A note on the nonexistence of spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

hypersurfaces with constant mean curvature in a lorentzian space form

Hypersurfaces of a Sasakian space form with recurrent shape operator

Let $(M^{2n},g)$ be a real hypersurface with recurrent shapeoperator and tangent to the structure vector field $xi$ of the Sasakian space form$widetilde{M}(c)$. We show that if the shape operator $A$ of $M$ isrecurrent then it is parallel. Moreover, we show that $M$is locally a product of two constant $phi-$sectional curvaturespaces.

Willmore spacelike submanifolds in a Lorentzian space form

Let N p (c) be an (n+p)-dimensional connected Lorentzian space form of constant sectional curvature c and φ : M → N p (c) an n-dimensional spacelike submanifold in N p (c). The immersion φ : M → N p (c) is called a Willmore spacelike submanifold in N p (c) if it is a critical submanifold to the Willmore functional W (φ) = ∫

Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying L_k(x)=Ax+b

We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k