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Keywords:
$f$-biharmonic maps; $f$-biharmonic hypersurface
$f$-biharmonic maps; $f$-biharmonic hypersurface
Summary:
In the present paper we give some properties of $f$-biharmonic hypersurfaces in real space forms. By using the $f$-biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$-biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$-biharmonic vertical cylinders in $S^{2}\times \mathbb{R}$.
References:
[1] Caddeo R., Montaldo S., Oniciuc C.: Biharmonic submanifolds of $S^3$. Internat. J. Math. 12 (2001), no. 8, 867–876. MR 1863283
[2] Chen B.-Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17 (1991), no. 2, 169–188. MR 1143504
[3] Cieśliński J., Sym A., Wesselius W.: On the geometry of the inhomogeneous Heisenberg ferromagnet: nonintegrable case. J. Phys. A. 26 (1993), no. 6, 1353–1364. DOI 10.1088/0305-4470/26/6/017 | MR 1212007
[4] Eells J., Lemaire L.: A report on harmonic maps. Bull. London Math. Soc. 10 (1978), no. 1, 1–68. DOI 10.1112/blms/10.1.1 | MR 0495450 | Zbl 0401.58003
[5] Eells J. Jr., Sampson J. H.: Harmonic mappings of the Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160. DOI 10.2307/2373037 | MR 0164306
[6] Jiang G. Y.: $2$-harmonic isometric immersions between Riemannian manifolds. Chinese Ann. Math. Ser. A 7 (1986), no. 2, 130–144 (Chinese); English summary in Chinese Ann. Math. Ser. B 7 (1986), no. 2, 255. MR 0858581
[7] Jiang G. Y.: $2$-harmonic maps and their first and second variation formulas. Chinese Ann. Math. Ser. A. 7 (1986), no. 4, 389–402 (Chinese); English summary in Chinese Ann. Math. Ser. B 7 (1986), no. 4, 523. MR 0886529
[8] Keleş S., Perktaş S. Y., Kiliç E.: Biharmonic curves in Lorentzian para-Sasakian manifolds. Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 2, 325–344. MR 2666434
[9] Li Y., Wang Y.: Bubbling location for $F$-harmonic maps and inhomogeneous Landau-Lifshitz equations. Comment. Math. Helv. 81 (2006), no. 2, 433–448. MR 2225633
[10] Lu W.-J.: On $f$-bi-harmonic maps and bi-$f$-harmonic maps between Riemannian manifolds. Sci. China Math. 58 (2015), no. 7, 1483–1498. DOI 10.1007/s11425-015-4997-1 | MR 3353985
[11] Montaldo S., Oniciuc C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Un. Mat. Argentina 47 (2006), no. 2, 1–22. MR 2301373
[12] Ou Y.-L.: Biharmonic hypersurfaces in Riemannian manifolds. Pacific J. Math. 248 (2010), no. 1, 217–232. DOI 10.2140/pjm.2010.248.217 | MR 2734173
[13] Ou Y.-L.: Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces. J. Geom. Phys. 62 (2012), no. 4, 751–762. DOI 10.1016/j.geomphys.2011.12.014 | MR 2888980
[14] Ou Y.-L.: On $f$-biharmonic maps and $f$-biharmonic submanifolds. Pacific J. Math. 271 (2014), no. 2, 461–477. DOI 10.2140/pjm.2014.271.461 | MR 3267537
[15] Ou Y.-L., Tang L.: On the generalized Chen's conjecture on biharmonic submanifolds. Michigan Math. J. 61 (2012), no. 3, 531–542. DOI 10.1307/mmj/1347040257 | MR 2975260
[16] Ou Y.-L., Wang Z.-P.: Constant mean curvature and totally umbilical biharmonic surfaces in $3$-dimensional geometries. J. Geom. Phys. 61 (2011), no. 10, 1845–1853. DOI 10.1016/j.geomphys.2011.04.008 | MR 2822453
[17] Perktaş S. Y., Kiliç E.: Biharmonic maps between doubly warped product manifolds. Balkan J. Geom. Appl. 15 (2010), no. 2, 159–170. MR 2608547
[18] Perktaş S. Y., Kiliç E., Keleş S.: Biharmonic hypersurfaces of LP-Sasakian manifolds. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) 57 (2011), no. 2, 387–408. MR 2933391
[19] Rimoldi M., Veronelli G.: Topology of steady and expanding gradient Ricci solitons via $f$-harmonic maps. Differetial. Geom. Appl. 31 (2013), no. 5, 623–638. DOI 10.1016/j.difgeo.2013.06.001 | MR 3093493
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