In this paper, we study hypersurfaces in E n+1 whose Gauss map G satisfies the equation L r G = f (G + C) for a smooth function f and a constant vector C, where L r is the linearized operator of the (r +1)-st mean curvature of the hypersurface, i.e., L r (fWe focus on hypersurfaces with constant (r + 1)-st mean curvature and constant mean curvature. We obtain some classification and characterization theorems for these classes of hypersurfaces.
We study some L k -finite-type Euclidean hypersurfaces. We classify L k -1-type Euclidean hypersurfaces and L k -null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We also prove that, under some mild restrictions, there exists no L k -null-3-type Euclidean hypersurface.
In this paper, we study hypersurfaces in E n+1 which Gauss map G satisfies the equation LrG = f (G + C) for a smooth function f and a constant vector C, where Lr is the linearized operator of the (r + 1)th mean curvature of the hypersurface, i.e., Lr(f ) = tr(Prwhere Pr is the rth Newton transformation, ∇ 2 f is the Hessian of f , LrG = (LrG 1 , . . . , LrG n+1 ), G = (G 1 , . . . , G n+1 ). We show that a rational hypersurface of revolution in a Euclidean space E n+1 has Lr-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.
In decade eighty, Bang-Yen Chen introduced the concept of biharmonic hypersurface in the Euclidean space. An isometrically immersed hypersurface x : M n → E n+1 is said to be biharmonic if ∆ 2 x = 0, where ∆ is the Laplace operator. We study the L r-biharmonic hypersurfaces as a generalization of biharmonic ones, where L r is the linearized operator of the (r + 1)th mean curvature of the hypersurface and in special case we have L 0 = ∆. We prove that L r-biharmonic hypersurface of L r-finite type and also L r-biharmonic hypersurface with at most two distinct principal curvatures in Euclidean spaces are r-minimal.
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