Some Results of $f$-Harmonic and Bi-$f$-Harmonic Maps with Potential

Year 2021, , 157 - 166, 15.04.2021

Zegga Kaddour

https://doi.org/10.36890/iejg.713254

Abstract

In this note, we characterize the $f$-harmonic maps and bi-$f$-harmonic maps with potential. We prove that every bi-$f$-harmonic map with potential from complete Riemannian manifold, satisfying some conditions is a $f$-harmonic map with potential. More, we study the case of conformal maps between equidimensional manifolds.

Keywords

f-harmonic maps with potential, bi-f-harmonic maps with potential, H-f-energy

Thanks

The author would like to thank the referee for his helpful suggestions and his valuable comments which helped to improve the manuscript.

References

• [1] Lichnerowicz, A.: Applications harmoniques et variétés Kähleriennes. Rend. Sem. Mat. Fis. Milano. 39, 186–195 (1969).
• [2] Cherif, A. M., Djaa, M.: On the bi-harmonic maps with potential. Arab J. Math. Sci. 24(1), 1–8 (2018).
• [3] Ratto, A.: Harmonic maps with potential. Proceedings of theWorkshop on Differential Geometry and Topology (Palermo, 1996). Rend. Circ. Mat. Palermo. (2) Suppl. No. 49, 229–242 (1997).
• [4] Zagane, A., Ouakass, S.: Some results and examples of the biharmonic maps with potential. Arab J. Math. Sci. 24(2), 182–198 (2018).
• [5] Zegga, K., Cherif, A. M., Djaa, M.: On the f-biharmonic maps and submanifolds. Kyungpook Math. J. 55(1), 157–168 (2015).
• [6] Ara, M. Geometry of F-harmonic maps. Kodai Math. J. 22(2), 243–263 (1999).
• [7] Djaa, M., Cherif, A. M., Zegga, K., Ouakkas, S.: On the generalized of harmonic and bi-harmonic maps. Int. Electron. J. Geom. 5(1), 90–100 (2012).
• [8] Cherif, A. M., Djaa, M., Zegga, K.: Stable f-harmonic maps on sphere. Commun. Korean Math. Soc. 30(4), 471–479 (2015).
• [9] Course N.: f-harmonic maps, Thesis, University ofWarwick, Coventry, CV4 7AL, UK,2004.
• [10] Baird, P.: Harmonic maps with symmetry, harmonic morphisms and deformations of metrics. Research Notes in Mathematics, 87. Pitman (Advanced Publishing Program), Boston, MA, 1983.
• [11] Chen, Q.: Harmonic maps with potential from complete manifolds. Chinese Sci. Bull. 43(21), 1780–1786 (1998).
• [12] Jiang, R.: Harmonic maps with potential from R2 into S2. Asian J. Math. 20(4), 597–627 (2016).
• [13] Ouakkas, S., Nasri, R., Djaa, M.: On the f-harmonic and f-biharmonic maps. JP J. Geom. Topol. 10(1), 11–27 (2010).
• [14] Branding, V.: The heat flow for the full bosonic string. Ann. Global Anal. Geom. 50(4), 347–365 (2016).