Classifications of quotion Yamabe solitons on Euclidean submanifolds with concurrent vector fields

The aim of this paper is to obtain some results for quotion Yamabe solitons with concurrent vector fields. We prove quotion Yamabe soliton [Formula: see text] on a hypersurface in Euclidean space [Formula: see text] contained either in a hyperplane or in a sphere [Formula: see text].

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

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 $E_1^4$.