Some results of F-biharmonic maps
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Abstract
In this paper, we give the notion of F-biharmonic maps, which is a generalization of biharmonic maps. We derive the first variation formula which yields F-biharmonic maps. Then we investigate the harmonicity of $F$-biharmonic maps under the curvature conditions on the target manifold (N; h). We also introduce the stress $F$-bienergy tensor $S_{F;2}$. Then by using the stress F-bienergy tensor $S_{F;2}$, we obtain some non existence results of proper $F$-biharmonic maps under the assumption that $S_{F;2} = 0$. Moreover, we derive some monotonicity formulas for the special case of biharmonic map, i.e. $F$-biharmonic map with $F(t) = t$. Then, by using these monotonicity formulas, we obtain new results on the non existence of proper biharmonic isometric immersions from complete manifolds.
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Han, Y., & Feng, S.
(2014).
Some results of F-biharmonic maps.
Acta Mathematica Universitatis Comenianae, 83(1), 47-66.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/77/43
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References
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[3] R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps, Contemp. Math., 288 (2001), 286-290.
[4] Y.X. Dong, S. S. Wei, On vanishing theorems for vector bundle valued p-forms and their applications, Comm. Math. Phys. 304 (2011), 329-368.
[5] Y.X. Dong, H.Z. Lin and G.L. Yang, Liouville theorems for F-harmonic maps and their applications, arXiv:1111.1882v1[math.DG] 8 Nov 2011.
[6] P. Hornung, R. Moser, Intrinsically p-biharmonic maps, preprint (Opus: University of Bath online publication store).
[7] J. Eells, L. Lemaire, Selected topics in harmonic maps, CBMS, 50, Amer. Math. Soc,1983.
[8] R.E. Greene, H. Wu, Function theory on manifolds which posses pole, in: Lecture Notes in Mathematics, V.699, Springer-veriag, Berlin, Heidelberg, New York,1979.
[9] G.Y. Jiang, 2-harmonic maps and their rst and second variational formulas, Chinese Ann. Math. 7A (1986), 388-402; the English translation, Note di Matematica, 28 (2009), 209-232.
[10] G.Y. Jiang, The conservation law for 2-harmonic maps between Riemannian manifolds. Acta Math. Sin. 30 (1987), 220-225.
[11] M. Kassi, A Liouville theorems for F-harmonic maps with nite F-energy, Electonic Journal Di Equa. 15 (2006), 1-9.
[12] J.C. Liu, Liouville theorems of stable F-harmonic maps for compact convex hypersurfaces, Hiroshima Math.J. 36 (2006), 221-234.
[13] E. Loubeau, C. Oniciuc, The index of biharmonic maps in spheres, Compositio Math. 141(2005),729-745.
[14] E. Loubeau, S. Montaldo and C. Oniciuc, The stress-energy tensor for biharmonic maps, Math.Z. 259 (2008), 503-524.
[15] N. Nakauchi, H. Urakawa and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, arXiv:1201.6457v4.
[16] C. Oniciuc, Biharmonic maps between Riemannian manifolds, Analele Stintice Ale Univer. Al.l.cuza Iasi, (2002), 237-248.
[17] S. Ouakkas, R. Nasri, and M. Djaa, On the f-harmonic and f-biharmonic maps, JP J. Geom. Topol. 10 (1), (2010), 11-27.
[18] C. Wang, Remarks on biharmonic maps into spheres, Calc.Var.Partial Differential Equations, 21 (2004), 221-242.
[19] C. Wang, Biharmonic maps from R^4 into Riemannian manifold. Math. Z 247 (2004), 65-87.