λ-Biharmonic hypersurfaces in the product space Lᵐ× R

In this paper, we study λ-biharmonic hypersurfaces in the product space Lᵐ, where Lᵐ is an Einstein space and R is a real line. We prove that λ-biharmonic hypersurfaces with constant mean curvature in Lᵐ are either minimal or vertical cylinders, and obtain some classification results for λ$-biharmonic hypersurfaces under various constraints. Furthermore, we investigate λ-biharmonic hypersurfaces in the product space Lᵐ(c), where Lᵐ(c) is a space form with constant sectional curvature c, and categorize hypersurfaces that are either totally umbilical or semi-parallel.