shi Miyagi, 980-8579 Japan
Address at time of publication: Nakakuki 3-10-9, Oyama-shi, Tochigi, Japan

Received by editor(s): September 19, 2010
Received by editor(s) in revised form: January 20, 2011, January 27, 2001, and January 29, 2011
Published electronically: September 26, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.