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Acta Mathematica Vietnamica

Lk-biharmonic Hypersurfaces in Space Forms

M. Aminian , icon-email S. M. B. Kashani

Abstract

In this paper, we introduce Lk-biharmonic hypersurfaces M in simply connected space forms Rn+1(c) and propose Lk-conjecture for them. For c=0,1, we prove the conjecture when hypersurface M has two principal curvatures with multiplicities 1,n1, or M is weakly convex, or M is complete with some constraints on it and on Lk. We also show that neither there is any Lk-biharmonic hypersurface Mn in Hn+1 with two principal curvatures of multiplicities greater than one, nor any Lk-biharmonic compact hypersurface Mn in Rn+1 or in Hn+1. As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, (Taiwan. J. Math., 19, 861–874, 5) in this context.