# Stability and Constant Boundary-Value Problems of f-Harmonic Maps with Potential

• Kacimi, Bouazza (Department of Mathematics, University Mustapha Stambouli) ;
• Cherif, Ahmed Mohammed (Department of Mathematics, University Mustapha Stambouli)
• Accepted : 2018.07.18
• Published : 2018.09.23
• 127 7

#### Abstract

In this paper, we give some results on the stability of f-harmonic maps with potential from or into spheres and any Riemannian manifold. We study the constant boundary-value problems of such maps defined on a specific Cartan-Hadamard manifolds, and obtain a Liouville-type theorem. It can also be applied to the static Landau-Lifshitz equations. We also prove a Liouville theorem for f-harmonic maps with finite f-energy or slowly divergent f-energy.

#### Keywords

f-harmonic maps with potential;stability;boundary-value

#### Acknowledgement

Supported by : National Agency Scientific Research of Algeria

#### References

1. P. Baird and J. C. Wood, Harmonic morphisms between Riemannain manifolds, Clarendon Press, Oxford University press, Oxford. 2003.
2. R. Caddeo and S. Montaldo, C. Oniciuc, Biharmonic submanifolds of ${\mathbb{S}}^3$, Internat. J. Math., 12(2001), 867-876. https://doi.org/10.1142/S0129167X01001027
3. Q. Chen, Stability and constant boundary-value problems of harmonic maps with potential, J. Austral. Math. Soc. Ser. A, 68(2000), 145-154.
4. A. M. Cherif and M. Djaa, Geometry of energy and bienergy variations between Riemannian Manifolds, Kyungpook Math. J., 55(2015), 715-730. https://doi.org/10.5666/KMJ.2015.55.3.715
5. A. M. Cherif, M. Djaa and K. Zegga, Stable f-harmonic maps on sphere, Commun. Korean Math. Soc., 30(4)(2015), 471-479. https://doi.org/10.4134/CKMS.2015.30.4.471
6. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86(1964), 109-160.
7. S. Feng and Y. Han, Liouville type theorems of f-harmonic maps with potential, Results Math., 66(2014), 43-64. https://doi.org/10.1007/s00025-014-0363-9
8. R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics 699, Springer-Verlag, Berlin, 1979.
9. M. C. Hong and L. Lemaire, Multiple solutions of the static Landau-Lifshitz equation from $B^2$ into $S^2$, Math. Z., 220(1995), 295-306.
10. J. Karcher and J. C. Wood, Nonexistence results and growth properties for harmonic maps and forms, J. Reine Angew. Math., 353(1984), 165-180.
11. B. O'Neil, Semi-Riemannian Geometry, Academic Press, New York. 1983.
12. S. Ouakkas, R. Nasri and M. Djaa, On the f-harmonic and f-biharmonic maps, JP J. Geom. Topol., 10(1)(2010), 11-27.
13. H. C. J. Sealey, Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory, Math. Proc. Cambridge Philos. Soc., 91(1982), 441-452. https://doi.org/10.1017/S030500410005948X
14. Y. Xin, Some results on stable harmonic maps, Duke Math. J., 47(1980), 609-613. https://doi.org/10.1215/S0012-7094-80-04736-5
15. Y. Xin, Harmonic maps of bounded symmetric domains, Math. Ann., 303(1995), 417-433. https://doi.org/10.1007/BF01460998
16. Y. Xin, Geometry of harmonic maps, Birkhauser Boston, Boston, MA, 1996.