The minimality of biharmonic hypersurfaces in pseudo-Euclidean spaces

Li Du; Xiaoqin Yuan; Li Du; Xiaoqin Yuan

doi:10.3934/era.2023081

Electronic Research Archive

2023, Volume 31, Issue 3: 1587-1595. doi: 10.3934/era.2023081

Previous Article Next Article

Research article

The minimality of biharmonic hypersurfaces in pseudo-Euclidean spaces

Li Du ^,,
Xiaoqin Yuan

School of Science, Chongqing University of Technology, Chongqing 400054, China

Received: 29 October 2022 Revised: 21 December 2022 Accepted: 29 December 2022 Published: 31 January 2023

In this paper, we investigate the minimality of biharmonic hypersurfaces with some recurrent operators in a pseudo-Euclidean space.
- pseudo-Euclidean spaces,
- biharmonic hypersurfaces,
- recurrent operators,
- minimal hypersurfaces
Citation: Li Du, Xiaoqin Yuan. The minimality of biharmonic hypersurfaces in pseudo-Euclidean spaces[J]. Electronic Research Archive, 2023, 31(3): 1587-1595. doi: 10.3934/era.2023081

Related Papers:

Abstract

In this paper, we investigate the minimality of biharmonic hypersurfaces with some recurrent operators in a pseudo-Euclidean space.

References

[1]	B. Y. Chen, S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math., 52 (1998), 167–185. https://doi.org/10.2206/kyushujm.52.167 doi: 10.2206/kyushujm.52.167
[2]	G. Y. Jiang, Some non-existence theorems of 2-harmonic isometric immersions into Euclidean spaces, Chin. Ann. Math. Ser. A, 8 (1987), 376–383.
[3]	F. D. O. Leuven, Hypersurfaces of $\mathbb{E}^4$ with harmonic mean curvature vector, Math. Nachr., 196 (1998), 61–69. https://doi.org/10.1002/mana.19981960104 doi: 10.1002/mana.19981960104
[4]	T. Hasanis, T. Vlachos, Hypersurface in $\mathbb{E}^4$ with harmonic mean curvature vector field, Math. Nachr., 172 (1995), 145–169. https://doi.org/10.1002/mana.19951720112 doi: 10.1002/mana.19951720112
[5]	Y. Fu, M. C. Hong, X. Zhan, On Chen's biharmonic conjecture for hypersurfaces in $\mathbb{R}^5$, Adv. Math., 383 (2021), 107697. https://doi.org/10.1016/j.aim.2021.107697 doi: 10.1016/j.aim.2021.107697
[6]	B. Y. Chen, M. I. Munteanu, Biharmonic ideal hypersurfaces in Euclidean spaces, Differ. Geom. Appl., 31 (2013), 1–16. https://doi.org/10.1016/j.difgeo.2012.10.008 doi: 10.1016/j.difgeo.2012.10.008
[7]	S. Montaldo, C. Oniciuc, A. Ratto, On cohomogeneity one biharmonic hypersurfaces into the Euclidean space, J. Geom. Phys., 106 (2016), 305–313. https://10.1016/j.geomphys.2016.04.012 doi: 10.1016/j.geomphys.2016.04.012
[8]	N. Mosadegh, E. Abedi, Biharmonic hypersurfaces with recurrent operators in the Euclidean space, arXiv preprint, (2021), arXiv: math/2108.06193v1. https://doi.org/10.48550/arXiv.2108.06193
[9]	Y. L. Ou, B. Y. Chen, Biharmonic Submanifolds and Biharmonic Maps in Riemannian Geometry, World Scientific, Hackensack, USA, 2020. https://doi.org/10.1142/11610
[10]	B. Y. Chen, S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A, Math., 45 (1991), 323–347. https://doi.org/10.2206/kyushumfs.45.323 doi: 10.2206/kyushumfs.45.323
[11]	F. Defever, G. Kaimakamis, V. Papantoniou, Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space $\mathbb{E}_s^4$, J. Math. Anal. Appl., 315 (2006), 276–286. https://doi.org/10.1016/j.jmaa.2005.05.049 doi: 10.1016/j.jmaa.2005.05.049
[12]	Y. Fu, Z. H. Hou, D. Yang, X. Zhan, Biharmonic hypersurface in $\mathbb{E}_s^5$, Acta Math. Sinica (Chinese Ser.), 65 (2022), 335–352.
[13]	J. C. Liu, C. Yang, Hypersurfaces in $\mathbb{E}^{n+1}_s$ satisfying $\Delta\vec{H} = \lambda\vec{H}$ with at most three distinct principal curvatures, J. Math. Anal. Appl., 419 (2014), 562–573. http://dx.doi.org/10.1016/j.jmaa.2014.04.066 doi: 10.1016/j.jmaa.2014.04.066
[14]	L. Du, On $\eta$-biharmonic hypersurfaces with constant scalar curvature in higher dimensional pseudo-Riemannian space forms, J. Math. Anal. Appl., 518 (2023), 126670. https://doi.org/10.1016/j.jmaa.2022.126670 doi: 10.1016/j.jmaa.2022.126670
[15]	B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Hackensack, USA, 2014.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)