L K -BIHARMONIC HYPERSURFACES IN SPACE FORMS WITH THREE DISTINCT PRINCIPAL CURVATURES

In this paper we consider Lk-conjecture introduced in [5, 6] for hypersurface Mn in space form Rn+1(c) with three principal curvatures. When c = 0,−1, we show that every L1-biharmonic hypersurface with three principal curvatures and H1 is constant, has H2 = 0 and at least one of the multiplicities of principal curvatures is one, where H1 and H2 are first and second mean curvature of M and we show that there is not L2-biharmonic hypersurface with three disjoint principal curvatures and, H1 and H2 is constant. For c = 1, by considering having three principal curvatures, we classify L1-biharmonic hypersurfaces with multiplicities greater than one, H1 is constant and H2 = 0, proper L1-biharmonic hypersurfaces which H1 is constant, and L2-biharmonic hypersurfaces which H1 and H2 is constant.