Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Geometry of (p, f)-bienergy variations between Riemannian manifolds

  • Embarka Remli (University of Mustapha Stambouli Mascara, Faculty of Exact Sciences, Department of Mathematics) ;
  • Ahmed Mohammed Cherif (University of Mustapha Stambouli Mascara, Faculty of Exact Sciences, Department of Mathematics)
  • Received : 2022.01.09
  • Accepted : 2023.02.08
  • Published : 2023.06.30
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Abstract

In this paper, we extend the definition of the Jacobi operator of smooth maps, and biharmonic maps via the variation of bienergy between two Riemannian manifolds. We construct an example of (p, f)-biharmonic non (p, f)-harmonic map. We also prove some Liouville type theorems for (p, f)-biharmonic maps.

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References

  1. P. Baird and J. C. Wood, Harmonic morphisms between Riemannain manifolds, Clarendon Press Oxford(2003).
  2. P. Baird and S. Gudmundsson, p-Harmonic maps and minimal submanifolds, Math. Ann., 294(1992), 611-624. https://doi.org/10.1007/BF01934344
  3. B. Bojarski and T. Iwaniec, p-Harmonic equation and quasiregular mappings, Banach Center Publ., 19(1)(1987), 25-38. https://doi.org/10.4064/-19-1-25-38
  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. Mohammed Cherif, K. Zagga and S. Ouakkas, On the generalized of harmonic and bi-harmonic maps, Int. Electron. J. Geom., 5(1)(2012), 90-100.
  6. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86(1964), 109-160. https://doi.org/10.2307/2373037
  7. A. Fardoun, On equivariant p-harmonic maps, Ann. Inst. Henri. Poincar'e, 15(1998), 25-72. https://doi.org/10.1016/s0294-1449(99)80020-1
  8. G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A., 7(4)(1986), 389-402.
  9. J. Liu, Liouville-type Theorems of p-harmonic Maps with free Boundary Values, Hiroshima Math., 40(2010), 333-342.
  10. A. Mohammed Cherif, On the p-harmonic and p-biharmonic maps, J. Geom., 109(2018), 11p.
  11. D. J. Moon, H. Liu and S. D. Jung, Liouville type theorems for p-harmonic maps, J. Math. Anal. Appl., 342(2008), 354-360. https://doi.org/10.1016/j.jmaa.2007.12.018
  12. N. Nakauchi, A Liouville type theorem for p-harmonic maps , Osaka J. Math., 35(1998), 303-312.
  13. S. Ouakkas, R. Nasri and M. Djaa, On the f-harmonic and f-biharmonic maps, J. P. Journal of Geom. and Top., 10(1)(2010), 11-27.
  14. E. Remli and A. Mohammed Cherif, On the Generalized of p-harmonic and f-harmonic Maps, Kyungpook Math. J., 61(2021), 169-179.