L k -biharmonic Hypersurfaces in Space Forms

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,n−1, 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 \( \mathbb {H}^{n+1} \) with two principal curvatures of multiplicities greater than one, nor any Lk-biharmonic compact hypersurface Mn in \( \mathbb {R}^{n+1} \) or in \( \mathbb {H}^{n+1} \). 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.