ON GENERALIZED f-HARMONIC MAPS AND LIOUVILLE TYPE THEOREM

Year 2016, Volume: 4 Issue: 1, 33 - 44, 01.04.2016

Mustapha Djaa Ahmed Mohamed Cherıf

Abstract

In this paper, we prove that every semi-conformal harmonic map between Riemannian manifolds is a generalized f-harmonic map. We also prove a Liouville type theorem for f-harmonic maps in general sense from IRm onto a Riemannian manifold N with non-positive sectional curvature, where f 2 C1(IRm  N) is a smooth positive function which satis es some suitable conditions.

Keywords

f-harmonic maps; f-conformal maps; Liouville Theorem.

References

• [1] Baird P., Fardoun A. and Ouakkas S., Liouville-type Theorems for Biharmonic Maps between Riemannian Manifolds, Advances in Calculus of Variations. 3, Issue 1 (2009), 4968.
• [2] Calderbank D. M. J., Gauduchon P. and Herzlich M., On the Kato inequality in Riemannian geometry, Sminaires et Congrs 4, SMF 2000, p. 95-113.
• [3] Cheng, S. Y., Liouville Theorem for Harmonic Maps, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, (1979), 147-151, Proc.Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980.
• [4] Djaa M., Mohamed Cherif A., Zegga K. And Ouakkas S., On the Generalized of Harmonic and Bi-harmonic Maps, international electronic journal of geometry, 5 no. 1(2012), 90-100.
• [5] Djaa M. and Mohamed Cherif A., On the Generalized f-Biharmonic Maps and Stress f- Bienergy Tensor, Journal of Geometry and Symmetry in Physics, JGSP 29(2013), 65-81.
• [6] Baird P., Wood J.C., Harmonic Morphisms between Riemannain Manifolds, Clarendon Press, Oxford, 2003.
• [7] P. Brard, A note on Bochner type theorems for complete manifolds, Manuscripta Math. 69 (1990) 261266.
• [8] Liu J., Liouville-type Theorems of p-harmonic Maps with free Boundary Values, Hiroshima Math.40 (2010), 333-342
• [9] Eells, J. Jr. and Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer.J. Math. 86 1964 109-160.
• [10] Ouakkas S., Nasri R. and Djaa M., On the f-harmonic and f-biharmonic Maps, J. P. Journal of Geometry. and Topology. 10 1 (2010), 11-27.
• [11] A.M. Cherif and M. Djaa, Geometry of energy and bienergy variations between Riemannian manifolds, Kyungpook Mathematical Journal, 55(2015), pp 715-730.
• [12] Mohammed Cherif A. and Djaa M.,On Generalized f-harmonic Morphisms, Com- ment.Math.Univ.Carolin. 55,1 (2014) 17-77.
• [13] Mohamed Cherif A., Elhendi H. and Terbeche M., On Generalized Conformal Maps, Bulletin of Mathematical Analysis and Applications, 4 Issue 4 (2012), 99-108.
• [14] Rimoldi M. and Veronelli G., f-Harmonic Maps and Applications to Gradient Ricci Solitons, arXiv:1112.3637, (2011).
• [15] Schoen R. M. and Yau, S.-T., Harmonic Maps and the Topology of Stable Hypersurfaces and Manifolds with Non-negative Ricci Curvature, Comment. Math. Helv. 51 (1976), no.3, 333-341.
• [16] Xu Wang D., Harmonic Maps from Sooth Metric Measure Spaces, Internat. J. Math. 23 (2012), no. 9, 1250095, 21.
• [17] Young W.C. , On the multiplication of successions of Fourier constants, Proc. Royal Soc. Lond. 87 (1912), 331-339.
• [18] Zegga K., Djaa M. and A.M. Cherif, On the f-biharmonic maps and submanifolds, Kyungpook Mathematical Journal, 55 (2015), pp 157-168.