On extended biharmonic hypersurfaces with three curvatures
DOI:
https://doi.org/10.22199/issn.0717-6279-5443Keywords:
Minkowski space, L₁-biharmonic, isoparametric, 1-minimalAbstract
The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics and differential geometry, arises from the variation problems of ordinary mean curvature vector field. Generally, harmonic submanifolds are biharmonic, but not vice versa. Of course, many examples of biharmonic hypersurfaces are harmonic. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every biharmonic submanifold is harmonic. Although the conjecture has not been proven (in general case), it has been affirmed in many cases, and this has led to its spread to various types of submanifolds. Inspired by the conjecture, we study the Lorentz submanifolds of the Lorentz-Minkowski spaces. We consider an advanced versión of the conjecture (namely, L1-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean 5-space L5 := E15 (i.e. the Minkowski 5-space). We confirm the extended conjecture on Lorentz hypersurfaces with three principal curvatures.
References
K. Akutagawa and S. Maeta, “Biharmonic properly immersed submanifolds in Euclidean spaces”, Geometriae Dedicata, vol. 164, pp. 351-355, 2013. https://doi.org/10.1007/s10711-012-9778-1
L. J. Alias and N. Gürbüz, “An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures”, Geometriae Dedicata, vol. 121, pp. 113-127, 2006. https://doi.org/10.1007/s10711-006-9093-9
A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and B. J. Papantoniou, “Biharmonic Lorentz hypersurfaces in E14”, Pacific Journal of Mathematics, vol. 229, pp. 293-306, 2007. https://doi.org/10.2140/pjm.2007.229.293
B. Y. Chen, “Some open problems and conjetures on submanifolds of finite type”, Soochow Journal of Mathematics, vol. 17, pp. 169-188, 1991.
F. Defever, “Hypersurfaces of E4 with harmonic mean curvature vector”, Mathematische Nachrichten, vol. 196, pp. 61-69, 1998. https://doi.org/10.1002/mana.19981960104
I. Dimitrić, Submanifolds of En with harmonic mean curvature vector”, Bulletin Institute Mathematics Academia Sinica, vol. 20, pp. 53-65, 1992. [On line]. Available: https://bit.ly/3W6tXhR
J. Eells and J. C. Wood, “Restrictions on harmonic maps of surfaces”, Topology, vol. 15, pp. 263-266, 1976. https://doi.org/10.1016/0040-9383(76)90042-2
R. S. Gupta, “Biharmonic hypersurfaces in E5”, Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica, Tom. 62, f. 2, vol. 2, pp. 585-593, 2016.
T. Hasanis and T. Vlachos, “Hypersurfaces in E4 with harmonic mean curvature vector field”, Mathematische Nachrichten, vol. 172, pp. 145-169, 1995. https://doi.org/10.1002/mana.19951720112
S. M. B. Kashani, “On some L1-finite type (hyper)surfaces in Rn+1”, Bulletin of the Korean Mathematical Society, vol. 46: 1, pp. 35-43, 2009. https://doi.org/10.4134/bkms.2009.46.1.035
P. Lucas and H. F. Ramirez-Ospina, “Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ = Aψ + b”, Geometriae Dedicata, vol. 153, pp. 151-175, 2011. https://doi.org/10.1007/s10711-010-9562-z
M. A. Magid, “Lorentzian isoparametric hypersurfaces”, Pacific Journal of Mathematic, vol. 118, no. 1, pp. 165-197, 1985. https://doi.org/10.2140/pjm.1985.118.165
F. Pashaie and A. Mohammadpouri, “Lk-biharmonic spacelike hypersurfaces in Minkowski 4-space E14 , Sahand Communications in Mathematical Analysis (SCMA), vol. 5: 1, pp. 21-30, 2017. [On line]. Available: https://bit.ly/3W2ctmN
B. O’Neill, Semi-Riemannian Geometry with Applicatins to Relativity. Academia Press Inc., 1983.
F. Pashaie and S. M. B. Kashani, “Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying Lkx = Ax + b”, Bulletin of the Iranian Mathematical Society, vol. 39, no. 1, pp. 195-213, 2013. [On line]. Available: https://bit.ly/3itEOVx
F. Pashaie and S. M. B. Kashani, “Timelike hypersurfaces in the Lorentzian standard space forms satisfying Lkx = Ax+b”, Mediterranean Journal of Mathematics, vol. 11, no. 2, pp. 755-773, 2014. https://doi.org/10.1007/s00009-013-0336-3
A. Z. Petrov, Einstein Spaces. Pergamon Press: Oxford, 1969.
N. C. Turgay, “Some classifications of biharmonic Lorentzian hyper-surfaces in Minkowski 5-space E51”, Mediterranean Journal of Mathematics, 2014, https://doi.org/10.1007/s00009-014-0491-1
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