An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

Abstract

We study hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ whose position vector x satisfies the condition L _k x  =  Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed $k=0,\ldots,n-1$, $A \in \mathbb{R}^{(n+1)\times (n+1)}$ is a constant matrix and $b\in\mathbb{R}^{n+1}$ 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 and open pieces of round hyperspheres and generalized right spherical cylinders of the form $\mathbb{S}^{m}(r)\times\mathbb{R}^{n-m}$, with $k+1\leq m \leq n-1$. This extends a previous classification for hypersurfaces in $\mathbb{R}^{n+1}$ satisfying $\Delta x=Ax+b$, where $\Delta=L_{0}$ is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].