scholarly journals On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

2020 ◽  
Vol 51 (4) ◽  
pp. 313-332
Author(s):  
Firooz Pashaie
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Principal Curvatures ◽  
First Variation ◽  
Chen’S Conjecture ◽  
The First Variation ◽  
Linearized Operator ◽  
Euclidean Submanifolds

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

scholarly journals A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $

2021 ◽  
Vol 7 (1) ◽  
pp. 39-53
Author(s):  
Dan Yang ◽  
◽  
Jinchao Yu ◽  
Jingjing Zhang ◽  
Xiaoying Zhu ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Curvature Vector ◽  
Shape Operator ◽  
Mean Curvature Vector ◽  
Principal Curvatures

<abstract><p>A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.</p></abstract>

scholarly journals Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1

2021 ◽  
Vol 40 (3) ◽  
pp. 711-719
Author(s):  
Firooz Pashaie
Keyword(s):  
Mean Curvature ◽  
Linear Part ◽  
Curvature Vector ◽  
Spacelike Hypersurface ◽  
Weakly Convex ◽  
Minkowski Space Time ◽  
Spacelike Hypersurfaces ◽  
The First Variation ◽  
Linearized Operator ◽  
Convex Hypersurfaces

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

scholarly journals Submanifolds of Euclidean space with parallel mean curvature vector

1991 ◽  
Vol 14 (3) ◽  
pp. 533-536
Author(s):  
Tahsin Ghazal ◽  
Sharief Deshmukh
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Characteristic Class ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Parallel Mean Curvature ◽  
Real Submanifold ◽  
Totally Real ◽  
Positively Curved

The object of the paper is to study some compact submanifolds in the Euclidean spaceRnwhose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist ann-dimensional compact simply connected totally real submanifold inR2nwhose mean curvature vector is parallel. Then we show that then-dimensional compact totally real submanifolds of constant curvature and parallel mean curvature inR2nare flat. Finally we show that compact Positively curved submanifolds inRnwith parallel mean curvature vector are homology spheres. The last result in particular for even dimensional submanifolds implies that their Euler poincaré characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric immersion with parallel mean curvature vector inRn, answers the problem of Chern and Hopf

Sign in / Sign up

Export Citation Format

Share Document