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

Abstract

We study hypersurfaces in Euclidean space 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, . . . , n − 1, A ∈ R (n+1)×(n+1) is a constant matrix and b ∈ 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 S m (r) × R n−m , with k + 1 ≤ m ≤ n − 1. This extends a previous classification for hypersurfaces in R n+1 satisfying x = Ax + b, where = L 0 is the Laplacian operator of the hypersurface, given independently by

HYPERSURFACES IN SPACE FORMS SATISFYING THE CONDITION $L_k x = Ax + b$

“…In [4] and inspired by Garay's extension of Takahashi theorem [18,6,7] and its subsequent generalizations and extensions [8,11,10,12,2,3], the first author jointly with Gürbuz started the study of hypersurfaces in the Euclidean space satisfying the general condition L k x = Ax + b, where A ∈ R (n+1)×(n+1) is a constant matrix and b ∈ R n+1 is a constant vector (we refer the reader to the Introduction of [4] for further details). In particular, the following classification result was given in [ In this paper, and as a natural continuation of the study started in [4], we consider the study of hypersurfaces M n immersed either into the sphere S n+1 ⊂ R n+2 or into the hyperbolic space H n+1 ⊂ R n+2 1 whose position vector x satisfies the condition L k x = Ax + b.…”

“…In particular, the following classification result was given in [ In this paper, and as a natural continuation of the study started in [4], we consider the study of hypersurfaces M n immersed either into the sphere S n+1 ⊂ R n+2 or into the hyperbolic space H n+1 ⊂ R n+2 1 whose position vector x satisfies the condition L k x = Ax + b. Here and for a fixed integer k = 0, .…”

CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP

“…Consider the n-th order non-linear ordinary differential equation given by (2.10) F (s, y, y (1) , y (2) , . .…”

“…In particular, L 1 is called the Cheng-Yau operator introduced in [8]. L k -finite type hypersurfaces have been studied in [1,15].…”

SURFACES IN <TEX>$\mathbb{E}^3$</TEX> WITH L1-POINTWISE 1-TYPE GAUSS MAP

“…Recently, in [1,18] the definition of finite type submanifolds is extended in a natural way, by replacing the Laplace operator ∆ with a sequence of operators L 0 , L 1 , L 2 , . .…”