On $\lambda $-biminimal conformal immersions

Abstract

In the present paper, we consider $\lambda $-biminimal conformal immersions. We find the Euler-Lagrange equation of $\lambda $-biminimal immersions under conformal change of metrics. We also consider $\lambda $-biminimal immersions from a surface $\left( M^{2},g\right) $ to a Riemannian manifold $\left( N^{3},h\right) $ under homothetic change of metric and give an example.

Keywords

• $\lambda $-biminimal immersion
• $\lambda $ -biminimal conformal immersion
• $\lambda $-biminimal homothetic immersion

References

1. [1] J.T. Cho and J.-E. Lee, Pseudohermitian biminimal Legendre surfaces in the 5- dimensional sphere, Osaka J. Math. 52 (4), 1063–1077, 2015.
2. [2] J.Jr. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 109–160, 1964.
3. [3] F. Gürler and C. Özgür, f-biminimal immersions, Turkish J. Math. 41 (3), 564–575, 2017.
4. [4] J. Inoguchi, Biminimal submanifolds in contact 3-manifolds, Balkan J. Geom. Appl. 12 (1), 56–67, 2007.
5. [5] J. Inoguchi and J.-E. Lee, Biminimal curves in 2-dimensional space forms, Commun. Korean Math. Soc. 27 (4), 771–780, 2012.
6. [6] G.Y. Jiang, 2-Harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A, 7 (4), 389–402, 1986.
7. [7] F. Karaca, f-biminimal submanifolds of generalized space forms, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 68 (2), 1301–1315, 2019.
8. [8] E. Loubeau and S. Montaldo, Biminimal immersions, Proc. Edinb. Math. Soc. 51 (2), 421–437, 2008.

9. [9] S. Maeta, Biminimal properly immersed submanifolds in the Euclidean spaces, J. Geom. Phys. 62 (11), 2288–2293, 2012.
10. [10] Y.-L. Ou, On conformal biharmonic immersions, Ann. Global Anal. Geom. 36 (2), 133–142, 2009.
11. [11] Y.-L. Ou, Biharmonic conformal immersions into three-dimensional manifolds, Mediterr. J. Math. 12 (2), 541–554, 2015.
12. [12] Y.-L. Ou and B.Y. Chen, Biharmonic submanifolds and biharmonic maps in Rie- mannian geometry, Hackensack, NJ: World Scientific, 2020.
13. [13] T. Sasahara, A classification result for biminimal Lagrangian surfaces in complex space forms, J. Geom. Phys. 60 (6-8), 884–895, 2010.
14. [14] T. Sasahara, Classification results for -biminimal surfaces in 2-dimensional complex space forms, Acta Math. Hungar. 144 (2), 433–448, 2014.
15. [15] K. Yano and M. Obata, Conformal changes of Riemannian metrics, J. Differential Geometry, 4, 53–72, 1970.