$L_k$-biharmonic hypersurfaces in space forms with three distinct principal curvatures

Commun. Korean Math. Soc. Published online August 5, 2020

Mehran Aminian
Faculty of Dept. of Math., Vali-e-Asr University of Rafsanjan

Abstract : In this paper we consider $ L_k $-conjecture introduced in [5,6] for hypersurface $ M^n $ in space form $ R^{n+1}(c) $ with three principal curvatures. When $ c=0, -1 $, we show that every $ L_1 $-biharmonic hypersurface with three principal curvatures and $ H_1 $ is constant, has $ H_2=0 $ and at least one of the multiplicities of principal curvatures is one, where $ H_1 $ and $ H_2 $ are first and second mean curvature of $ M $ and we show that there is not $ L_2 $-biharmonic hypersurface with three disjoint principal curvatures and, $ H_1 $ and $ H_2 $ is constant.
For $ c=1 $,
by considering having three principal curvatures,
we classify $L_1$-biharmonic hypersurfaces with multiplicities greater than one, $ H_1 $ is constant and $ H_2=0 $,
proper $L_1$-biharmonic hypersurfaces which $ H_1 $ is constant, and $ L_2 $-biharmonic hypersurfaces which $ H_1 $ and $ H_2 $ is constant.