Hypersurfaces in the Lorentz-Minkowski space satisfying L _k ψ = Aψ + b

Abstract

We study hypersurfaces in the Lorentz-Minkowski space ${\mathbb{L}^{n+1}}$ whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, ${A\in\mathbb{R}^{(n+1)\times(n+1)}}$ is a constant matrix and ${b\in\mathbb{L}^{n+1}}$ is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces ${\mathbb{S}^n_1(r)}$ or ${\mathbb{H}^n(-r)}$, and open pieces of generalized cylinders ${\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}$, ${\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}$, with k + 1 ≤ m ≤ n − 1, or ${\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}$, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in ${\mathbb{R}^{n+1}}$ given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).