Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


$ 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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. B. Y. Chen, Some open problems and conjectures on submanfolds of finite type, Soochow J. Math. 17 (1991), 169-188. MR 1143504 (92m:53091)
  • 2. B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math. 52 (1998), no. 1, 167-185. MR 1609044 (99b:53078)
  • 3. I. Dimitric, Submanifolds of $ E^n$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20 (1992), 53-65. MR 1166218 (93g:53087)
  • 4. J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS, 50, Amer. Math. Soc., 1983. MR 703510 (85g:58030)
  • 5. T. Ichiyama, J. Inoguchi and H. Urakawa, Bi-harmonic map and bi-Yang-Mills fields, Note di Matematica 28 (2009), 233-275. MR 2640583
  • 6. G. Y. Jiang, $ 2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math., 7A (1986), 388-402; English translation, Note di Mathematica 28 (2008), 209-232. 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. H. Urakawa, Calculus of variations and harmonic maps, Transl. Math. Monograph. 132, Amer. Math. Soc., 1993. MR 1252178 (95c:58050)
  • 10. S. B. Wang, The first variation formula for $ K$-harmonic mapping, Journal of Nanchang University 13, No. 1, 1989.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 58E20, 53C43

Retrieve articles in all journals with MSC (2010): 58E20, 53C43

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.