Geodesic
$f$-biminimal maps between Riemannian manifolds
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 893-905
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We give the definition of $f$-biminimal submanifolds and derive the equation for $f$-biminimal submanifolds. As an application, we give some examples of $f$-biminimal manifolds. Finally, we consider $f$-minimal hypersurfaces in the product space $\mathbb {R}^{n}\times \mathbb {S}^{1}(a)$ and derive two rigidity theorems.
We give the definition of $f$-biminimal submanifolds and derive the equation for $f$-biminimal submanifolds. As an application, we give some examples of $f$-biminimal manifolds. Finally, we consider $f$-minimal hypersurfaces in the product space $\mathbb {R}^{n}\times \mathbb {S}^{1}(a)$ and derive two rigidity theorems.
DOI : 10.21136/CMJ.2019.0328-17
Classification : 53B25, 53C40
Keywords: variational vector field; hypersurface; $f$-biminimal submanifold; mean curvature vector 
@article{10_21136_CMJ_2019_0328_17,
     author = {Zhao, Yan and Liu, Ximin},
     title = {$f$-biminimal maps between {Riemannian} manifolds},
     journal = {Czechoslovak Mathematical Journal},
     pages = {893--905},
     year = {2019},
     volume = {69},
     number = {4},
     doi = {10.21136/CMJ.2019.0328-17},
     mrnumber = {4039608},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0328-17/}
}
TY  - JOUR
AU  - Zhao, Yan
AU  - Liu, Ximin
TI  - $f$-biminimal maps between Riemannian manifolds
JO  - Czechoslovak Mathematical Journal
PY  - 2019
SP  - 893
EP  - 905
VL  - 69
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0328-17/
DO  - 10.21136/CMJ.2019.0328-17
LA  - en
ID  - 10_21136_CMJ_2019_0328_17
ER  - 
%0 Journal Article
%A Zhao, Yan
%A Liu, Ximin
%T $f$-biminimal maps between Riemannian manifolds
%J Czechoslovak Mathematical Journal
%D 2019
%P 893-905
%V 69
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0328-17/
%R 10.21136/CMJ.2019.0328-17
%G en
%F 10_21136_CMJ_2019_0328_17
Zhao, Yan; Liu, Ximin. $f$-biminimal maps between Riemannian manifolds. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 893-905. doi: 10.21136/CMJ.2019.0328-17

[1] Balmuş, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. Isr. J. Math. 168 (2008), 201-220. | DOI | MR | JFM

[2] Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications. These de Doctorat, Université Joseph-Fourier, Grenoble French (2003).

[3] Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of $\mathbb S^{3}$. Int. J. Math. 12 (2001), 867-876. | DOI | MR | JFM

[4] Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in spheres. Isr. J. Math. 130 (2002), 109-123. | DOI | MR | JFM

[5] Chen, B.-Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17 (1991), 169-188. | MR | JFM

[6] Chen, B.-Y., Ishikawa, S.: Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. Kyushu J. Math. 52 (1998), 167-185. | DOI | MR | JFM

[7] Cheng, X., Mejia, T., Zhou, D.: Eigenvalue estimate and compactness for closed $f$-minimal surfaces. Pac. J. Math. 271 (2014), 347-367. | DOI | MR | JFM

[8] Cheng, X., Mejia, T., Zhou, D.: Stability and compactness for complete $f$-minimal surfaces. Trans. Am. Math. Soc. 367 (2015), 4041-4059. | DOI | MR | JFM

[9] Dimitrić, I.: Submanifolds of $E^{m}$ with harmonic mean curvature vector. Bull. Inst. Math., Acad. Sin. 20 (1992), 53-65. | MR | JFM

[10] J. Eells, Jr., J. H. Sampson: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86 (1964), 109-160. | DOI | MR | JFM

[11] Fetcu, D., Oniciuc, C., Rosenberg, H.: Biharmonic submanifolds with parallel mean curvature in $\mathbb S^{n}\times \mathbb R$. J. Geom. Anal. 23 (2013), 2158-2176. | DOI | MR | JFM

[12] Hasanis, T., Vlachos, T.: Hypersurfaces in $E^{4}$ with harmonic mean curvature vector field. Math. Nachr. 172 (1995), 145-169. | DOI | MR | JFM

[13] Jiang, G.: 2-harmonic maps and their first and second variational formulas. Chin. Ann. Math., Ser. A 7 (1986), 389-402 Chinese. | DOI | MR | JFM

[14] Jiang, G.: Some nonexistence theorems on 2-harmonic and isometric immersions in Euclidean space. Chin. Ann. Math., Ser. A 8 (1987), 377-383 Chinese. | MR | JFM

[15] Li, X. X., Li, J. T.: The rigidity and stability of complete $f$-minimal hypersurfaces in $\mathbb{R}\times\mathbb{S}^{1}(a)$. (to appear) in Proc. Am. Math. Soc. | MR

[16] Liu, G.: Stable weighted minimal surfaces in manifolds with non-negative Bakry-Emery Ricci tensor. Commun. Anal. Geom. 21 (2013), 1061-1079. | DOI | MR | JFM

[17] Lu, W. J.: On $f$-bi-harmonic maps and bi-$f$-harmonic maps between Riemannian manifolds. Sci. China, Math. 58 (2015), 1483-1498. | DOI | MR | JFM

[18] Ou, Y.-L., Wang, Z.-P.: Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries. J. Geom. Phys. 61 (2011), 1845-1853. | DOI | MR | JFM

[19] Ouakkas, S., Nasri, R., Djaa, M.: On the $f$-harmonic and $f$-biharmonic maps. JP J. Geom. Topol. 10 (2010), 11-27. | MR | JFM

Cité par Sources :