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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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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.