# SOME RESULTS ON STABLE f-HARMONIC MAPS

• Embarka, Remli (Department of Mathematics University Mustapha Stambouli) ;
• Cherif, Ahmed Mohammed (Department of Mathematics University Mustapha Stambouli)
• Accepted : 2017.09.26
• Published : 2018.07.31
• 271 7

#### Abstract

In this paper, we prove that any stable f-harmonic map from sphere ${\mathbb{S}}^n$ to Riemannian manifold (N, h) is constant, where f is a smooth positive function on ${\mathbb{S}}^n{\times}N$ satisfying one condition with n > 2. We also prove that any stable f-harmonic map ${\varphi}$ from a compact Riemannian manifold (M, g) to ${\mathbb{S}}^n$ (n > 2) is constant where, in this case, f is a smooth positive function on $M{\times}{\mathbb{S}}^n$ satisfying ${\Delta}^{{\mathbb{S}}^n}(f){\circ}{\varphi}{\leq}0$.

#### Keywords

harmonic maps;f-harmonic maps;stable f-harmonic maps

#### References

1. P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, 29, The Clarendon Press, Oxford University Press, Oxford, 2003.
2. R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of $S^3$, Internat. J. Math. 12 (2001), no. 8, 867-876. https://doi.org/10.1142/S0129167X01001027
3. A. M. Cherif, M. Djaa, and K. Zegga, Stable f-harmonic maps on sphere, Commun. Korean Math. Soc. 30 (2015), no. 4, 471-479. https://doi.org/10.4134/CKMS.2015.30.4.471
4. N. Course, f-harmonic maps which map the boundary of the domain to one point in the target, New York J. Math. 13 (2007), 423-435.
5. M. Djaa, A. M. Cherif, K. Zagga, and S. Ouakkas, On the generalized of harmonic and bi-harmonic maps, Int. Electron. J. Geom. 5 (2012), no. 1, 90-100.
6. J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037
7. R. Howard and S. W. Wei, Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space, Trans. Amer. Math. Soc. 294 (1986), no. 1, 319-331. https://doi.org/10.1090/S0002-9947-1986-0819950-4
8. Y. Ohnita, Stability of harmonic maps and standard minimal immersions, Tohoku Math. J. (2) 38 (1986), no. 2, 259-267. https://doi.org/10.2748/tmj/1178228492
9. B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
10. S. Ouakkas, R. Nasri, and M. Djaa, On the f-harmonic and f-biharmonic maps, J. P. J. Geom. Topol. 10 (2010), no. 1, 11-27.
11. Y. Xin, Geometry of Harmonic Maps, Progress in Nonlinear Diﬀerential Equations and their Applications, 23, Birkhauser Boston, Inc., Boston, MA, 1996.
12. Y. L. Xin, Some results on stable harmonic maps, Duke Math. J. 47 (1980), no. 3, 609-613. https://doi.org/10.1215/S0012-7094-80-04736-5