On λ-biharmonic hypersurfaces in $\boldsymbol{L}^{\boldsymbol{m}} \times \mathbb{R}$

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper investigated λ-biharmonic hypersurfaces in $\boldsymbol{L}^{\boldsymbol{m}} imes \mathbb{R}$, where L^m represents an Einstein space, and ℝ denotes the real line. We demonstrated that such hypersurfaces with some curvature conditions are either of two types: minimal or vertical cylinders over λ-biharmonic hypersurfaces in L^m. Particularly, when the Einstein space L^m has constant sectional curvature, we classify λ-biharmonic hypersurfaces as totally umbilical or semi-parallel.

Keywords

λ-biharmonic hypersurfaces / product manifold / minimal / vertical cylinders

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type , 2nd edition, World Scientific, Hackensack, 2014.

[2] B. Y. Chen, Null 2-type surfaces in E³ are circular cylinders , Kodai Math. J., 11 (1988), 295-299. https://doi.org/10.2996/kmj/1138038880

[3] A. Ferrández, P. Lucas, Null finite type hypersurfaces in space forms, Kodai Math. J., 14 (1991), 406-419. https://doi.org/10.2996/kmj/1138039464

[4] B. Y. Chen, O. J. Garay, δ(2)-ideal null 2-type hypersurfaces of Euclidean space are spherical cylinders, Kodai Math. J., 35 (2012), 382-391. https://doi.org/10.2996/kmj/1341401058

[5] L. Du, Submanifolds Satisfying the Condition τ₂(φ)=ητ(φ) in Pseudo-Riemannian Space Forms , Ph.D thesis, Northwest Normal University, 2016.

[6] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, Hypersurfaces of E_s⁴ with proper mean curvature vector , J. Math. Soc. Jpn., 59 (2007), 797-809. https://doi.org/10.2969/jmsj/05930797

[7] Y. Fu, Null 2-type hypersurfaces with at most three distinct principal curvatures in Euclidean space, Taiwan. J. Math., 19 (2015), 519-533. https://doi.org/10.11650/tjm.19.2015.4847

[8] Y. Fu, M. C. Hong, X. Zhan, On Chen’s biharmonic conjecture for hypersurfaces in R⁵ , Adv. Math., 383 (2021), 107697. https://doi.org/10.1016/j.aim.2021.107697

[9] T. Hasanis, T. Vlachos, Hypersurfaces with constant scalar curvature and constant mean curvature, Ann. Glob. Anal. Geom., 13 (1995), 69-77. https://doi.org/10.1007/BF00774569

[10] J. C. Liu, C. Yang, Hypersurfaces in E_sⁿ⁺¹ satisfying ΔH=λH with at most three distinct principal curvatures , J. Math. Anal. Appl., 419 (2014), 562-573. https://doi.org/10.1016/j.jmaa.2014.04.066

[11] A. L. Albujer, J. A. Aledo, L. J. Alias, On the scalar curvature of hypersurfaces in spaces with a Killing field, Adv. Geom., 10 (2010), 478-503. https://doi.org/10.1515/advgeom.2010.017

[12] F. Dillen, J. Fastenakels, J. V. der Veken, Rotation hypersurfaces in Sⁿ×R and Hⁿ×R , Note Mat., 29 (2009), 41-54. https://doi.org/10.1285/i15900932v29n1p41

[13] S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670. https://doi.org/10.1512/iumj.1976.25.25051

[14] Y. Luo, S. Maeta, Biharmonic hypersurfaces in a sphere, Proc. Am. Math. Soc., 145 (2017), 3109-3116. https://doi.org/10.1090/proc/13320

[15] D. Fetcu, C. Oniciuc, H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in Sⁿ×R , J. Geom. Anal., 23 (2013), 2158-2176. https://doi.org/10.1007/s12220-012-9323-3

[16] S. Andronic, S. Nistor, Gap results for biharmonic submanifolds in spheres, J. Math. Anal. Appl., 548 (2025), 129378. https://doi.org/10.1016/j.jmaa.2025.129378

[17] Y. Fu, S. Maeta, Y. L. Ou, Biharmonic hypersurfaces in a product space L^m×R , Math. Nachr., 294 (2021), 1724-1741. https://doi.org/10.1002/mana.201900457

[18] G. Calvaruso, D. Kowalczyk, J. V. der Veken, On extrinsically symmetric hypersurfaces of Hⁿ×R , Bull. Aust. Math. Soc., 82 (2010), 390-400. https://doi.org/10.1017/S0004972710001760

[19] J. V. der Veken, L. Vrancken, Parallel and semi-parallel hypersurfaces of Sⁿ×R , Bull. Braz. Math. Soc., 39 (2008), 355-370. https://doi.org/10.1007/s00574-008-0010-8