On L1-biharmonic hypersurfaces with constant mean curvature in the Lorentz-Minkowski space

Firooz Pashaie

Abstract


In this paper, we study isometrically immersed spacelike hypersurfaces in the Lorentz-
Minkowski space, whose position vector eld satises an extended version of biharmonicity
condition, named L1-biharmonicity, where L1 stands for the linearized operator of the rst variation
of the 2-th mean curvature arising from normal variations of Hypersurface. A well-known
conjecture of Bang-Yen Chen says that any biharmonic Euclidean submanifold is minimal. We
introduce and verify an advanced version of Chen's conjecture, replacing the Laplace operator
by L1. For any spacelike hypersurface, having assumed that Mn has three distinct principal
curvatures and constant ordinary mean curvature, we prove that it must be 1-maximal.


Full Text:

PDF

References


Airy, G. B., On the strains in the interior of beams, Philos. Trans. R. Soc. London Ser. A, 153 (1863),

-79.

Akutagawa, K., Maeta, S., Biharmonic properly immersed submanifolds in Euclidean spaces, Geom.

Ded., 164 (2013), 351-355.

Alias, L. J., Gurbuz, N., An extension of Takahashi theorem for the linearized operators of the higher

order mean curvatures, Geom. Ded., 121 (2006), pp. 113-127.

Aminian, M., Kashani, S.M.B., Lk-biharmonic hypersurfaces inthe Euclidean space, Taiwan. J. Math.,

Online (DOI:10.11650/tjm.18.2014.4830).

Caminha, A., On spacelike hypersurfaces of constant sectional curvature lorentz manifolds, J. Geom.

phys., 56 (2006), pp. 1144-1174.

Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, Ser. Pure Math., World Sci. Pub.

Co., Singapore (2014).

Chen, B. Y., Some open problems and conjetures on submanifolds of nite type, Soochow J. Math., 17

(1991), pp. 169-188.

Dimitric, I., Submanifolds of En with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sin., 20

(1992), pp. 53-65.

Eells, J., Wood, J. C., Restrictions on harmonic maps of surfaces, Topology, 15 (1976), pp. 263-266.

Fu, Y., Biharmonic hypersurfaces with three distinct principal curvatures Euclidean 5-space, J. Geom.

Phys., 75 (2014), pp. 113-119.

Hasanis, T., Vlachos, T., Hypersurfaces in E4 with harmonic mean curvature vector eld, Math. Nachr.,

(1995), pp. 145-169.

Kashani, S.M.B., On some L1-nite type (hyper)surfaces in Rn+1, Bull. Kor. Math. Soc., 46 (1), (2009),

pp. 35-43.

Lucas, P., Ramirez-Ospina, H.F., Hypersurfaces in the Lorentz-Minkowski space satisfying Lk = A +

b, Geom. Dedicata, 153 (2011), 151-175.

J. Marsdan, F. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general

relativity, Phys. Rep. 66 (1980) pp. 109-139.

Maxwell, J. C., On reciprocal diagrams in space, and their relation to Airy's function of stress, Proc.

London. Math. Soc., 2 (1868), pp. 102-104.

Mohammadpouri, A., Kashani, S.M.B., Pashaie, F., On some L1- nite type Euclidean surfaces, Acta

Math. Vietnam., 38 (2013), pp. 303316.

Mohammadpouri, A., Pashaie, F., L1-biharmonic hypersurfaces with three distinct principal curvatures

in Euclidean 5-spaces, Func. Anal. Appl. comp., 7:1 (2015) 67 - 75.

O'Neill, B., Semi-Riemannian geometry with applicatins to relativity, Academic Press Inc. (1983).

Pashaie, F., Kashani, S.M.B., Spacelike hypersurfaces in Riemannian or Lorentzian space forms satis-

fying Lkx = Ax + b, Bull. Iran. Math. Soc., 39 (1), (2013), pp. 195-213.

Pashaie, F., Mohammadpouri, A., Lk-biharmonic hypersurfaces in pseudo-Euclidean space E4

, TWMS

J. Pure Appl. Math. (To appear).


Refbacks

  • There are currently no refbacks.