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On some Lr-biharmonic Euclidean hypersurfaces

Mohammadpouri A., Pashaie F.

Journal of Mathematics and Applications

2016

Vol. 39

91--104

In decade eighty, Bang-Yen Chen introduced the concept of biharmonic hypersurface in the Euclidean space. An isometrically immersed hypersurface x : Mn → En+1 is said to be biharmonic if ∆2x = 0, where ∆ is the Laplace operator. We study the Lr-biharmonic hypersurfaces as a generalization of biharmonic ones, where Lr is the linearized operator of the (r + 1)th mean curvature of the hypersurface and in special case we have L0 = ∆. We prove that Lr-biharmonic hypersurface of Lr-finite type and also Lr-biharmonic hypersurface with at most two distinct principal curvatures in Euclidean spaces are r-minimal.