Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Generalized of p-harmonic and f-harmonic Maps

  • Remli, Embarka (Mascara University, Faculty of Exact Sciences, Department of Mathematics) ;
  • Cherif, Ahmed Mohammed (Mascara University, Faculty of Exact Sciences, Department of Mathematics)
  • Received : 2020.02.15
  • Accepted : 2020.08.18
  • Published : 2021.03.31
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Abstract

In this paper, we extend the definition of p-harmonic maps between two Riemannian manifolds. We prove a Liouville type theorem for generalized p-harmonic maps. We present some new properties for the generalized stress p-energy tensor. We also prove that every generalized p-harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a homothetic vector field satisfying some condition is constant.

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References

  1. P. Baird and S. Gudmundsson, p-Harmonic maps and minimal submanifolds, Math. Ann., 294(1992), 611-624. https://doi.org/10.1007/BF01934344
  2. P. Baird and J. C. Wood, Harmonic morphisms between Riemannain manifolds, Clarendon Press Oxford, 2003.
  3. B. Bojarski and T. Iwaniec, p-Harmonic equation and quasiregular mappings, Partial differential equations (Warsaw, 1984), 25-38, Banach Center Publ. 19, PWN, Warsaw, 1987.
  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 and A. Mohammed Cherif, On generalized f-harmonic maps and liouville type theorem, Konuralp J. Math., 4(1)(2016), 33-44.
  6. 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.
  7. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86(1964), 109-160. https://doi.org/10.2307/2373037
  8. A. Fardoun, On equivariant p-harmonic maps, Ann. Inst. Henri. Poincare Anal. Non Lineaire, 15(1998), 25-72. https://doi.org/10.1016/s0294-1449(99)80020-1
  9. W. Kuhnel and H. Rademacher, Conformal vector fields on pseudo-Riemannian spaces, Differential Geom. Appl., 7(1997), 237-250. https://doi.org/10.1016/S0926-2245(96)00052-6
  10. J. Liu, Liouville-type theorems of p-harmonic maps with free boundary values, Hiroshima Math. J., 40(2010), 333-342. https://doi.org/10.32917/hmj/1291818848
  11. A. Mohammed Cherif, Some results on harmonic and bi-harmonic maps, Int. J. Geom. Methods Mod. Phys., 14(2017), 1750098, 8 pp. https://doi.org/10.1142/S0219887817500980
  12. A. Mohammed 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
  13. 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
  14. N. Nakauchi, A Liouville type theorem for p-harmonic maps, Osaka J. Math., 35(1998), 303-312.
  15. S. Ouakkas, R. Nasri and M. Djaa, On the f-harmonic and f-biharmonic maps, JP J. Geom. Topol., 10(1)(2010), 11-27.
  16. M. Rimoldi and G.Veronelli, f-harmonic maps and applications to gradient Ricci solitons, arXiv:1112.3637, (2011). https://doi.org/10.1016/j.difgeo.2013.06.001
  17. Z. P. Wang, Y. L. Ou, and H. C. Yang, Biharmonic maps from tori into a 2-sphere, Chin. Ann. Math. Ser. B, 39(5)(2018), 861878.
  18. Y. Xin, Geometry of harmonic maps, Fudan University, 1996.
  19. K. Yano and T. Nagano, The de Rham decomposition, isometries and affine transformations in Riemannian space, Japan. J. Math., 29(1959), 173-184. https://doi.org/10.4099/jjm1924.29.0_173