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

Authors:	Luis J Alías Nevin Gürbüz

Institution:	(1) Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain;(2) Department of Mathematics, Osmangazi University, 26480 Eskişehir, Turkey

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)].

Keywords:	Takahashi theorem Higher order mean curvatures Linearized operators L k Newton transformations
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